Well another year has passed and it’s that time of the year when one starts to think about the future. This year promises to be a busy one although I shall be reducing my OU commitments to at the most one 60 point course per year.

I’m currently doing the Number Theory and logic course and completion of that will conclude my second open degree. If I get grade 2 it will put me on line for a 2:1. I then want a break of at least a year from OU work.

The courses will consist of

Level 1 MST121 Introducing Mathematics Pass

Leve 2 MST221 Exploring Mathematics Distinction

M208 Pure Maths Grade 2

A208 Philosophy and the Human Condition Distinction

A224 Music Grade 3

Level 3 M338 Topology Grade 4

MST324 Waves Diffusion etc Grade 3

SM358 Quantum Mechanics Grade 2

Then M381 number theory and logic

The only thing that would persuade me to change my mind is if A303 Philosophy of the Mind gets a reprieve and is presented next year. I wont know till about March. I could replace MST324 and M338 with it and would hopefully get a better grade we’ll see.

Not that I wont be busy I want to concentrate on music and my own physics and maths studies for a while. I intend to finally get a piano tutor with the aim of doing grade 1 piano in June and possibly Grade 2 by the end of the year. Also to practice the theory doing grade 5 and 6 theory this year alongside the OCA composition courses

http://www.oca-uk.com/subjects/music.html

I like the idea that eventually I’ll be able to compose a symphony (construct is probably a more accurate term). As I said before A224 is fine for an overview, but it is short on hammering the basics. grade 5 theory whilst quite simple harmonically is quite demanding in terms of getting transposition, the names of the keys and intervals at your finger tips. All necessary pre-requisites for a budding composer. One of the most reliable methods of workling out intervals and transpositon is to use modulo aritthmetic which I will expand on in another post

I also want to get on with my physics I left hanging in the air about 18 months ago the derivation of the Friedmann equations which govern the expansion of the universe from General relativitry and I’m taking the current break as an opportunity to get back into it. Any calculatiion in general relativity, even the most simplest such as those for the expansion of the universe or the Schwarzschild metric involves pages and pages of tedious algebra and it nearly broke my heart round about June of last year. Still having done the spade work I can now get on to the physics, watch this space.

Also I need to crack on with the second part of number theory and logic the computablity part of which awaits me. Given Duncan’s and Daniel’s experiences of it I’m not looking forward to it. Still as it is an important part of maths then it is necessary.

I’ll be in two minds whether to continue with the Open University after June. Part of me wants to do the Maths MSc, the new third level course in music and philosophy and the new pure maths course. The other part wants to concentrate on my maths physics and music. Finish the big bang calculation, then get up to speed with modern cosmology, do the three Peskin and Schroeder research projects, understand the functional analysis formulation of quantum mechanics and also the Hawking and Penrose singularity theorems. I’m in my mid fifties and if I want to do all that and get to grade 8 piano and compose a symphony or two before I die then time is ticking on.

Anyway seasons greetings to you all and I hope 2014 brings you nearer to your goals.

Best wishes Chris

## Friday, 27 December 2013

## Tuesday, 26 November 2013

### Result for M358 Quantum Mechanics

Well
Results are out for SM358 quantum mechanics I got a grade 2 (just) actually the
examiners this time were on the generous side

As I got
67% for the exam which technically should have been a grade 3 and 90% for the
Assessment.

It does seem a bit arbirtrary looking over the
past courses

M208 I
got 84% in the exam and averaged over 85% for the assessment and didn't not get
distinction

MS324 I got 67% in the exam and got over 80% for
the assessment but did not get grade 2

M338 Topology just scrapped a pass at 40%
although I think my marks were below that in the exam so passed

MST326 got 37% in the exam and above 80% for the
assessment but failed.

It's a bit worrying that depending on the whim
of an examiner you can miss out on a grade.

## Tuesday, 19 November 2013

### M381 Number theory and logic First TMA

It’s been a while since my last post for which apologies. I have just sent off what I have managed to complete, for the first TMA for M381, Number theory and logic. Daniel and Duncan on their blogs have given a good overview of the topics so far and also the fact that at this level one is expected to think for one self a lot more. Duncan is way ahead as usual but then he did start early last year whilst I was busy on Fluids, Music and Quantum Mechanics. I started looking at the units about 5 weeks ago and I have just about completed the units for TMA01, two number theory units and an introductory one on computability. Not much to add to what Duncan and Daniel have already said although I do find it strange that the course has decided to put computablity ahead of logic, a relatively gentle break in, would have been to tackle at least the basic elements of logic first, truth tables etc which most people are probabily aware of, before embarking on what is quite a challenging introduction to computability.

Again rather than introduce computability via Turing machines, the course has decided to use URM (Unlimited Register Machines) which were introduced in the 1960’s I think. As Duncan has pointed out URM machines are based around four simple commands

Z(n) make the contents of the given register zero

C(n,m) copy the contents of register n to m (Not the other way around which it is very easy to confuse)

S(n) increment the contents of register n by 1

J(m,n,q) jump to instruction q if the contents of register m are the same as register n.

URM machines are meant to simulate computer programming, but as a caveat I would add that the Jump instruction seems very similar to the now frowned upon GOTO statement, that has lead to piles and piles of spaghetti code and provides quite a great deal of diificulty in interpreting a URM code. Still it can be useful in providing loops until two registers are made equal simply by including an instruction of the form J(n,n,q) where q is a command earlier in the list. I dare say however a new formulation of computability, reflecting current programming practice, could be devised.

Anyway onto the TMA, it fell into two parts 6 questions on Number theory and 4 questions on computability

Question 1 A relatively straightforward one on the Eulidean Algorithm familiar from the last part of MS221

Question 2 A question on induction which had actually changed from the initial one, a bit tricky, but once one had seen how to proceed it was relatively straightforward

Question 3. A question on proving a condition on the last three digits of any number to be satisfied if the number that is to be divisible by 8. This was quite tricky at first but once you realised what was going on it became obvious. This question is really clever, the sort of thing that doesn’t involve any complicated maths, but requires some thought. I hope there are more questions like this as the course proceeds

Question 4. Another question on divisibility involving factorials divided into three parts first calculating the possible remainders of n^2 when expressed as 1Ok+r, Then proving what factorials are divisible by 10. Finally putting parts one and two together to find out what factorials can be expressed as squares Think I got most of this out.

Question 5 was on Fibonacci Numbers and as I hadn’t had time to look at this topic in any great detail I left it

Question 6 was a question on showing that there are infiinitely many primes of a certain form. As this was very similar to a theorem in the unit it didn’t require that much adaptation to generate a similar proof.

So Apart from question 5 feel reasonably confident that I have got most of this correct.

Second Part Computability part 1

Question 7 Asked us to demonstrate what would happen to a given URM program draw a flow diagram and work out what the code was trying to do. I think I got it out but it would have been easier if one of the copy instructions had their registers swapped ie C(n,m) instead of C(m,n) not sure whether this was a misprint or not.

Question 8 asked us to provide URM codes for two functions Think I got this out although for more complicated codes one has to make sure all the inputs are covered especially if one or more of the inputs are zero.

Question 9 Involved generating a primitive recursive function from two others. The definition of primitive recursion is a bit of a mouthful to say the least. Still after a few examples it begins to make sense. In general if a function is defined by recursion it means that a sequence can be generated such that the next term in the sequence can be generated from the others. That is quite straightforward but the definition appears like gobbldeygook and bears at first sight little relation to our intuitive ideas of recursion. However once one realises that the definition of primitive recursion as given in the text is a recipe for generating a primitively recursive function h from two other functions f and g then it makes a lot more sense.

Finally Question 10 was a question asking us to generate the URM code for a function h defined by primitive recursion from two other codes for f and g. There is a somewhat complicated recipe for doing this however if one follows the steps then one eventually gets there. The final part asked us to work out what the function h was Fingers crossed I think I got there.

Overall then should get a grade two, but having left out question 5, unlikely to get distinction.

So first impressions on the whole quite interesting, the number theory part (so far) seems simpler than the computablilty part. But I do wonder whether number theory really counts as a theory, rather than it being a set of miscellenous facts about numbers. It’s not like group theory or analysis, or even quantum mechanics, where given a few axioms, everything follows from them.

As for computability, so far have just scratched the surface, the meat will come in the next two parts. Fortunately I have the Christmas holiday to get to grips with it. I suspect it will be the hardest part conceptually of this course. But it’s good to be challenged.

Bye for now Chris

Again rather than introduce computability via Turing machines, the course has decided to use URM (Unlimited Register Machines) which were introduced in the 1960’s I think. As Duncan has pointed out URM machines are based around four simple commands

Z(n) make the contents of the given register zero

C(n,m) copy the contents of register n to m (Not the other way around which it is very easy to confuse)

S(n) increment the contents of register n by 1

J(m,n,q) jump to instruction q if the contents of register m are the same as register n.

URM machines are meant to simulate computer programming, but as a caveat I would add that the Jump instruction seems very similar to the now frowned upon GOTO statement, that has lead to piles and piles of spaghetti code and provides quite a great deal of diificulty in interpreting a URM code. Still it can be useful in providing loops until two registers are made equal simply by including an instruction of the form J(n,n,q) where q is a command earlier in the list. I dare say however a new formulation of computability, reflecting current programming practice, could be devised.

Anyway onto the TMA, it fell into two parts 6 questions on Number theory and 4 questions on computability

Question 1 A relatively straightforward one on the Eulidean Algorithm familiar from the last part of MS221

Question 2 A question on induction which had actually changed from the initial one, a bit tricky, but once one had seen how to proceed it was relatively straightforward

Question 3. A question on proving a condition on the last three digits of any number to be satisfied if the number that is to be divisible by 8. This was quite tricky at first but once you realised what was going on it became obvious. This question is really clever, the sort of thing that doesn’t involve any complicated maths, but requires some thought. I hope there are more questions like this as the course proceeds

Question 4. Another question on divisibility involving factorials divided into three parts first calculating the possible remainders of n^2 when expressed as 1Ok+r, Then proving what factorials are divisible by 10. Finally putting parts one and two together to find out what factorials can be expressed as squares Think I got most of this out.

Question 5 was on Fibonacci Numbers and as I hadn’t had time to look at this topic in any great detail I left it

Question 6 was a question on showing that there are infiinitely many primes of a certain form. As this was very similar to a theorem in the unit it didn’t require that much adaptation to generate a similar proof.

So Apart from question 5 feel reasonably confident that I have got most of this correct.

Second Part Computability part 1

Question 7 Asked us to demonstrate what would happen to a given URM program draw a flow diagram and work out what the code was trying to do. I think I got it out but it would have been easier if one of the copy instructions had their registers swapped ie C(n,m) instead of C(m,n) not sure whether this was a misprint or not.

Question 8 asked us to provide URM codes for two functions Think I got this out although for more complicated codes one has to make sure all the inputs are covered especially if one or more of the inputs are zero.

Question 9 Involved generating a primitive recursive function from two others. The definition of primitive recursion is a bit of a mouthful to say the least. Still after a few examples it begins to make sense. In general if a function is defined by recursion it means that a sequence can be generated such that the next term in the sequence can be generated from the others. That is quite straightforward but the definition appears like gobbldeygook and bears at first sight little relation to our intuitive ideas of recursion. However once one realises that the definition of primitive recursion as given in the text is a recipe for generating a primitively recursive function h from two other functions f and g then it makes a lot more sense.

Finally Question 10 was a question asking us to generate the URM code for a function h defined by primitive recursion from two other codes for f and g. There is a somewhat complicated recipe for doing this however if one follows the steps then one eventually gets there. The final part asked us to work out what the function h was Fingers crossed I think I got there.

Overall then should get a grade two, but having left out question 5, unlikely to get distinction.

So first impressions on the whole quite interesting, the number theory part (so far) seems simpler than the computablilty part. But I do wonder whether number theory really counts as a theory, rather than it being a set of miscellenous facts about numbers. It’s not like group theory or analysis, or even quantum mechanics, where given a few axioms, everything follows from them.

As for computability, so far have just scratched the surface, the meat will come in the next two parts. Fortunately I have the Christmas holiday to get to grips with it. I suspect it will be the hardest part conceptually of this course. But it’s good to be challenged.

Bye for now Chris

## Tuesday, 8 October 2013

### SM358 exam debrief

As promised here is my summary (off the top of my head) of the questions and my responses to them

I think I'm on target for the top end of Grade 3 or if I'm lucky Grade 2 but it will only just get there.

The first part was 10 out of 12 questions of which I answered 9 but some were only sketches

Question 1 was about the degeneracies of energy levels in an infinite square well and the calculation of the wave length of light emitted when the electron fell from the higher level to the other.

Got most of this out 4 marks say

Question 2 Concerned the normalisation of of a given wave function and a calculation of the energy levels again got most of this out 4 marks say

Question 3 I ignored think it involved calculation of quantum oscillator properties using annhilation and creation operators

Question 4 On the Born interpretation this had a different twist to the normal ones and threw me so I wrote down the born intepretation 1 mark out of 5

Question 6 Involved calculating the Energy eigenvalues of Lz for a given wave function then a question using the generalised uncertainty principle for Ly and Lx most out say 4 out of 5

Question 7 Involved calculation of the maximum separation of proton distance from a given radial wavefunction got most of this out again say 4 out of 5

Question 8 was fairly straightforward whether or not statements about the Bell Inequalities were true or false I think I got most of theses so 4 marks

I can't really remember most of the other questions in part 1 but they involved calculation of the expectation value of various quantites such as x or x^2. The calculation of bond order for a given molecular configuration and so forth. If my memory serves me correctly I got about 4 marks for two and say 2 marks for the other one

So pessimistically total marks for part 1

so overall for part 1 I got 4+4+1+4+4+4+8+2 = 31 marks (Some may include full marks so possibly 35 marks )

Then parts 2,3 and 4 long questions

Part 2 I chose a question on eigenfunctions of a square well I think I got most of this so say 15/17

Part 3 was a question on spin functions the first part was really strange so I left it.losing 3 marks went onto the question in part 4 before coming to this question at the end..

The rest should have been quite straightforward as it involved calculating the coefficients of a given spinor in another basis and then the expectation value of Sy. I got the first part out but chose the wrong matrix to calculate the expectation value. I might just scrape 8 or nine marks here

Part 4 was a question on eigenfunctions of the angular momentum operator in spherical coordinates

Got most of the first part out but then there were two part questions on of which involved showing a given wave function was an eigenfunction of J = Lz + Sz I couldn't quite get this. The last part should have been straightforward involving transition selection rules but my mind went blank so possibly 12 out of 17 here.

So part two is about 35 marks as well

35 + 35 is 70 so might just get grade 2 but a more realistic assessment would be for grade 3.

So overall not bad (certainly better than last june's car crash of Fluid mechanics) I decided not to do the resit as it clashed with my revision for quantum mechanics.

Number theory and logic awaits and I want to consolidate my music by doing the Grade theory exams I should be able to do Grade 5 in March and also piano lessons again I want to do grade 1 piano again in March. I will also get back to my own physics calculations. My first project is to examine the X ray diffraction theory behind the discovery of DNA. Then back to general relativity and also the physics of deep inelastic scattering which lead to the discovery of quarks. I will also do an Edinburgh dept continuing education course on Plato come January Enough to keep me busy for a while yet :)

Bye for now

I think I'm on target for the top end of Grade 3 or if I'm lucky Grade 2 but it will only just get there.

The first part was 10 out of 12 questions of which I answered 9 but some were only sketches

Question 1 was about the degeneracies of energy levels in an infinite square well and the calculation of the wave length of light emitted when the electron fell from the higher level to the other.

Got most of this out 4 marks say

Question 2 Concerned the normalisation of of a given wave function and a calculation of the energy levels again got most of this out 4 marks say

Question 3 I ignored think it involved calculation of quantum oscillator properties using annhilation and creation operators

Question 4 On the Born interpretation this had a different twist to the normal ones and threw me so I wrote down the born intepretation 1 mark out of 5

Question 6 Involved calculating the Energy eigenvalues of Lz for a given wave function then a question using the generalised uncertainty principle for Ly and Lx most out say 4 out of 5

Question 7 Involved calculation of the maximum separation of proton distance from a given radial wavefunction got most of this out again say 4 out of 5

Question 8 was fairly straightforward whether or not statements about the Bell Inequalities were true or false I think I got most of theses so 4 marks

I can't really remember most of the other questions in part 1 but they involved calculation of the expectation value of various quantites such as x or x^2. The calculation of bond order for a given molecular configuration and so forth. If my memory serves me correctly I got about 4 marks for two and say 2 marks for the other one

So pessimistically total marks for part 1

so overall for part 1 I got 4+4+1+4+4+4+8+2 = 31 marks (Some may include full marks so possibly 35 marks )

Then parts 2,3 and 4 long questions

Part 2 I chose a question on eigenfunctions of a square well I think I got most of this so say 15/17

Part 3 was a question on spin functions the first part was really strange so I left it.losing 3 marks went onto the question in part 4 before coming to this question at the end..

The rest should have been quite straightforward as it involved calculating the coefficients of a given spinor in another basis and then the expectation value of Sy. I got the first part out but chose the wrong matrix to calculate the expectation value. I might just scrape 8 or nine marks here

Part 4 was a question on eigenfunctions of the angular momentum operator in spherical coordinates

Got most of the first part out but then there were two part questions on of which involved showing a given wave function was an eigenfunction of J = Lz + Sz I couldn't quite get this. The last part should have been straightforward involving transition selection rules but my mind went blank so possibly 12 out of 17 here.

So part two is about 35 marks as well

35 + 35 is 70 so might just get grade 2 but a more realistic assessment would be for grade 3.

So overall not bad (certainly better than last june's car crash of Fluid mechanics) I decided not to do the resit as it clashed with my revision for quantum mechanics.

Number theory and logic awaits and I want to consolidate my music by doing the Grade theory exams I should be able to do Grade 5 in March and also piano lessons again I want to do grade 1 piano again in March. I will also get back to my own physics calculations. My first project is to examine the X ray diffraction theory behind the discovery of DNA. Then back to general relativity and also the physics of deep inelastic scattering which lead to the discovery of quarks. I will also do an Edinburgh dept continuing education course on Plato come January Enough to keep me busy for a while yet :)

Bye for now

## Thursday, 3 October 2013

### Exam Time again

Well it's that time of year again for me (twice in one year) I have my quantum mechanics exam on Monday

I did my first practice past paper under exam conditions today and got borderline grade3/grade2 part of the problem is there is just not enough time available. For part 1 you have to answer 10 so called short questions in 1 hour and 30 minutes and then there are three long questions which you are supposed to allocate 30 mins to. This means that you almost have to download the already prepared answers in your head if you are to stand any chance of a grade 2.

The short questions seem to range over a variety of topics whereas the long questions can be grouped into 3 quite distinct groups

PArt 1 A question on a particle in a potential usually square but sometimes they will throw a wobbly and ask about a non square potential or just some general type questions. This is a banker question

Then there appears to be a choice between a question involving Transmission or reflection at a boundary or one on Harmonic oscillator raising and lowering operators. Will hope the question on potential is reasonable

Part 2 A question on spin states in a magnetic field again a banker question the alternative is usually a question on the Bell inequalities

Part 3 A question on hydrogen atom wave functions which could get messy however it's a topic I feel reasonably confident in although in my trial I couldn't for the life of me see how to get the normalisation of a given hydrogen atom wave function correct.

The second question is usually on something involving molecules, solid state physics, perturbation theory or Transistions induced by radiation. I've not paid much attention to this part which comes at the end of the course so I'll go for the question on Hydrogen atom wavefunctions. If it looks really tough then if a question on perturbation theory comes up I'm reasonably confident I could tackle that one.

In part 1 typical questions include normalising a wave function via the coefficient rule. Calculating a probability density function from Born's rule. A bit on hermitian operators, one or two questions on the Bell Inequalities and spin states, Then a whole bunch of questions on either molecular or atomic orbitals

Selection rules between states induced by radiation one or two waffly questions on solid state physics and so forth

Ok prediction top end of grade 3 or if I'm lucky grade 2

On another note I've got the material for Number Theory and Logic which has officially started but so far apart from a quick glance I've not had any chance to look at it. The first unit looks reasonably straightforward covering Triangular numbers, Induction and Divisibilty some of which was covered in the last unit of MST221 so hopefully I will be able to get up to speed,

Post match analysis on Tuesday till then that's all folks

I did my first practice past paper under exam conditions today and got borderline grade3/grade2 part of the problem is there is just not enough time available. For part 1 you have to answer 10 so called short questions in 1 hour and 30 minutes and then there are three long questions which you are supposed to allocate 30 mins to. This means that you almost have to download the already prepared answers in your head if you are to stand any chance of a grade 2.

The short questions seem to range over a variety of topics whereas the long questions can be grouped into 3 quite distinct groups

PArt 1 A question on a particle in a potential usually square but sometimes they will throw a wobbly and ask about a non square potential or just some general type questions. This is a banker question

Then there appears to be a choice between a question involving Transmission or reflection at a boundary or one on Harmonic oscillator raising and lowering operators. Will hope the question on potential is reasonable

Part 2 A question on spin states in a magnetic field again a banker question the alternative is usually a question on the Bell inequalities

Part 3 A question on hydrogen atom wave functions which could get messy however it's a topic I feel reasonably confident in although in my trial I couldn't for the life of me see how to get the normalisation of a given hydrogen atom wave function correct.

The second question is usually on something involving molecules, solid state physics, perturbation theory or Transistions induced by radiation. I've not paid much attention to this part which comes at the end of the course so I'll go for the question on Hydrogen atom wavefunctions. If it looks really tough then if a question on perturbation theory comes up I'm reasonably confident I could tackle that one.

In part 1 typical questions include normalising a wave function via the coefficient rule. Calculating a probability density function from Born's rule. A bit on hermitian operators, one or two questions on the Bell Inequalities and spin states, Then a whole bunch of questions on either molecular or atomic orbitals

Selection rules between states induced by radiation one or two waffly questions on solid state physics and so forth

Ok prediction top end of grade 3 or if I'm lucky grade 2

On another note I've got the material for Number Theory and Logic which has officially started but so far apart from a quick glance I've not had any chance to look at it. The first unit looks reasonably straightforward covering Triangular numbers, Induction and Divisibilty some of which was covered in the last unit of MST221 so hopefully I will be able to get up to speed,

Post match analysis on Tuesday till then that's all folks

## Monday, 2 September 2013

### SM358 quantum physics TMA04

So final TMA submitted for SM358 but still have 3 on line assessments to complete one nearly finished and of course I missed out on TMA02 as was bogged down in completing the EMA for the music course and desperately attempting to do revision for Fluids

TMA04 concentrates on the first part of the 3rd block quantum mechanics of matter this involves an overview of the hydrogen atom and it's wave functions, and it's extension to many electron atoms plus an introduction to pertirbation theory. The rest of the unit is (IMHO) a rather inadequate introdution to molecular physics, Solid state physics and quantum optics. Each of these deserve at least a 30 point unit to themselves and again I feel the OU is selling it's physics graduates short. Anyone thinking that by completion of this course, electromagnetism and say the relativistic universe has a degree equivalent to that of a standard course in undergraduate physics would be kidding themselves. Still I don't write the rules and it is not the point of this post to beef (yet again ) about the gulf between OU courses and those of brick univesities.

So the questions on this TMA divided into 3

1) A question based on one particular wave function of the hydrogen atom, calculating various quantities such as the expection value of 1/r and 1/r^2 and their uncertainty. As they say in the books this was tedious but straightforward

2) Again more questions on the hydrogen atom wave functions involving a hypothesised modification to the coulombic interaction. In the context of the hydrogen atom this is obviously unphysical however it may correspond to the force between two nucleons such as the potential between two deuterons. Any way the point was to test our ability to calculate the first order corrections to a change in the potential. Again this was straightforward but tedious and it took a couple of goes before all the signs and factors of pi cancelled out to get the correct answer.

One of the skills one needs in solving problems concerning hydrogen atom wavefunctions is patience and the ability to obtain the correct numerical factor. This is somewhat tedious but a necessary skill just hope the exam questions aren't so pernickity and fiddly.

3) The final question probably the most straightforward involved calculation of the various atomic terms associated with a multi electron atom and their degenaricies. Just one point that wasn't really mentioned in the text namely how to correlate the total number of States given with a given L and S nuimber with the individual l and s numbers of the valence electrons. Fortunately an example in the additional exercise book gave a clue

So all in all I think I've done reasonably well on this TMA. I omitted an essay type question at the start of question 3.

So that apart from completing the exam and the online assessments is that as far as quantum mechanics goes. It really doesn't seem all that long ago since I started and now it's more or less finished the feeling does seem a bit weird to say the least but the same is true of any OU course. It is just rather a pity to say the least that there are no follow up courses as far as the OU is concerned.

TMA04 concentrates on the first part of the 3rd block quantum mechanics of matter this involves an overview of the hydrogen atom and it's wave functions, and it's extension to many electron atoms plus an introduction to pertirbation theory. The rest of the unit is (IMHO) a rather inadequate introdution to molecular physics, Solid state physics and quantum optics. Each of these deserve at least a 30 point unit to themselves and again I feel the OU is selling it's physics graduates short. Anyone thinking that by completion of this course, electromagnetism and say the relativistic universe has a degree equivalent to that of a standard course in undergraduate physics would be kidding themselves. Still I don't write the rules and it is not the point of this post to beef (yet again ) about the gulf between OU courses and those of brick univesities.

So the questions on this TMA divided into 3

1) A question based on one particular wave function of the hydrogen atom, calculating various quantities such as the expection value of 1/r and 1/r^2 and their uncertainty. As they say in the books this was tedious but straightforward

2) Again more questions on the hydrogen atom wave functions involving a hypothesised modification to the coulombic interaction. In the context of the hydrogen atom this is obviously unphysical however it may correspond to the force between two nucleons such as the potential between two deuterons. Any way the point was to test our ability to calculate the first order corrections to a change in the potential. Again this was straightforward but tedious and it took a couple of goes before all the signs and factors of pi cancelled out to get the correct answer.

One of the skills one needs in solving problems concerning hydrogen atom wavefunctions is patience and the ability to obtain the correct numerical factor. This is somewhat tedious but a necessary skill just hope the exam questions aren't so pernickity and fiddly.

3) The final question probably the most straightforward involved calculation of the various atomic terms associated with a multi electron atom and their degenaricies. Just one point that wasn't really mentioned in the text namely how to correlate the total number of States given with a given L and S nuimber with the individual l and s numbers of the valence electrons. Fortunately an example in the additional exercise book gave a clue

So all in all I think I've done reasonably well on this TMA. I omitted an essay type question at the start of question 3.

So that apart from completing the exam and the online assessments is that as far as quantum mechanics goes. It really doesn't seem all that long ago since I started and now it's more or less finished the feeling does seem a bit weird to say the least but the same is true of any OU course. It is just rather a pity to say the least that there are no follow up courses as far as the OU is concerned.

## Sunday, 18 August 2013

### Statisical and Solid State Physics

Well it's nearly the end of SM358 unfortunately the open university has no where else to go as far as quantum physics is concerned. A next logical step is Statistical Physics and Solid State Physics A search through the internet has yielded these two courses which seem to give a good over view of the subject and are based on courses given to those in their third year at Oxford and Cambridge whats more they have problem sheets associated with them

Statistical Physics

http://www.damtp.cam.ac.uk/user/tong/statphys.html

Solid State Physics

http://www-thphys.physics.ox.ac.uk/people/SteveSimon/condmat2013/condmat.html

If any one is interested in forming a study group to work through these courses after the exams could they please let me know. Obviously we would have to share any solutions amongst our selves as if they were published on the internet then the authors might well restrict access to the lecture notes.

Two suitable textbooks to accompany both might be

http://www.amazon.co.uk/Concepts-Thermal-Physics-Stephen-Blundell/dp/0199562105

and by the author of the solid state course himself (and honour would suggest the least we could do is buy his book)

http://www.amazon.co.uk/Oxford-Solid-State-Basics/dp/0199680779/ref=sr_1_1?s=books&ie=UTF8&qid=1376838763&sr=1-1&keywords=Simon+Solid+State+physics

I have also been looking at the diffraction theory behind the discovery of DNA it really is quite clever and I hope to write up a paper which covers the theory and goes through the actual calculations. Watch this space

Statistical Physics

http://www.damtp.cam.ac.uk/user/tong/statphys.html

Solid State Physics

http://www-thphys.physics.ox.ac.uk/people/SteveSimon/condmat2013/condmat.html

If any one is interested in forming a study group to work through these courses after the exams could they please let me know. Obviously we would have to share any solutions amongst our selves as if they were published on the internet then the authors might well restrict access to the lecture notes.

Two suitable textbooks to accompany both might be

http://www.amazon.co.uk/Concepts-Thermal-Physics-Stephen-Blundell/dp/0199562105

and by the author of the solid state course himself (and honour would suggest the least we could do is buy his book)

http://www.amazon.co.uk/Oxford-Solid-State-Basics/dp/0199680779/ref=sr_1_1?s=books&ie=UTF8&qid=1376838763&sr=1-1&keywords=Simon+Solid+State+physics

I have also been looking at the diffraction theory behind the discovery of DNA it really is quite clever and I hope to write up a paper which covers the theory and goes through the actual calculations. Watch this space

## Sunday, 4 August 2013

### Coursera

Ok so there is an alternative to the OU which I’m going to use to supplement my current OU studies. I’m sure most people reading this blog will have heard of Coursera which is a free on-line set of courses provided by mainly American Universities. It’s like being a kid in a sweet shop.
The courses are short.

I’m doing a) Mathematical Philosophy whiich started last week and I’m grateful for Gavin who pointed this out to me in a comment he made on the last post

https://class.coursera.org/mathphil-001/wiki/view?page=syllabus

b) Introduction to Logic which starts in September and should hopefully complement the Logic part of M381 https://www.coursera.org/#course/intrologic?utm_classid=971129&utm_nottype=class.welcome.before&utm_notid=-1&utm_linknum=1

c) Finally (for now) Analysis of a Complex Kind which starts end of October this hopefully will remind me about complex analysis and help me revise the topic

https://www.coursera.org/#course/complexanalysis?utm_classid=971056&utm_nottype=class.welcome.before&utm_notid=-1&utm_linknum=1

It might also be a good introduction for those who are contemplating doing M337 next year

I really am begining to think the OU days are numbered for those who want to do courses mainly for interest given the cost of fees.Ok one gets more support but as Free internet provision of University level courses continues to grow it's difficult to see how the OU can justify it's expensive fee structure.

Incidentally I've just had a hell of a nightmare trying to publish this post I was using Internet Explorer 8 however blogger in their infinite wisdom have decided to withdraw support from Internet Explorer 8 so I've had to change to FireFox a case of if it's working we'll break it. Most inconvenient and tedious. Also in order to access the Coursera courses I had to download VLC media and access the videos that way but that came with a whole load of crap such as sweetsearch so I had to remove these manually. I hate computers.

I’m doing a) Mathematical Philosophy whiich started last week and I’m grateful for Gavin who pointed this out to me in a comment he made on the last post

https://class.coursera.org/mathphil-001/wiki/view?page=syllabus

b) Introduction to Logic which starts in September and should hopefully complement the Logic part of M381 https://www.coursera.org/#course/intrologic?utm_classid=971129&utm_nottype=class.welcome.before&utm_notid=-1&utm_linknum=1

c) Finally (for now) Analysis of a Complex Kind which starts end of October this hopefully will remind me about complex analysis and help me revise the topic

https://www.coursera.org/#course/complexanalysis?utm_classid=971056&utm_nottype=class.welcome.before&utm_notid=-1&utm_linknum=1

It might also be a good introduction for those who are contemplating doing M337 next year

I really am begining to think the OU days are numbered for those who want to do courses mainly for interest given the cost of fees.Ok one gets more support but as Free internet provision of University level courses continues to grow it's difficult to see how the OU can justify it's expensive fee structure.

Incidentally I've just had a hell of a nightmare trying to publish this post I was using Internet Explorer 8 however blogger in their infinite wisdom have decided to withdraw support from Internet Explorer 8 so I've had to change to FireFox a case of if it's working we'll break it. Most inconvenient and tedious. Also in order to access the Coursera courses I had to download VLC media and access the videos that way but that came with a whole load of crap such as sweetsearch so I had to remove these manually. I hate computers.

## Friday, 26 July 2013

### Results out

Just a quick post to inform readers that my results are out

I got grade 3 for music with 61% OCAS (TMA's not bad since I only submitted 5 out of 6 ( and there is no substitution ) and 65% for the EMA (Extended Module Assessment) so reasonable there is as yet no feedback for the EMA but it's forthcoming

As for Fluids I failed with 37% for the exam and 71% for the continuous assessment I've been offered a resit but it will be capped most I can get is grade 4 .

Still as I would like the opportunity to hone my exam technique and reconsolidate the material it will be worth doing. Also it's another excuse to keep up with partial differenctial equations and mathematical methods.

My course schedule for this year is

SM358 Quatum mechanics exam in October

M381 logic and number theory starting in October

Physics project on entanglement Feb 2014

Also for music

I will enrol for grade 4 theory exam in November with a theory exam every 6 months or so till I get to grade 8 start piano lessons in September and hopefully do grade 1 piano exams in March 2014

I decided it would be far to much for me to take on board AA308 the third level course in philosophy this year round. It's allegedly in it's last presentation this year with a replacement in Oct2014 but going on previous form I can't see this happening so hopefully it will be available in October I really hope so as from a preliminary survey I don't really like the prospectus for the new course without being to harsh it fits into "waffle philosophy".That is philosophy good for discussion at dinner parties but hardly cutting edge philosophy that one would expect for a third level course in the subject. On the other hand AA308 whilst not perfect does cover philosophy of the mind at a level comparable to that of other 3rd year undergraduate brick level courses.

I am seriously begining to think the only way in which to get a rigorous training in philosophy is to bite the bullet and do the external London BA in philosophy even though I wont be able to start for another two years. Watch this space.

I got grade 3 for music with 61% OCAS (TMA's not bad since I only submitted 5 out of 6 ( and there is no substitution ) and 65% for the EMA (Extended Module Assessment) so reasonable there is as yet no feedback for the EMA but it's forthcoming

As for Fluids I failed with 37% for the exam and 71% for the continuous assessment I've been offered a resit but it will be capped most I can get is grade 4 .

Still as I would like the opportunity to hone my exam technique and reconsolidate the material it will be worth doing. Also it's another excuse to keep up with partial differenctial equations and mathematical methods.

My course schedule for this year is

SM358 Quatum mechanics exam in October

M381 logic and number theory starting in October

Physics project on entanglement Feb 2014

Also for music

I will enrol for grade 4 theory exam in November with a theory exam every 6 months or so till I get to grade 8 start piano lessons in September and hopefully do grade 1 piano exams in March 2014

I decided it would be far to much for me to take on board AA308 the third level course in philosophy this year round. It's allegedly in it's last presentation this year with a replacement in Oct2014 but going on previous form I can't see this happening so hopefully it will be available in October I really hope so as from a preliminary survey I don't really like the prospectus for the new course without being to harsh it fits into "waffle philosophy".That is philosophy good for discussion at dinner parties but hardly cutting edge philosophy that one would expect for a third level course in the subject. On the other hand AA308 whilst not perfect does cover philosophy of the mind at a level comparable to that of other 3rd year undergraduate brick level courses.

I am seriously begining to think the only way in which to get a rigorous training in philosophy is to bite the bullet and do the external London BA in philosophy even though I wont be able to start for another two years. Watch this space.

## Saturday, 20 July 2013

### SM358 quantum mechanics TMA03

Ok so back to quantum mechanics I have just submitted the third TMA and am reasonably confident I've done enough to get a grade two pass not that this matters for the assessment as one only needs above 30% in seven of the assessments of which at least two must be TMA's. Due to the intensity of revising for both my music course and fluid mechanics I was unable to complete the second TMA but will look at it for the exam.

Anyway a breakdown of the TMA questions is as follows

1) A question on the time dependence of spin states the basic theory of which under pins magnetic resonance imaging. One had to calculate how a composite spinor would vary with time the expectation values of the Sx and Sy spin components and check that the results were consistent with the generalised Ehrenfest theorem I got consistent answers so am reasonably confident I got most of this correct.

2) A question involving the symmetries of a system of two particles in a square well. Given a spin state one with a given Spin quantum number S one had to determine the symmetry of the spatial wavefunction. So if the spin state is zero say for a pair of fermions with a wavefunction that is asymmetric if the spin part is antisymmetric, the spatial part of the wavefunction must be symmetric to maintain the overall antisymmetry of the wave function. This was again relatively straightforward although one had to calculate the probability that both particles would be in the left hand side of the box from the joint wavefunction and this was quite tedious also causing word to crash a number of times as the equation editor can't handle to much copying of equaitons without it crashing. Always remember to save the document after writing complicated equations.

3) A question on the polarisation states of a pair of photons and a gentle run through of the steps leading to the violation of the Bell inequalities or in our case to be more precise the Clauser Horne Shimony (CHSH) inequalities. Plus a short description of what a non local hidden variable theory was. All this was straightforward and satisfying. Although I do feel that the way the text has phrased the two conditions of realism and non realism slightly misleading to say the least

It defines the features of a local hidden variable theory as follows

a) Realism implies that observables have values independent of meausrement

With of course the implication that if the so called collapse of the wavefunction is seen as a physical process observables in quantum mechanics are not independent of measurement. That is not strictly true its only non commuting variables that on the standard view do no have values independent of measurement. Quantities such as mass and charge are commutative and so do have values of measurement. Also if as I have argued in many posts before the wavefunction is just a means for calculating the correct probabilities and nothing physical then nothing can be said to collapse.

b) Locality implies that events at any location cannot influence what happens at another location before a light signal could travel between the two locations

This is of course the standard view with the implication that in situations such as the Aspect experiment when a measurement is made on one photon it causes the other photon to jump into the opposite spin state even though they may be miles away thus implying some faster than speed of light influence between the particles.

Of course the formalism of quantum mechanics says nothing about what might be causing the correlation but that hasn't stopped all sorts of weird and wonderful ideas about it.

A more prosaic view would say that non locality is essentially the fact that the probability distributiion function for a joint pair of variables is non separable whereas for local theories they are. All that happens in the Aspect experiment is that because the pair of particles have to have a net angular momentum of zero and this dicatates the form of the wavefunction for the joint pair of particles leading to the correlation. But no more can be said certainly there is no need to invoke faster than light signalling as some people are prone to. That is an addition to the formalism not warranted by the facts.

Finally whilst it's true to say that local hidden variable theories have been refuted the current form of hidden variable theories are in fact non local. The issue is still wide open, What one can't have is the usual view that quantum mechanics is both non realist and non local. If you claim that quantum mechaincs is non realist with respect to non commuting variables then there is no need to claim that quantum mechanics is non local. And also the non realism associated with quantum mechanics is only a limited form of non realism.

I didn't have the space to go into this in any detail in the assignment. It does concern me slighthly that misleading interpretations are being passed of as fact when in fact the situation is not as black and white as some people make out. Watch this space for when I embark on the entanglement project.

Anyway a breakdown of the TMA questions is as follows

1) A question on the time dependence of spin states the basic theory of which under pins magnetic resonance imaging. One had to calculate how a composite spinor would vary with time the expectation values of the Sx and Sy spin components and check that the results were consistent with the generalised Ehrenfest theorem I got consistent answers so am reasonably confident I got most of this correct.

2) A question involving the symmetries of a system of two particles in a square well. Given a spin state one with a given Spin quantum number S one had to determine the symmetry of the spatial wavefunction. So if the spin state is zero say for a pair of fermions with a wavefunction that is asymmetric if the spin part is antisymmetric, the spatial part of the wavefunction must be symmetric to maintain the overall antisymmetry of the wave function. This was again relatively straightforward although one had to calculate the probability that both particles would be in the left hand side of the box from the joint wavefunction and this was quite tedious also causing word to crash a number of times as the equation editor can't handle to much copying of equaitons without it crashing. Always remember to save the document after writing complicated equations.

3) A question on the polarisation states of a pair of photons and a gentle run through of the steps leading to the violation of the Bell inequalities or in our case to be more precise the Clauser Horne Shimony (CHSH) inequalities. Plus a short description of what a non local hidden variable theory was. All this was straightforward and satisfying. Although I do feel that the way the text has phrased the two conditions of realism and non realism slightly misleading to say the least

It defines the features of a local hidden variable theory as follows

a) Realism implies that observables have values independent of meausrement

With of course the implication that if the so called collapse of the wavefunction is seen as a physical process observables in quantum mechanics are not independent of measurement. That is not strictly true its only non commuting variables that on the standard view do no have values independent of measurement. Quantities such as mass and charge are commutative and so do have values of measurement. Also if as I have argued in many posts before the wavefunction is just a means for calculating the correct probabilities and nothing physical then nothing can be said to collapse.

b) Locality implies that events at any location cannot influence what happens at another location before a light signal could travel between the two locations

This is of course the standard view with the implication that in situations such as the Aspect experiment when a measurement is made on one photon it causes the other photon to jump into the opposite spin state even though they may be miles away thus implying some faster than speed of light influence between the particles.

Of course the formalism of quantum mechanics says nothing about what might be causing the correlation but that hasn't stopped all sorts of weird and wonderful ideas about it.

A more prosaic view would say that non locality is essentially the fact that the probability distributiion function for a joint pair of variables is non separable whereas for local theories they are. All that happens in the Aspect experiment is that because the pair of particles have to have a net angular momentum of zero and this dicatates the form of the wavefunction for the joint pair of particles leading to the correlation. But no more can be said certainly there is no need to invoke faster than light signalling as some people are prone to. That is an addition to the formalism not warranted by the facts.

Finally whilst it's true to say that local hidden variable theories have been refuted the current form of hidden variable theories are in fact non local. The issue is still wide open, What one can't have is the usual view that quantum mechanics is both non realist and non local. If you claim that quantum mechaincs is non realist with respect to non commuting variables then there is no need to claim that quantum mechanics is non local. And also the non realism associated with quantum mechanics is only a limited form of non realism.

I didn't have the space to go into this in any detail in the assignment. It does concern me slighthly that misleading interpretations are being passed of as fact when in fact the situation is not as black and white as some people make out. Watch this space for when I embark on the entanglement project.

## Sunday, 7 July 2013

### What is an explanation ?

This post is motivated by some debates I've been having on the OU quantum physics fora about the nature of explanation especially in physics. In quantum mechanics we have what I consider to be, a rather bizzare claim that despite the fact that trained physicists are able to use the formalism to predict measurable quantities about atomic systems such as the energy levels or particle decay rates. The claim is made in many circles that no one is able to understand quantum physics. Yet clearly those who can use the formalism of quantum mechanics to successfully model say the properties of solids, lasers, the properties of stars etc obviously do understand a great deal about how qunatum physics works.

I suppose the problem is that quantum physics (at least in it;s Non-relativistic formalism) uses a quantity called the wavefunction, but, as I've argued before in previous posts better seen as a probability state vector who's modulus squared gives rise to a probability density function and whos eigenvalues can be related to the energy levels or decay rates of atoms, molecules or solids. The problem is that for an N body system the so called wavefunction becomes a function of the 3 N coordinates of the system. So if it is seen as a field analagous to an electromagnetic field or the gravitational field. It is a field in the 3N+1 configuration space of the system rather than our normal 3 dimensional space. Matters are even more complicated when extra variables such as spin are also included. Spin has no spatial or time component so what is spin really ?

The other key issue is the notorious collapse of the wave function if there is a possibility that a particle can be in one of two or more states then it is said to be in limbo until a measurement is made and then it collapses into one of it;s states and in situations like the Aspect experiment this influence is said to occur at speed's faster than the speed of light because given a pair of photons emitted from a common source then so the story goes if I measure the spin of one photon it immediately causes the other photon which could be miles away to have the opposite spin.

So alleged mystery on mystery even if the formalism gives a precise mathematical description of the wave function and there are precise rules which can be learnt by anyone prepared to put the effort in learning the maths how to use it to predict the essential features of quantum systems.

As I've explained before much of the mystery can be dissolved if the wave function is seen for what it is mathematically namely the complex square root of a probability distribution function or a complex probability amplitude. The configuration space of an N body system is simply the probability sample space associated with the system and one can include spin variables in this with out any worry about the existence of such an entity as a field in configuration space. The only difference between qunatum probability and every day probability is that one needs to use complex probability amplitudes and the Born rule in order to obtain the correct probability density function for a given situation.

I explain how all this works here

http://chrisfmathsphysicsmusic.blogspot.co.uk/search?q=Two+state+

I can not stress to highly how it really is quite instructive to cast the language of classical probability in terms of quantum probablilty. No one claims that the probability state vector of a coin or dice is a real superposition of head or tail states why should we do the same for quantum systems.

Accepting the intrinsic statisitical nature of quantum physics seems to me to dissolve a lot of the conceptual problems associated with quantum physics. The superposition of quantum states is a superposition of quantum probablities and the quantum state is a mathematical description of the possibilities open to a given system. Nothing collapses physically when a measurement is made and nothing exists in (3N+1)*S configuration space where S is the sum of all the allowable spin states.

OK we still don't know why we have to use complex probability amplitudes, but once we accept that then the rest follows. I would say the successful use of the formalism by physicists to model more and more complicated situations does mean that we do understand how quantum physics works and those who are able to use it can genuinely be said to understand the phenomenon they are trying to model.

As a final point even Classical physics which was claimed to be understood used quantities which were quite mysterious, No one knew what gravity was apart from the fact that it obeyed an inverse square law. No one knew how electromagnetic waves could propagate through free space or what entropy was. Of course that didn't stop people (as today with the wave function) trying to visualise what an electric field was but these speculations got no where. Helmholtz in a move, seen as desperate by a lot of the people at the time. getting so tired of the endless speculation, claimed that Maxwell's theory was Maxwell's equations.

A statement very similar to the 'Shut up and calculate' approach of Feynman.and other successful physicists who want to develop the applications of quantum mechanics rather than indulge in pretty fruitless speculation about the real nature of the wave-function. If you want to understand quantum physics learn how it is used engage with the maths and learn how to apply it to the phenomenon you are interested in. That way you will begin to get a feel for how quantum mechanics works, and just as no one claimed we didn't understand classical physics even if we couldn't visualise what an electric field was so we should stop claiming that we don't understand how quantum physics works.

I suppose the problem is that quantum physics (at least in it;s Non-relativistic formalism) uses a quantity called the wavefunction, but, as I've argued before in previous posts better seen as a probability state vector who's modulus squared gives rise to a probability density function and whos eigenvalues can be related to the energy levels or decay rates of atoms, molecules or solids. The problem is that for an N body system the so called wavefunction becomes a function of the 3 N coordinates of the system. So if it is seen as a field analagous to an electromagnetic field or the gravitational field. It is a field in the 3N+1 configuration space of the system rather than our normal 3 dimensional space. Matters are even more complicated when extra variables such as spin are also included. Spin has no spatial or time component so what is spin really ?

The other key issue is the notorious collapse of the wave function if there is a possibility that a particle can be in one of two or more states then it is said to be in limbo until a measurement is made and then it collapses into one of it;s states and in situations like the Aspect experiment this influence is said to occur at speed's faster than the speed of light because given a pair of photons emitted from a common source then so the story goes if I measure the spin of one photon it immediately causes the other photon which could be miles away to have the opposite spin.

So alleged mystery on mystery even if the formalism gives a precise mathematical description of the wave function and there are precise rules which can be learnt by anyone prepared to put the effort in learning the maths how to use it to predict the essential features of quantum systems.

As I've explained before much of the mystery can be dissolved if the wave function is seen for what it is mathematically namely the complex square root of a probability distribution function or a complex probability amplitude. The configuration space of an N body system is simply the probability sample space associated with the system and one can include spin variables in this with out any worry about the existence of such an entity as a field in configuration space. The only difference between qunatum probability and every day probability is that one needs to use complex probability amplitudes and the Born rule in order to obtain the correct probability density function for a given situation.

I explain how all this works here

http://chrisfmathsphysicsmusic.blogspot.co.uk/search?q=Two+state+

I can not stress to highly how it really is quite instructive to cast the language of classical probability in terms of quantum probablilty. No one claims that the probability state vector of a coin or dice is a real superposition of head or tail states why should we do the same for quantum systems.

Accepting the intrinsic statisitical nature of quantum physics seems to me to dissolve a lot of the conceptual problems associated with quantum physics. The superposition of quantum states is a superposition of quantum probablities and the quantum state is a mathematical description of the possibilities open to a given system. Nothing collapses physically when a measurement is made and nothing exists in (3N+1)*S configuration space where S is the sum of all the allowable spin states.

OK we still don't know why we have to use complex probability amplitudes, but once we accept that then the rest follows. I would say the successful use of the formalism by physicists to model more and more complicated situations does mean that we do understand how quantum physics works and those who are able to use it can genuinely be said to understand the phenomenon they are trying to model.

As a final point even Classical physics which was claimed to be understood used quantities which were quite mysterious, No one knew what gravity was apart from the fact that it obeyed an inverse square law. No one knew how electromagnetic waves could propagate through free space or what entropy was. Of course that didn't stop people (as today with the wave function) trying to visualise what an electric field was but these speculations got no where. Helmholtz in a move, seen as desperate by a lot of the people at the time. getting so tired of the endless speculation, claimed that Maxwell's theory was Maxwell's equations.

A statement very similar to the 'Shut up and calculate' approach of Feynman.and other successful physicists who want to develop the applications of quantum mechanics rather than indulge in pretty fruitless speculation about the real nature of the wave-function. If you want to understand quantum physics learn how it is used engage with the maths and learn how to apply it to the phenomenon you are interested in. That way you will begin to get a feel for how quantum mechanics works, and just as no one claimed we didn't understand classical physics even if we couldn't visualise what an electric field was so we should stop claiming that we don't understand how quantum physics works.

## Thursday, 13 June 2013

### A224 Inside Music Review

Of course the other thing that has been going on in my OU life has been the study of A224 Inside Music. I would say on the whole this is a great course. However I do have my reservations but I'm not sure they could be resolved given the nature of the course.

The course covers a lot of a ground in a very short space of time. In terms of theory you are taken up to about ABRSM grade 6 with a mention of a bit more. It assumes as a pre-requisite that you have a basic ability to read music up to about ABRSM grade three. There is an introduction to basic music theory available on Open learn for those who don't have the basics

http://www.open.edu/openlearn/history-the-arts/culture/music/introduction-music-theory/content-section-0

But there is a lot in this course that is not even mentioned in the ABRSM grade exams, such as the discussion of Sonata form and you are taught the basics of analysis of quite complicated pieces.such as Mozart's piano concerto in C minor and Brahm's third symphony.

The core of the course is a crash course in the composition of songs and it really is a crash course. These days any one with a laptop and access to a music notation software such as Sibelius can compose. One no longer needs access to a piano or the ability to hear music in ones head before it's written down or play a musical instrument, something this course takes advantage of. Unfortunately in my opinion Sibelius is really hard to use fluently and to write pieces of music. You have to search through various menus which I never did get the hang of how to use properly. Yes it can be simplified by use of key-board short cuts, but the keyboard short cuts only take you so far. Unfortunately Sibelius has rapidly become the industry standard, so if you aspire to take composing further as I do then it's a necessary evil just like Microsoft Office.

The course material is very interesting, but the composition material is only accessible on-line I guess the OU feel that they have something unique here and don't want outsiders to access it via second hand copies. Even so I would have thought that for those who want to refer to the material once the course is finished it would have only been fair to give the students a hard copy of the material. Especially if they have paid £2500 for it as those in England have to it seems a bit mean to say the least. Ok you can download pdfs of the material but as the material would be quite bulky if printed out, it's hardly the most convenient. However each online unit tried to cover far too much material in the short space of a week. However that is probably true of most OU courses in general, if one were to diligently do all the suggested exercises it would take about two - three weeks per unit. In my case I only managed to seriously look at the first four composition units out of 6 and I was able to draw on my previous knowledge of music.

In terms of other content the course is an odd mix of pop, classical (Art music) and world (Ethnic music) I guess it's trying to be all things to all people. One minute you will be analysing a song by Oasis, then a keyboard piece by Bach, then some jazz and then some world music. All of this is not treated very systematically and most of it to be quite frank just wont be read or listened to by most students as they will be desparately trying to just complete the assignments which are just relentless. Thus although the set piece was nominally Mozarts piano concerto in C minor I more or less ignored it as it wasn't assessed in the assignments.

Also the assignments seem to go from fairly basic ones to really advanced ones in the short space of time. I realise that by only completing 8 assignments you can't cover everything. However I think a real shortcoming was the serious lack of drill exercises as part of the formal assessment. No where were we asked to harmonise a melody, apart from in our songs. As a consequence I have for example a vague idea of what an Augmented 6th is but I lack confidence in how to introduce into my compositions. The course material really did not help here. In this the course is quite different from the ABRSM grade theory exams which start from the basics and build up step by step. I feel the need to do these to actually learn the theory rather than just retain a vague idea of it.

So overall as an introduction to music this course is really quite good, but I doubt I have really learnt the basics in a way that will stick. It does need to be complemented by a detailed study of the mechanics of music in a way similar to that of the ABRSM exams. If done in conjunction with such rigour then this course nicely complements them but byiteslf it is only part of the whole jigsaw. Perhaps it was trying too much, certainly if one were to compare this course with what happens at the first year of a degree level music course at a conventional university this would be sadly lacking. Where is the systematic introduction to harmony and counterpoint? Where is the systematic historic introduction to classical music and the various classical styles? Absolutely nowhere, the older course A214, whilst not so exciting was more systematic in teaching the basics of music and there were drill exercises (although probably not enough). Also a basic overview of the the three main styles of classical music, Baroque, classical and Romantic was given. A214 was more focused than this course and probably better than A224 for it.

As part of a general humanities degree this course fits quite well and it is good that such a course is available but anyone thinking they have studied music at degree level or even A level or higher by doing this course would be kidding themselves. On the whole I enjoyed this course as it helped get me thinking about music again and the composition part was something new to me. However I will certainly need to do a lot more. I will put myself forward for the Grade 5 ABRSM exam in November then all the way up to grade 8 and hopefully start piano lessons so I can consolidate the material here before embarking on the OCA composition courses which seems the next logical step.

The course covers a lot of a ground in a very short space of time. In terms of theory you are taken up to about ABRSM grade 6 with a mention of a bit more. It assumes as a pre-requisite that you have a basic ability to read music up to about ABRSM grade three. There is an introduction to basic music theory available on Open learn for those who don't have the basics

http://www.open.edu/openlearn/history-the-arts/culture/music/introduction-music-theory/content-section-0

But there is a lot in this course that is not even mentioned in the ABRSM grade exams, such as the discussion of Sonata form and you are taught the basics of analysis of quite complicated pieces.such as Mozart's piano concerto in C minor and Brahm's third symphony.

The core of the course is a crash course in the composition of songs and it really is a crash course. These days any one with a laptop and access to a music notation software such as Sibelius can compose. One no longer needs access to a piano or the ability to hear music in ones head before it's written down or play a musical instrument, something this course takes advantage of. Unfortunately in my opinion Sibelius is really hard to use fluently and to write pieces of music. You have to search through various menus which I never did get the hang of how to use properly. Yes it can be simplified by use of key-board short cuts, but the keyboard short cuts only take you so far. Unfortunately Sibelius has rapidly become the industry standard, so if you aspire to take composing further as I do then it's a necessary evil just like Microsoft Office.

The course material is very interesting, but the composition material is only accessible on-line I guess the OU feel that they have something unique here and don't want outsiders to access it via second hand copies. Even so I would have thought that for those who want to refer to the material once the course is finished it would have only been fair to give the students a hard copy of the material. Especially if they have paid £2500 for it as those in England have to it seems a bit mean to say the least. Ok you can download pdfs of the material but as the material would be quite bulky if printed out, it's hardly the most convenient. However each online unit tried to cover far too much material in the short space of a week. However that is probably true of most OU courses in general, if one were to diligently do all the suggested exercises it would take about two - three weeks per unit. In my case I only managed to seriously look at the first four composition units out of 6 and I was able to draw on my previous knowledge of music.

In terms of other content the course is an odd mix of pop, classical (Art music) and world (Ethnic music) I guess it's trying to be all things to all people. One minute you will be analysing a song by Oasis, then a keyboard piece by Bach, then some jazz and then some world music. All of this is not treated very systematically and most of it to be quite frank just wont be read or listened to by most students as they will be desparately trying to just complete the assignments which are just relentless. Thus although the set piece was nominally Mozarts piano concerto in C minor I more or less ignored it as it wasn't assessed in the assignments.

Also the assignments seem to go from fairly basic ones to really advanced ones in the short space of time. I realise that by only completing 8 assignments you can't cover everything. However I think a real shortcoming was the serious lack of drill exercises as part of the formal assessment. No where were we asked to harmonise a melody, apart from in our songs. As a consequence I have for example a vague idea of what an Augmented 6th is but I lack confidence in how to introduce into my compositions. The course material really did not help here. In this the course is quite different from the ABRSM grade theory exams which start from the basics and build up step by step. I feel the need to do these to actually learn the theory rather than just retain a vague idea of it.

So overall as an introduction to music this course is really quite good, but I doubt I have really learnt the basics in a way that will stick. It does need to be complemented by a detailed study of the mechanics of music in a way similar to that of the ABRSM exams. If done in conjunction with such rigour then this course nicely complements them but byiteslf it is only part of the whole jigsaw. Perhaps it was trying too much, certainly if one were to compare this course with what happens at the first year of a degree level music course at a conventional university this would be sadly lacking. Where is the systematic introduction to harmony and counterpoint? Where is the systematic historic introduction to classical music and the various classical styles? Absolutely nowhere, the older course A214, whilst not so exciting was more systematic in teaching the basics of music and there were drill exercises (although probably not enough). Also a basic overview of the the three main styles of classical music, Baroque, classical and Romantic was given. A214 was more focused than this course and probably better than A224 for it.

As part of a general humanities degree this course fits quite well and it is good that such a course is available but anyone thinking they have studied music at degree level or even A level or higher by doing this course would be kidding themselves. On the whole I enjoyed this course as it helped get me thinking about music again and the composition part was something new to me. However I will certainly need to do a lot more. I will put myself forward for the Grade 5 ABRSM exam in November then all the way up to grade 8 and hopefully start piano lessons so I can consolidate the material here before embarking on the OCA composition courses which seems the next logical step.

### Not my greatest day MST326 exam debrief part 1

Well I had MST326 exam this morning and it was a bit of a disaster quite frankly. Things started to go wrong right from the start when I got bogged down in two 'easy' part 1 questions. Only managed to do about half of each question needed. Might just scrape a pass but thats about it. I'll give a fuller debrief in a couple of days but the OU have asked us not to discuss the exam in any detail for a couple of days so I'll do it after the weekend. I can't remember the questions in any great detail anyway. Just that nothing seemed to come out for me. I think I'm getting a bit too old to do detailed tricky calculation type questions under exam conditions admittedly my revision schedule wasn't that great. I do have serious reservations about whether or not to do the MSc in maths if every year it's going to end up with me struggling to do the TMA's on time and being ill prepared for an exam. Indeed I'm beginning to have doubts about the whole OU experience don't get me wrong the courses are interesting and the material is great it's just that I feel under real pressure to do the TMA's and focus on the exam and so I'm just cherry picking the bits of the course relevant to the TMA and the exam and not really learning the material this applies to all the courses I've done since M208 the last OU course I've really enjoyed and made a reasonable stab at.

In the grand scheme of things I don't have to count this course for anything so will quietly drop it and hope I do better on my other courses which will be

Quantum Mechanics exam in October

Then I've registered for Number theory and logic and also the third philosophy level course AA308 Thought and experience and I intend to do the physics project. It will be good to get back into philosophy again. I will have an effective degree in Maths physics and philosophy the courses being

MST121 and MS221 Introducing Maths and exploring mathematics 60 points

M208 Pure Maths 60 points

A211 Introducing Philosophy 60 points

M358 Quantun Physics 30 points

M381 Number theory and logic 30 points

Quantum Physics project 30 points

AA308 Thought and experience Philosophy of Mind 60 poiints

and my least worse of MS324 Waves diffusion etc or MST326 Fluids. If I get grade 2 in quantum physics, and Philosophy of Mind then I should be in a reasonable position to get a 2/I for my second OU degree which should put me in a reasonable postion to do the MA in European Philosophy at St Davids.

Not that I intend to drop my study of maths I really want to get back to studying my own subjects in my own way without a TMA looming ahead. Matters haven't helped that there has been no effective break since Feb 2012 and I'm feeliing quite fatigued with it all. Certainly I don't want to have to be asked to solve a tricky separation of variables question in a ridiculously short time scale as MST326 expected us to do. This is a pity as that was for me the most interesting bit of the course but I just can't do this sort of thing accurately and under time pressure. When they come out it is one of the most satisfying of all experiences but not under exam conditions.

Amongst maths/physics projects I want to take up in the lull are

1) Finally complete the derivation of the Friedmann equations from the equations of General Relativity and apply it to the current standard model of the universe. Part of this will involve solving the resulting differential equations numerically and so I will look at the Cambridge computing projects

2) Dig a bit deeper into Galois theory a topic I started a couple of years ago the book I have in mind is

http://www.amazon.com/Galois-Theory-Beginners-Mathematical-Matehmatical/dp/0821838172

This starts off with explicit solutions for cubic and quartic equations. There is obviously a lot of interest in Galois theory as a search through the statistics for this blog shows that any posts I have that refer to the topic seem to get a large number of hits.

3) Solve Schrodinger's equation in parabolic coordinates for both the bound state problem and the scattering problem. This will be a tour de force of all the techniques used to solve partial differential equations, the resulting solutions go by the ridiculous name of the confluent hyper-geometric function however what is of interest is that the scattering problem can be solved exactly even if the resulting expressions are a tad obscure to extract meaning from.

Plus the small matter of getting back on track with my quantum mechanics course.

In the grand scheme of things I don't have to count this course for anything so will quietly drop it and hope I do better on my other courses which will be

Quantum Mechanics exam in October

Then I've registered for Number theory and logic and also the third philosophy level course AA308 Thought and experience and I intend to do the physics project. It will be good to get back into philosophy again. I will have an effective degree in Maths physics and philosophy the courses being

MST121 and MS221 Introducing Maths and exploring mathematics 60 points

M208 Pure Maths 60 points

A211 Introducing Philosophy 60 points

M358 Quantun Physics 30 points

M381 Number theory and logic 30 points

Quantum Physics project 30 points

AA308 Thought and experience Philosophy of Mind 60 poiints

and my least worse of MS324 Waves diffusion etc or MST326 Fluids. If I get grade 2 in quantum physics, and Philosophy of Mind then I should be in a reasonable position to get a 2/I for my second OU degree which should put me in a reasonable postion to do the MA in European Philosophy at St Davids.

Not that I intend to drop my study of maths I really want to get back to studying my own subjects in my own way without a TMA looming ahead. Matters haven't helped that there has been no effective break since Feb 2012 and I'm feeliing quite fatigued with it all. Certainly I don't want to have to be asked to solve a tricky separation of variables question in a ridiculously short time scale as MST326 expected us to do. This is a pity as that was for me the most interesting bit of the course but I just can't do this sort of thing accurately and under time pressure. When they come out it is one of the most satisfying of all experiences but not under exam conditions.

Amongst maths/physics projects I want to take up in the lull are

1) Finally complete the derivation of the Friedmann equations from the equations of General Relativity and apply it to the current standard model of the universe. Part of this will involve solving the resulting differential equations numerically and so I will look at the Cambridge computing projects

2) Dig a bit deeper into Galois theory a topic I started a couple of years ago the book I have in mind is

http://www.amazon.com/Galois-Theory-Beginners-Mathematical-Matehmatical/dp/0821838172

This starts off with explicit solutions for cubic and quartic equations. There is obviously a lot of interest in Galois theory as a search through the statistics for this blog shows that any posts I have that refer to the topic seem to get a large number of hits.

3) Solve Schrodinger's equation in parabolic coordinates for both the bound state problem and the scattering problem. This will be a tour de force of all the techniques used to solve partial differential equations, the resulting solutions go by the ridiculous name of the confluent hyper-geometric function however what is of interest is that the scattering problem can be solved exactly even if the resulting expressions are a tad obscure to extract meaning from.

Plus the small matter of getting back on track with my quantum mechanics course.

## Wednesday, 5 June 2013

### Two sides of Hume's Fork Weinstein's Geometric Theory

My attention was drawn to an attempt by an outsider to come up with (yet another) Grand Unified theory.

http://www.guardian.co.uk/science/2013/may/23/eric-weinstein-answer-physics-problems

http://www.guardian.co.uk/science/blog/2013/may/23/roll-over-einstein-meet-weinstein

What is unsual about this is that Mr Weinstein hasn't even published a paper describing his theory not even on ArXiv so that other people can examine it and test it's credibility. It would seem that his friendship with Marcus du Sautoy enabled him to give a highly prestigious lecture. I'm sure the guy has come up with some clever maths but like say the current state of superstrings or the multiverse the fact that the theory can't make any predictions and hasn't even been written down means it's definitely in the not even wrong theory.

It does make one question the credibility of Marcus du Sautoy who seems to be latching onto every possible break through in physics (not that Weinstein's attempt could be seen as a breakthrough) and over egg the pudding to say the least. He claimed in a Horizon programme last year that the alleged violation of the speed of light by neutrino's proved superstring theory and M theory. When in fact the result was shown to be due to a faulty cable. I did not see him eating his boxer shorts as Jim Al Khalili offered to if the result was shown to be correct. Marcus du Sautoy righthly has come in for some intense criticism in the way he has handled this issues some links to which can be seen here

http://blogs.scientificamerican.com/cocktail-party-physics/2013/05/24/dear-guardian-youve-been-played/

Anyway this is not the main point of this post. I want to examine what seems to be a misconception by the likes of Marcus du Sautoy about how physics works. People like him seem to think there is a one-one correspondence betweem mathematical structures and the natural world. However any mathematical construction when applied to the real world can only be an approximation not the real thing.

Hume put the dilemma quite precisely in his Fork. Humans use two types of reasoning deductive and inductive. Deductive reasoning such as mathematics applies to the realm of ideas whereas we use inductive reasoning to apply to matters of fact.

As I pointed out a couple of years back

http://www.blogger.com/blogger.g?blogID=7897235423812277683#editor/target=post;postID=7025597943261709647;onPublishedMenu=allposts;onClosedMenu=allposts;postNum=59;src=postname

Hume points out that although we use inductive reasoning all the time there can be no rational justification for it as it involves a circular argument. How do I know that if confronted with a glass of claret it will taste the same as another one by an appeal to experience. How do I ground my experiences by an appeal to the uniformity of nature. How do I know that nature is uniform by an appeal to experience. Reasoning based on induction requires many observations to establish a truth and again not every glass of claret will taste exactly the same. On the other hand inductive reasoning is used all the time.

In contrast in the realm of ideas mathematical and logical deductions once proven are true for all time it only needs one demonstration that eg the gap between two prime numbers is less than 7.5 billion to show that this is the case.

When it comes to applying mathematical ideas to the natural world although the laws of physics can be expressed mathematically they are only approximations albeit really good ones for a lot of cases. In what Hume calls mixed maths which we nowadays call Applied maths the laws are grounded in empirical observation. To test a theory requires a lot of hard work in relating the general principles to a concerte prediction. For example the predicition of the properties of the Higg's boson required many calculations by whole teams of people based on reasonable approximations from the Standard model using the techniques of quantum field theory and even more people to test the predctions by measuring the decay rates and scattering cross sections. This involves calculatiing Hundreds of Feynman diagrams and summing up their individual contributions to the given decay rate or scattering cross-section. Messy, tedious prone to error but thats how real calculations are made in physics even then the actual value of the given parameter will only be an approximation albeit a good one.

Another point is that the laws of phyiscs when expressed mathematically will always have an empirical constant associated with it. The accuracy of the application of say Schrodinger's equation which involves the masses and charges of the particles involved will only be as good as the measured values. Even if one day we do reduce everything to 1 coupling constant, that coupling constant will have to be measured and one measurement will not suffice. . All this points to a degree of approximation associated with the laws of physics expressed mathematically and far removed from the pristine unmessy platonic world that du Sautoy and Weinstein live in.

The symmetries of the Standard model are only approximate to take an example the masses of the quarks in each generation are only approximately the same, indeed as I pointed out it here

http://www.blogger.com/blogger.g?blogID=7897235423812277683#editor/target=post;postID=3500862736577281127;onPublishedMenu=allposts;onClosedMenu=allposts;postNum=3;src=postname

It stretches credulity to suggest that the top quark has the same mass as the bottom quark. Or that neutrino's are massless. Nevertheless as an empirical summary of the current status of particle physics the Standard model is probably the most empirically adequate theory we have. Even if trying to predict results from it is quite a messy busimess.

Such a world is far removed from the Pristine Platonic world of Marcus du Sautoy and Weinstein. It is not enough to come up with some elegant mathematics one has to show how it applies to the real world and acknowledge that at best it will be an ideal approximation. What people like Weinstein and Marcus du Sautoy do is confuse two quite separate areas and types of reasoning. Seduced by the elegance of their mathematics they think they have found the ultimate secret of reality. Unfortunately for them the natural world will always end up blowing a great raspberry at their naive idealism as the approximation will sooner or later breakdown.

http://www.guardian.co.uk/science/2013/may/23/eric-weinstein-answer-physics-problems

http://www.guardian.co.uk/science/blog/2013/may/23/roll-over-einstein-meet-weinstein

What is unsual about this is that Mr Weinstein hasn't even published a paper describing his theory not even on ArXiv so that other people can examine it and test it's credibility. It would seem that his friendship with Marcus du Sautoy enabled him to give a highly prestigious lecture. I'm sure the guy has come up with some clever maths but like say the current state of superstrings or the multiverse the fact that the theory can't make any predictions and hasn't even been written down means it's definitely in the not even wrong theory.

It does make one question the credibility of Marcus du Sautoy who seems to be latching onto every possible break through in physics (not that Weinstein's attempt could be seen as a breakthrough) and over egg the pudding to say the least. He claimed in a Horizon programme last year that the alleged violation of the speed of light by neutrino's proved superstring theory and M theory. When in fact the result was shown to be due to a faulty cable. I did not see him eating his boxer shorts as Jim Al Khalili offered to if the result was shown to be correct. Marcus du Sautoy righthly has come in for some intense criticism in the way he has handled this issues some links to which can be seen here

http://blogs.scientificamerican.com/cocktail-party-physics/2013/05/24/dear-guardian-youve-been-played/

Anyway this is not the main point of this post. I want to examine what seems to be a misconception by the likes of Marcus du Sautoy about how physics works. People like him seem to think there is a one-one correspondence betweem mathematical structures and the natural world. However any mathematical construction when applied to the real world can only be an approximation not the real thing.

Hume put the dilemma quite precisely in his Fork. Humans use two types of reasoning deductive and inductive. Deductive reasoning such as mathematics applies to the realm of ideas whereas we use inductive reasoning to apply to matters of fact.

As I pointed out a couple of years back

http://www.blogger.com/blogger.g?blogID=7897235423812277683#editor/target=post;postID=7025597943261709647;onPublishedMenu=allposts;onClosedMenu=allposts;postNum=59;src=postname

Hume points out that although we use inductive reasoning all the time there can be no rational justification for it as it involves a circular argument. How do I know that if confronted with a glass of claret it will taste the same as another one by an appeal to experience. How do I ground my experiences by an appeal to the uniformity of nature. How do I know that nature is uniform by an appeal to experience. Reasoning based on induction requires many observations to establish a truth and again not every glass of claret will taste exactly the same. On the other hand inductive reasoning is used all the time.

In contrast in the realm of ideas mathematical and logical deductions once proven are true for all time it only needs one demonstration that eg the gap between two prime numbers is less than 7.5 billion to show that this is the case.

When it comes to applying mathematical ideas to the natural world although the laws of physics can be expressed mathematically they are only approximations albeit really good ones for a lot of cases. In what Hume calls mixed maths which we nowadays call Applied maths the laws are grounded in empirical observation. To test a theory requires a lot of hard work in relating the general principles to a concerte prediction. For example the predicition of the properties of the Higg's boson required many calculations by whole teams of people based on reasonable approximations from the Standard model using the techniques of quantum field theory and even more people to test the predctions by measuring the decay rates and scattering cross sections. This involves calculatiing Hundreds of Feynman diagrams and summing up their individual contributions to the given decay rate or scattering cross-section. Messy, tedious prone to error but thats how real calculations are made in physics even then the actual value of the given parameter will only be an approximation albeit a good one.

Another point is that the laws of phyiscs when expressed mathematically will always have an empirical constant associated with it. The accuracy of the application of say Schrodinger's equation which involves the masses and charges of the particles involved will only be as good as the measured values. Even if one day we do reduce everything to 1 coupling constant, that coupling constant will have to be measured and one measurement will not suffice. . All this points to a degree of approximation associated with the laws of physics expressed mathematically and far removed from the pristine unmessy platonic world that du Sautoy and Weinstein live in.

The symmetries of the Standard model are only approximate to take an example the masses of the quarks in each generation are only approximately the same, indeed as I pointed out it here

http://www.blogger.com/blogger.g?blogID=7897235423812277683#editor/target=post;postID=3500862736577281127;onPublishedMenu=allposts;onClosedMenu=allposts;postNum=3;src=postname

It stretches credulity to suggest that the top quark has the same mass as the bottom quark. Or that neutrino's are massless. Nevertheless as an empirical summary of the current status of particle physics the Standard model is probably the most empirically adequate theory we have. Even if trying to predict results from it is quite a messy busimess.

Such a world is far removed from the Pristine Platonic world of Marcus du Sautoy and Weinstein. It is not enough to come up with some elegant mathematics one has to show how it applies to the real world and acknowledge that at best it will be an ideal approximation. What people like Weinstein and Marcus du Sautoy do is confuse two quite separate areas and types of reasoning. Seduced by the elegance of their mathematics they think they have found the ultimate secret of reality. Unfortunately for them the natural world will always end up blowing a great raspberry at their naive idealism as the approximation will sooner or later breakdown.

## Wednesday, 22 May 2013

### One for the number theorists

Ok here is something which should interest the number theorists who read this blog (Duncan Neil possibly Daniel) . Some guy has shown that the gap between any two prime numbers is bounded to be less than 7.5*10^7. This was drawn to my attention by an article in todays independent

http://www.independent.co.uk/news/science/that-figures-professor-who-had-to-work-at-subway-dazzles-world-of-maths-after-solving-centuriesold-prime-number-riddle-8625637.html

A bit more detail can be found here

http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

If ever I do the analytical number theory options for the MSc I might get a glimmer of what is going on.

For those who have access to the Open University a draft of his paper can be found in the Annals of Maths forthcoming papers just search for Zhang. For those who don't I can download a copy for them just send me an e-mail chrisf19572002@yahoo.co.uk

What is heartwarming about this story is that it shows the virtues of persistance and being willing to spend a lot of time on ones own in relative obscurity not publishing many papers. In these days of corporate research where everything is governed by how many papers you can churn out (a bit like meeting TMA deadlines) its great that some people still have enough independence to pursue their dream. Alas as my fellow blogger Nilo has pointed out this is all to rare.

http://mathematics-diary.blogspot.co.uk/2013/05/what-is-mathematical-research.html

Will give an update on progress in Fluids over the weekend completed about 60% worth of the last TMA which should be enough to enable me to get a grade 2 as far as OCAS is concerned but just missed the lunchtime post so don't know if I'll make the deadline Music this weekend.

http://www.independent.co.uk/news/science/that-figures-professor-who-had-to-work-at-subway-dazzles-world-of-maths-after-solving-centuriesold-prime-number-riddle-8625637.html

A bit more detail can be found here

http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

If ever I do the analytical number theory options for the MSc I might get a glimmer of what is going on.

For those who have access to the Open University a draft of his paper can be found in the Annals of Maths forthcoming papers just search for Zhang. For those who don't I can download a copy for them just send me an e-mail chrisf19572002@yahoo.co.uk

What is heartwarming about this story is that it shows the virtues of persistance and being willing to spend a lot of time on ones own in relative obscurity not publishing many papers. In these days of corporate research where everything is governed by how many papers you can churn out (a bit like meeting TMA deadlines) its great that some people still have enough independence to pursue their dream. Alas as my fellow blogger Nilo has pointed out this is all to rare.

http://mathematics-diary.blogspot.co.uk/2013/05/what-is-mathematical-research.html

Will give an update on progress in Fluids over the weekend completed about 60% worth of the last TMA which should be enough to enable me to get a grade 2 as far as OCAS is concerned but just missed the lunchtime post so don't know if I'll make the deadline Music this weekend.

## Sunday, 12 May 2013

### Cutting one's losses

Well I've had to default on a couple of TMA's as I'm sinking in them

The TMA's I've had to drop are TMA06 for the music course and TMA02 for quantum mechanics, the reason is really the lack of time between now and exam week which is on 13th June for me. I have to complete the Final TMA for the Fluids course and also the final EMA assignment for the music course before this. There really is far too short a gap between the deadlines for the final assigments and the exams normally one would expect 5-6 weeks for revison but we only have three.

Part of the problem is of course being caught between change over from Febuary starts to October starts so I really don't feel to bad about doing this. For Music I have alrady got 60% for the OCAS so on target for a grade three. I possibly would have got grade 2 had I submitted TMA06 but as I don't need this course for anything else I don't mind cutting my losses

As for QM the TMA's are formative and as I need to get 30% in at least seven of them I can drop one unfortunate as it is. I'll suspend my study of quantum mechanics till after the exam,

I've completed half the TMA for Fluids (more next week) and should finish by Monday next week, So I can start to look at past papers I should do at least 3 under exam conditions to get up to speed,

As for the EMA for music I hope to make a good start on it next week and should have completed it in time. It consists of two parts an analysis of the first movement of Beethoven's Ghost Trio and completion of the composition which was started in TMA05.

All of this is unsatisfactory but I don't really have any choice and the consequences aren't that fateful

If I can get up to speed for the Fluids I'm looking at grade 2 but as I tend to make silly slips in signs etc especially when under pressure a high end Grade 3just as with MS324 is more likely.

Must admit the pressure has been quite relentless the fact that there was no break between this batch of courses and the previous batch has meant a continous set of 18 months intensive study since Feb 2012, and I'm really looking forward to having to concentrate on one course after June.

Footnote added 13th May

My quantum mechanics tutor has offered me a three week extension so I should be able to complete the second TMA by then. One of the things about the third level physics courses is that there is a staggering 10 assignments to complete. 4 conventioonal ones and 6 Online ones. The Online ones are actually quite tricky and fiddly but there is no room to show how you arrived at your answer so I spent some rather tedious hours simply because I had got a sign wrong. OK I appreciate accuracy is important but this is a bit over the top. Anyway at least I should make a decent stab at the second TMA by early June.

The TMA's I've had to drop are TMA06 for the music course and TMA02 for quantum mechanics, the reason is really the lack of time between now and exam week which is on 13th June for me. I have to complete the Final TMA for the Fluids course and also the final EMA assignment for the music course before this. There really is far too short a gap between the deadlines for the final assigments and the exams normally one would expect 5-6 weeks for revison but we only have three.

Part of the problem is of course being caught between change over from Febuary starts to October starts so I really don't feel to bad about doing this. For Music I have alrady got 60% for the OCAS so on target for a grade three. I possibly would have got grade 2 had I submitted TMA06 but as I don't need this course for anything else I don't mind cutting my losses

As for QM the TMA's are formative and as I need to get 30% in at least seven of them I can drop one unfortunate as it is. I'll suspend my study of quantum mechanics till after the exam,

I've completed half the TMA for Fluids (more next week) and should finish by Monday next week, So I can start to look at past papers I should do at least 3 under exam conditions to get up to speed,

As for the EMA for music I hope to make a good start on it next week and should have completed it in time. It consists of two parts an analysis of the first movement of Beethoven's Ghost Trio and completion of the composition which was started in TMA05.

All of this is unsatisfactory but I don't really have any choice and the consequences aren't that fateful

If I can get up to speed for the Fluids I'm looking at grade 2 but as I tend to make silly slips in signs etc especially when under pressure a high end Grade 3just as with MS324 is more likely.

Must admit the pressure has been quite relentless the fact that there was no break between this batch of courses and the previous batch has meant a continous set of 18 months intensive study since Feb 2012, and I'm really looking forward to having to concentrate on one course after June.

Footnote added 13th May

My quantum mechanics tutor has offered me a three week extension so I should be able to complete the second TMA by then. One of the things about the third level physics courses is that there is a staggering 10 assignments to complete. 4 conventioonal ones and 6 Online ones. The Online ones are actually quite tricky and fiddly but there is no room to show how you arrived at your answer so I spent some rather tedious hours simply because I had got a sign wrong. OK I appreciate accuracy is important but this is a bit over the top. Anyway at least I should make a decent stab at the second TMA by early June.

## Tuesday, 23 April 2013

### MST326 Fluids TMA03

I've jsut completed the above ready to hand to my tutor tomorrow evening after work. This is probably the most hardcore Applied maths block that the OU offers much more involved mathematically than MST324 wonder if any of my colleagues who initially thought MST324 is harder than MST326 still think so.

Anyway the topic is a familiar one solution of Partial differential equations by the Separation of variables but this goes much further than either MST209 or MST326

The first question was on classifying a partial differential equation with mixed coefficents in terms of its type namely hyperbolic a well known example being the wave equation. Parabolic of which the diffusion equation is an example and elliptic which Laplaces equation is an example.

The second part of the question asked us to transform this complicated equation into a simpler form by the chain rule. Which is OK for first derirvatives but for second order partial derivatives the algebra gets quite messy still 5 pages later I transformed the equation into it's simple form and got the general solution.

The last part asked us to find a particular solution for a given boundary conditions I have to say i fouund this quite tricky and potentially confusing so had to leave most of the question. Rough estimate 18/25

Quesiton 2 was solving the Diffusion equation for a given boundaty condition by separation of variables I got most of this out but had to leave a couple of questions at the end so rough guess 20/25

Question 3 was a similar question to question 2 only for the Laplace equation on a rectangular region with variable boundary conditions again got most of this out but had to leave one or two tricky questions. so again about 20/25

Finally question 4. This was an odd's and sod's type question the first question was a relatively straightforward one which could have almost come out of an A level physics question calculating the frequency, wavelength and speed of a composite wave

The second part for just 1 extra mark from part 1 asked us to solve the wave equation using D'Alembert's solution as I've almost lost the will to live after the heavy algebra associatied with questions 2 and 3 I left this

The final part of question 4 involved expressing a Polynomical in terms of Legendre Polynomials and then using the solution to solve a heat conduction problem in a sphere with a variable boundary condition on the surface. Think I got most of this out so about 18/25 overall.

So just under 3/4 of the assignment done looking at about grade 2 or just under for this one. This will probably be my lowest score so far. However being cynical I should get grade 2 overall for the OCAS part of this course. The exam is looming and I still have another TMA coming up before revision starts. There is only a gap of about two weeks between the deadling for the TMA and revision. As I want to start looking at papers by early may so I can do 1 per week then I need to continue the momentum as far as fluids is concerned. If I can get up to speed then I'm looking for a grade 2 pass, but exams have a habit of slipping away.

Those of my colleagues (Duncan Daniel) reading this blog who have deserted Applied maths for Pure maths might like to consider doing MST326 to complement their pure maths.

As far as the other courses are going I got 90% for my quantum mechanics TMA but was slightly disappointed that my emphasis on the statistical interpretation of quantum mechanics barely got a mention.

For the music the last TMA involved setting some lyrics to music and showing that we could modulate effectively I got 76% for this which is reasonable but need to work on a few things. This will be embedded in a fuller setting for the final assessment.

Anyway No rest for the wicked another music assignment and an interactive quantum mechanics assignment looms and also I'll snip away at the last TMA for the fluids course.

Bye for now

Anyway the topic is a familiar one solution of Partial differential equations by the Separation of variables but this goes much further than either MST209 or MST326

The first question was on classifying a partial differential equation with mixed coefficents in terms of its type namely hyperbolic a well known example being the wave equation. Parabolic of which the diffusion equation is an example and elliptic which Laplaces equation is an example.

The second part of the question asked us to transform this complicated equation into a simpler form by the chain rule. Which is OK for first derirvatives but for second order partial derivatives the algebra gets quite messy still 5 pages later I transformed the equation into it's simple form and got the general solution.

The last part asked us to find a particular solution for a given boundary conditions I have to say i fouund this quite tricky and potentially confusing so had to leave most of the question. Rough estimate 18/25

Quesiton 2 was solving the Diffusion equation for a given boundaty condition by separation of variables I got most of this out but had to leave a couple of questions at the end so rough guess 20/25

Question 3 was a similar question to question 2 only for the Laplace equation on a rectangular region with variable boundary conditions again got most of this out but had to leave one or two tricky questions. so again about 20/25

Finally question 4. This was an odd's and sod's type question the first question was a relatively straightforward one which could have almost come out of an A level physics question calculating the frequency, wavelength and speed of a composite wave

The second part for just 1 extra mark from part 1 asked us to solve the wave equation using D'Alembert's solution as I've almost lost the will to live after the heavy algebra associatied with questions 2 and 3 I left this

The final part of question 4 involved expressing a Polynomical in terms of Legendre Polynomials and then using the solution to solve a heat conduction problem in a sphere with a variable boundary condition on the surface. Think I got most of this out so about 18/25 overall.

So just under 3/4 of the assignment done looking at about grade 2 or just under for this one. This will probably be my lowest score so far. However being cynical I should get grade 2 overall for the OCAS part of this course. The exam is looming and I still have another TMA coming up before revision starts. There is only a gap of about two weeks between the deadling for the TMA and revision. As I want to start looking at papers by early may so I can do 1 per week then I need to continue the momentum as far as fluids is concerned. If I can get up to speed then I'm looking for a grade 2 pass, but exams have a habit of slipping away.

Those of my colleagues (Duncan Daniel) reading this blog who have deserted Applied maths for Pure maths might like to consider doing MST326 to complement their pure maths.

As far as the other courses are going I got 90% for my quantum mechanics TMA but was slightly disappointed that my emphasis on the statistical interpretation of quantum mechanics barely got a mention.

For the music the last TMA involved setting some lyrics to music and showing that we could modulate effectively I got 76% for this which is reasonable but need to work on a few things. This will be embedded in a fuller setting for the final assessment.

Anyway No rest for the wicked another music assignment and an interactive quantum mechanics assignment looms and also I'll snip away at the last TMA for the fluids course.

Bye for now

## Tuesday, 2 April 2013

### SM358 Quantum Physics TMA01

I completed the first TMA for the quantum physics yesterday. On the whole quite straightforward anyway here is a break down of the questions

1) A question on the energy levels of an infinite square well, the first part asked us to calculate the frequency of radiation emitted when an electron jumped from 1 level to another the second part asked us to calculate the degeneracy of the energy levels mainly numerical tedious but straightforward.

2) The only really mathematical part of the TMA. Given a wavefunction we had to show that it was normalised calculate the expectation value of its momentum and also the probability of finding it in a ground state which is calculated by integrating the product of the ground state wave function and the original wave function and then taking the modulus squared of the integral. The integrals were Gaussian Integrals and would have been quite tricky to solve unaided but the question gave us the key integrals. Also one of the integrals was an odd function so could immediately be set to zero. I think I got all of this out.

3) An essay question about the nature of predictions in quantum physics and how they could be tested. I stressed the fact the quantum mechanics is essentially a statistical theory albeit a novel one as it involves the use of complex probability amplitudes rather than real numbers. It follows that in order to check the predictions of quantum mechanics one has to make many measurements under the same conditions and that a single measurement has just as much relevance as a single dice throw does in classical statistics. One point that struck me as I was writng down the full version of Schrodinger's equation is that the time derivative is first order thus mathematically Schrodinger's equation is similar to the diffusion equation and not a wave equation which has second order time derivatives. Strictly speaking we should be talking about Schrodinger's diffusion equation and not Schrodinger's wave equation, More evidence that the solutions to Schrodinger's equation are not classical waves.

I also pointed out that in say the two slit experiment the interference pattern is a cumulative effect and the emphasis on the behaviour of a single particle much beloved by many textbook accounts is irrelevant.

Finally I quoted Einstein who whilst well known for his quote that God does not play dice later said

"The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations (Eg Schrodinger's cat and the collapse of the wavefunction seen as a physical process (my comments)), which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems'

It would appear that even Einstein came to accept a statistical interpretation of quantum mechanics,

I gave some references to my favourite books and papers which regular readers of this blog will know however for convenience I repeat them here.

1) Silverman Quantum Superposition

http://www.amazon.co.uk/Quantum-Superposition-Counterintuitive-Consequences-Entanglement/dp/3540718834/ref=sr_1_1?ie=UTF8&qid=1364902238&sr=8-1

The first two chapters of this book should be essential reading for anyone who has been seduced by the alleged mysterious aspects of quantum physics. The point being that quantum superpostion is a superposition of probablity amplitudes and not a superposition of real waves or fields.

2) Ballentine Quantum Mechanics A Modern development

http://www.amazon.co.uk/Quantum-Mechanics-Development-Leslie-Ballentine/dp/9810241054/ref=sr_1_1?s=books&ie=UTF8&qid=1364902399&sr=1-1

This book whilst covering most of the standard content of any quantum physics course also introduces the Ensemble interpretation which following the hint from the Einstein quote above Ballentine has done much to develop.

3) Finally my favourite paper on the two slit experiment which alas is little known. Here Marcella shows how the Born rule and the use of complex probability amplitudes enables one to predict the essential features of the two slit experiment. Showing that the wave like aspects are essentially statistical and that one can speak of a single particle traveliing through a single slit. The point being that the slits act as measuring devices the uncertainty in position giving rise to a corresponding uncertainty in momentum. Something not usually covered in most quantum text books which tend to impose a classical interpretation on to an essentially quantum phenomenon.

http://arxiv.org/abs/quant-ph/0703126

I wonder if my tutor is aware of these references and if so what he makes of them.

It should be pointed out that whilst the TMA is part of the assessment there are also a whole load of on online activities which enable the topics covered to be treated in more detail. Some are actually quite tricky and also because of bad eyesight when it comes to small print on computer screens I tend to confuse chains of operators thus for example I wasted aeons of time on a couple of questions involving the number of creation and annhilation operators associated with the harmonic oscillator simply because I miscounted the number of A's and A hats involved. Fortunately for these type of questions it is possible to make many attempts. What I hadn't realised until recently was that even after three attempts one doesn't have to submit the final answer so you are allowed multiple goes for each question and then only after one has got the correct answer first time round do you have to submit. Had I realised that I would have got higher marks than I did for the questions I submitted. Still none of these really count all one has to do is get 40% overall for the assesment. But as there are a total of 10 of them it is worth doing them as thoroughly as one can and all of them. If you only did 5 say you would have to guarantee getting 80% for all of them to pass.

1) A question on the energy levels of an infinite square well, the first part asked us to calculate the frequency of radiation emitted when an electron jumped from 1 level to another the second part asked us to calculate the degeneracy of the energy levels mainly numerical tedious but straightforward.

2) The only really mathematical part of the TMA. Given a wavefunction we had to show that it was normalised calculate the expectation value of its momentum and also the probability of finding it in a ground state which is calculated by integrating the product of the ground state wave function and the original wave function and then taking the modulus squared of the integral. The integrals were Gaussian Integrals and would have been quite tricky to solve unaided but the question gave us the key integrals. Also one of the integrals was an odd function so could immediately be set to zero. I think I got all of this out.

3) An essay question about the nature of predictions in quantum physics and how they could be tested. I stressed the fact the quantum mechanics is essentially a statistical theory albeit a novel one as it involves the use of complex probability amplitudes rather than real numbers. It follows that in order to check the predictions of quantum mechanics one has to make many measurements under the same conditions and that a single measurement has just as much relevance as a single dice throw does in classical statistics. One point that struck me as I was writng down the full version of Schrodinger's equation is that the time derivative is first order thus mathematically Schrodinger's equation is similar to the diffusion equation and not a wave equation which has second order time derivatives. Strictly speaking we should be talking about Schrodinger's diffusion equation and not Schrodinger's wave equation, More evidence that the solutions to Schrodinger's equation are not classical waves.

I also pointed out that in say the two slit experiment the interference pattern is a cumulative effect and the emphasis on the behaviour of a single particle much beloved by many textbook accounts is irrelevant.

Finally I quoted Einstein who whilst well known for his quote that God does not play dice later said

"The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations (Eg Schrodinger's cat and the collapse of the wavefunction seen as a physical process (my comments)), which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems'

It would appear that even Einstein came to accept a statistical interpretation of quantum mechanics,

I gave some references to my favourite books and papers which regular readers of this blog will know however for convenience I repeat them here.

1) Silverman Quantum Superposition

http://www.amazon.co.uk/Quantum-Superposition-Counterintuitive-Consequences-Entanglement/dp/3540718834/ref=sr_1_1?ie=UTF8&qid=1364902238&sr=8-1

The first two chapters of this book should be essential reading for anyone who has been seduced by the alleged mysterious aspects of quantum physics. The point being that quantum superpostion is a superposition of probablity amplitudes and not a superposition of real waves or fields.

2) Ballentine Quantum Mechanics A Modern development

http://www.amazon.co.uk/Quantum-Mechanics-Development-Leslie-Ballentine/dp/9810241054/ref=sr_1_1?s=books&ie=UTF8&qid=1364902399&sr=1-1

This book whilst covering most of the standard content of any quantum physics course also introduces the Ensemble interpretation which following the hint from the Einstein quote above Ballentine has done much to develop.

3) Finally my favourite paper on the two slit experiment which alas is little known. Here Marcella shows how the Born rule and the use of complex probability amplitudes enables one to predict the essential features of the two slit experiment. Showing that the wave like aspects are essentially statistical and that one can speak of a single particle traveliing through a single slit. The point being that the slits act as measuring devices the uncertainty in position giving rise to a corresponding uncertainty in momentum. Something not usually covered in most quantum text books which tend to impose a classical interpretation on to an essentially quantum phenomenon.

http://arxiv.org/abs/quant-ph/0703126

I wonder if my tutor is aware of these references and if so what he makes of them.

It should be pointed out that whilst the TMA is part of the assessment there are also a whole load of on online activities which enable the topics covered to be treated in more detail. Some are actually quite tricky and also because of bad eyesight when it comes to small print on computer screens I tend to confuse chains of operators thus for example I wasted aeons of time on a couple of questions involving the number of creation and annhilation operators associated with the harmonic oscillator simply because I miscounted the number of A's and A hats involved. Fortunately for these type of questions it is possible to make many attempts. What I hadn't realised until recently was that even after three attempts one doesn't have to submit the final answer so you are allowed multiple goes for each question and then only after one has got the correct answer first time round do you have to submit. Had I realised that I would have got higher marks than I did for the questions I submitted. Still none of these really count all one has to do is get 40% overall for the assesment. But as there are a total of 10 of them it is worth doing them as thoroughly as one can and all of them. If you only did 5 say you would have to guarantee getting 80% for all of them to pass.

## Sunday, 24 March 2013

### Yet More plans

Well its that time of the year when one contemplates what one will be doing in the future. As anyone involved in this OU lark will know this is a perpetual problem. Matters not being helped by the fact that the OU seems to be engaged in a perpetual revision of their courses and the fee structure changing etc. Anyway my immediate plans after my current OU courses have finished are

Oct 2013 - June 2014 M381 Number theory and logic and the quantum entanglement project

Oct 2014 - June 2015 Third level Philosophy course

This will complete my second Open degree which has concentrated on mathematics and philosophy after I have decided to discount Topology as I only got grade 4

In the meantime I have opened up my third :) Open degree with my current OU course in Music and I intend to do at least another third level course in Music and the new third level pure maths course M303 that is due to start in October 2014 but I will probably do it in October 2015.

After that I shall (finally) embark on the maths MSc at one course per year

However there is lots of other stuff for me to do. I want to get piano lessons at the end of June and embark on the grade exams, Hopefully at least 1 a year and maybe 1 every 6 months after again I finally get started.

Also grade 5 - 8 theory in the next year or two so that I can start the OCA compositon courses

http://www.oca-uk.com/subjects/music.html

with the aim in the next 5 years or so of completing those and hopefully up to grade 5 or 6 piano,

Finally in the long term I haven't given up my hopes of doing a philosophy MA but I think the only reasonable way to do it is via St Davids

http://www.trinitysaintdavid.ac.uk/en/courses/postgraduatecourses/maeuropeanphilosophy/

This means focusing on Continental philosophy rather than my favourite branch Analytic philosophy but this does not look feasible as the fees for part time postgraduate study at Edinburgh University are circa £4000 a year and rising. Whereas my budget is about £2000 a year and St Davids seem to have the price correct.

The easy option would be to do the OU MA in philosophy but the focus of that is social and political not really my thing. With St Davids I would at least get to understand the background to say the relationship between Schopenhauer, Nietszche and Wagner, or the ideas of the Frankfurt school especially Adorno on a critical approach to Music. Then possibly onto a Phd which take me close to seventy. Then this education lark might finally be over. I'm hoping also that the break between June and September will give me time to get back into my big bang calculation with the aim of finishing that by end 2014 Watch this space.

Oct 2013 - June 2014 M381 Number theory and logic and the quantum entanglement project

Oct 2014 - June 2015 Third level Philosophy course

This will complete my second Open degree which has concentrated on mathematics and philosophy after I have decided to discount Topology as I only got grade 4

In the meantime I have opened up my third :) Open degree with my current OU course in Music and I intend to do at least another third level course in Music and the new third level pure maths course M303 that is due to start in October 2014 but I will probably do it in October 2015.

After that I shall (finally) embark on the maths MSc at one course per year

However there is lots of other stuff for me to do. I want to get piano lessons at the end of June and embark on the grade exams, Hopefully at least 1 a year and maybe 1 every 6 months after again I finally get started.

Also grade 5 - 8 theory in the next year or two so that I can start the OCA compositon courses

http://www.oca-uk.com/subjects/music.html

with the aim in the next 5 years or so of completing those and hopefully up to grade 5 or 6 piano,

Finally in the long term I haven't given up my hopes of doing a philosophy MA but I think the only reasonable way to do it is via St Davids

http://www.trinitysaintdavid.ac.uk/en/courses/postgraduatecourses/maeuropeanphilosophy/

This means focusing on Continental philosophy rather than my favourite branch Analytic philosophy but this does not look feasible as the fees for part time postgraduate study at Edinburgh University are circa £4000 a year and rising. Whereas my budget is about £2000 a year and St Davids seem to have the price correct.

The easy option would be to do the OU MA in philosophy but the focus of that is social and political not really my thing. With St Davids I would at least get to understand the background to say the relationship between Schopenhauer, Nietszche and Wagner, or the ideas of the Frankfurt school especially Adorno on a critical approach to Music. Then possibly onto a Phd which take me close to seventy. Then this education lark might finally be over. I'm hoping also that the break between June and September will give me time to get back into my big bang calculation with the aim of finishing that by end 2014 Watch this space.

## Sunday, 10 March 2013

### MST326 Fluids 2nd TMA

Well as promised here is my review of the 2nd Fluids TMA

Block 2 can be described as a crash course in the essentials of Fluid dynamics it covers in 4 short units, Streamlines and path lines. The Euler equation, the equation of continuity, Bernoullis equation and it's application to flows where the diameter changes, Vorticity and flow around shapes such as cylinders and last but not least the Navier Stokes equations which deals with viscous flow. As Feynman said in his lectures treating Fluid flow without consideration of viscosity is like treating the flow of dry fluids.

Anyway the questions covered the following topics

Question 1 We had to calculate the pathlines and streamlines for a two dimensional flow and sketch these for various questions. As the books say Straightforward but tedious especially the sketching. Got most of it out

Question 2. A relatively straightforward but again tedious question involving the solution of Bernoullis equation for the flow of water over a hump. We had to calculate the variation in height along certain points. One slightly interesting aspect of all this is that quite often one has to solve a cubic equation so a potential hint of Galois theory here, But the question usually involves solving the resultant equation by the Newton Raphson method. Unfortunately by the later half of the question I had lost the will to live so about 2/3 of the marks here

Question 3 Calculating the flow and vorticity for flow around a segment of a cylinder for a given stream function the vorticity is the curl of the velocity vector. The meat of the question involved the calculation of the force on the cylinder I got this out but again by the end of the question I had lost the wiil to live so again about 3/4 of the marks

Question 4 Involved the setting up of the Navier Stokes equation and the appropriate boundary conditions for the flow of two layers of liquid between two plates one of which was moving with constant velocity. This question was a test of both the physics and the maths. One had to justify the various approximations. Again the meat of the question involved the solving of the Navier Stokes equations for the problem this was straightforward but tedious especially as the form of the equation required in the final answer involved recasting some of the constants in terms of another constant. This just added to the tedium but I got there in the end. and got most of the marks I think

So overall I think I'll just get enough for a grade 2 pass. There was a lot of tedious curve sketching for this TMA and I think I prefer the mathematical aspects. Next block gets back to the nitty gritty of solving partial differential equations including a brief look at one of my favourite topics Sturm Liouville theory of which I've been having a little discussion on the quantum mechanics forum about.

In general Fluid mechanics is simply a matter of balancing equations and Newton's laws of motion. On the other hand who would have thought some thing quite conceptually straightforward would lead to the complications of something like the Navier Stokes equations

http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

Fluid mechanics is just Conservation of Energy and Newtons second law but you have to know them really well.

As a footnote I got a respectable 72% for my last music TMA.

Thats all folks till next time. Will speak about the first quantum mechanics TMA next.

Block 2 can be described as a crash course in the essentials of Fluid dynamics it covers in 4 short units, Streamlines and path lines. The Euler equation, the equation of continuity, Bernoullis equation and it's application to flows where the diameter changes, Vorticity and flow around shapes such as cylinders and last but not least the Navier Stokes equations which deals with viscous flow. As Feynman said in his lectures treating Fluid flow without consideration of viscosity is like treating the flow of dry fluids.

Anyway the questions covered the following topics

Question 1 We had to calculate the pathlines and streamlines for a two dimensional flow and sketch these for various questions. As the books say Straightforward but tedious especially the sketching. Got most of it out

Question 2. A relatively straightforward but again tedious question involving the solution of Bernoullis equation for the flow of water over a hump. We had to calculate the variation in height along certain points. One slightly interesting aspect of all this is that quite often one has to solve a cubic equation so a potential hint of Galois theory here, But the question usually involves solving the resultant equation by the Newton Raphson method. Unfortunately by the later half of the question I had lost the will to live so about 2/3 of the marks here

Question 3 Calculating the flow and vorticity for flow around a segment of a cylinder for a given stream function the vorticity is the curl of the velocity vector. The meat of the question involved the calculation of the force on the cylinder I got this out but again by the end of the question I had lost the wiil to live so again about 3/4 of the marks

Question 4 Involved the setting up of the Navier Stokes equation and the appropriate boundary conditions for the flow of two layers of liquid between two plates one of which was moving with constant velocity. This question was a test of both the physics and the maths. One had to justify the various approximations. Again the meat of the question involved the solving of the Navier Stokes equations for the problem this was straightforward but tedious especially as the form of the equation required in the final answer involved recasting some of the constants in terms of another constant. This just added to the tedium but I got there in the end. and got most of the marks I think

So overall I think I'll just get enough for a grade 2 pass. There was a lot of tedious curve sketching for this TMA and I think I prefer the mathematical aspects. Next block gets back to the nitty gritty of solving partial differential equations including a brief look at one of my favourite topics Sturm Liouville theory of which I've been having a little discussion on the quantum mechanics forum about.

In general Fluid mechanics is simply a matter of balancing equations and Newton's laws of motion. On the other hand who would have thought some thing quite conceptually straightforward would lead to the complications of something like the Navier Stokes equations

http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

Fluid mechanics is just Conservation of Energy and Newtons second law but you have to know them really well.

As a footnote I got a respectable 72% for my last music TMA.

Thats all folks till next time. Will speak about the first quantum mechanics TMA next.

## Sunday, 24 February 2013

### Odds and Sods

This is more of a general catch up post rather than anything specific.

Current Status of A224

I've just finished my 4th TMA for my music course. Not much to say really it involved analysing an Allegro for String Quartet and a Scherzo and Trio for piano. Musical analysis in many ways is a bit like going through a mathematical proof. The main thing to look for are the Key changes and how they are implemented for simple pieces this is quite straightforward as one simply moves from the Tonic Key to the Dominant in the first half and then back again. Or if it is a minor key one usually ends up in the relative major. For more complicated pieces where there is a transition passage exploring many keys the key changes almost on a bar by bar basis and it can be quite tricky to work out what is going on. I think whilst I got the major key changes I fudged the transition keys. Other things to look out for are contrasts in the melody and texture of the piece. Unlike TMA02 at least we had the scores and so didn't have to rely on purely aural effects. So this is in hand but can be quite time consuming.

Piano Playing

Well last January a colleague of mine at work has lent me his sons Yamaha Clavinova and I've got used to playing it. It's important to get something which feels like a real piano. Keyboards just don't hack it. I can play more or less all the scales for Grades 1 and 2 and the Arpeggios and broken chords. I wish I could say the same for the pieces though I'm struggling with the Grade 1 book of pieces although as I practice more I'm stumbling less I guess it's just a matter of time really. My plan is after June when the music course has finished to start getting lessons and hopefully be fit to do at least grade 1 in November maybe even grade 1 and 2. We'll see. My practice routine most days is to go through the scales in the morning before work and play the pieces a couple of times. Then in the evening if I have time spend 15 minutes per piece. I think my practice is skewed more to scales but as they form the building blocks It's best to concentrate on those first.

Fluids I really need to spend the next week polishing off the second TMA more next week

Quantum Mechanics I have done the first two assignments ICMA's as they are called, I got above 80% for both so decided to leave it at that. They don't really count I need to do the first proper TMA by the end of March so have plenty of time. On a related note I just invested in what would be the first text book account of the Many Worlds interpretation that isn't just popular science

http://www.amazon.co.uk/The-Emergent-Multiverse-according-Interpretation/dp/0199546967/ref=sr_1_1?ie=UTF8&qid=1361728117&sr=8-1

Not that I'm convinced but it's useful to have as a definitive account so one knows what one is arguing about. I have written the first review of it on Amazon.

I also took time out to see Lincoln and Master Django. Lincoln is an absolutely brilliant film and I hope it gets the Oscars. I hadn't realised that after the Emancipation proclamation Lincoln struggled to get the bill through the house of representatives. Sounds as if things haven't changed much Obama seems to have similar problems. The film shows just how much compromise and dirty deals are needed to get any progressive legislation through. The best part as far as I was concerned was that of Thaddeus Stephens played by Timothy Lee Jones. I hope he gets a best supporting actor role. I've ordered the book on which the film was based apparently it was one of President Obama's favourite books

http://www.amazon.co.uk/The-Emergent-Multiverse-according-Interpretation/dp/0199546967/ref=sr_1_1?ie=UTF8&qid=1361728117&sr=8-1

Also a book I read about 15 years ago on the American Civil War by McPherson

http://www.amazon.co.uk/Battle-Cry-Freedom-Penguin-history/dp/0140125183/ref=sr_1_1?s=books&ie=UTF8&qid=1361731662&sr=1-1

Master Django Unchained is also in it's way a good film quite a contrast to Lincoln really brutal in parts and doesn't flinch about the racism of the Southern States. Tarantino injects some humour especially in a scene where the Ku Klux Clan start complaining that they can't see through the badly cut holes in their pillow sheets. Whilst Lincoln is more my type of film this is still definitely worth watching if only to bring home just how brutally slaves were treated in the South during that period. Again the supporting role played by Christopher Waltz (the baddie from Inglorious basterds) was very good.

So that about wraps it up for this time Next post will be on the second TMA for Fluids.

Current Status of A224

I've just finished my 4th TMA for my music course. Not much to say really it involved analysing an Allegro for String Quartet and a Scherzo and Trio for piano. Musical analysis in many ways is a bit like going through a mathematical proof. The main thing to look for are the Key changes and how they are implemented for simple pieces this is quite straightforward as one simply moves from the Tonic Key to the Dominant in the first half and then back again. Or if it is a minor key one usually ends up in the relative major. For more complicated pieces where there is a transition passage exploring many keys the key changes almost on a bar by bar basis and it can be quite tricky to work out what is going on. I think whilst I got the major key changes I fudged the transition keys. Other things to look out for are contrasts in the melody and texture of the piece. Unlike TMA02 at least we had the scores and so didn't have to rely on purely aural effects. So this is in hand but can be quite time consuming.

Piano Playing

Well last January a colleague of mine at work has lent me his sons Yamaha Clavinova and I've got used to playing it. It's important to get something which feels like a real piano. Keyboards just don't hack it. I can play more or less all the scales for Grades 1 and 2 and the Arpeggios and broken chords. I wish I could say the same for the pieces though I'm struggling with the Grade 1 book of pieces although as I practice more I'm stumbling less I guess it's just a matter of time really. My plan is after June when the music course has finished to start getting lessons and hopefully be fit to do at least grade 1 in November maybe even grade 1 and 2. We'll see. My practice routine most days is to go through the scales in the morning before work and play the pieces a couple of times. Then in the evening if I have time spend 15 minutes per piece. I think my practice is skewed more to scales but as they form the building blocks It's best to concentrate on those first.

Fluids I really need to spend the next week polishing off the second TMA more next week

Quantum Mechanics I have done the first two assignments ICMA's as they are called, I got above 80% for both so decided to leave it at that. They don't really count I need to do the first proper TMA by the end of March so have plenty of time. On a related note I just invested in what would be the first text book account of the Many Worlds interpretation that isn't just popular science

http://www.amazon.co.uk/The-Emergent-Multiverse-according-Interpretation/dp/0199546967/ref=sr_1_1?ie=UTF8&qid=1361728117&sr=8-1

Not that I'm convinced but it's useful to have as a definitive account so one knows what one is arguing about. I have written the first review of it on Amazon.

I also took time out to see Lincoln and Master Django. Lincoln is an absolutely brilliant film and I hope it gets the Oscars. I hadn't realised that after the Emancipation proclamation Lincoln struggled to get the bill through the house of representatives. Sounds as if things haven't changed much Obama seems to have similar problems. The film shows just how much compromise and dirty deals are needed to get any progressive legislation through. The best part as far as I was concerned was that of Thaddeus Stephens played by Timothy Lee Jones. I hope he gets a best supporting actor role. I've ordered the book on which the film was based apparently it was one of President Obama's favourite books

http://www.amazon.co.uk/The-Emergent-Multiverse-according-Interpretation/dp/0199546967/ref=sr_1_1?ie=UTF8&qid=1361728117&sr=8-1

Also a book I read about 15 years ago on the American Civil War by McPherson

http://www.amazon.co.uk/Battle-Cry-Freedom-Penguin-history/dp/0140125183/ref=sr_1_1?s=books&ie=UTF8&qid=1361731662&sr=1-1

Master Django Unchained is also in it's way a good film quite a contrast to Lincoln really brutal in parts and doesn't flinch about the racism of the Southern States. Tarantino injects some humour especially in a scene where the Ku Klux Clan start complaining that they can't see through the badly cut holes in their pillow sheets. Whilst Lincoln is more my type of film this is still definitely worth watching if only to bring home just how brutally slaves were treated in the South during that period. Again the supporting role played by Christopher Waltz (the baddie from Inglorious basterds) was very good.

So that about wraps it up for this time Next post will be on the second TMA for Fluids.

## Saturday, 9 February 2013

### SM358 Quantum Mechanics First Impressions

I have started SM358 and here is a quick first impression of the course it seems to cover the basics pretty well there are three units

1) Wave Mechanics, this sets the background to the development of Schrodinger's equation and solves it for simple problems such as the square well potential, transmission and reflection at a barrier and the simple harmonic oscillator which it solves by annhilation and creation operators as well as giving a brief overview of the solution by series using Hermite's polynomials. As this is a physics course, not one addressed to mathematicians, then it does tend to skip over some mathematical details. Certainly one is not likley to be asked to solve Schrodinger's equation in parabolic coordinates or solve many complicated problems. For that one would need a course based on a book such as Landau and Lifshitz volume 3

http://www.amazon.co.uk/Quantum-Mechanics-non-relativistic-theory-Theoretical/dp/0750635398/ref=sr_1_1?ie=UTF8&qid=1360429881&sr=8-1

If I get time as a tour de-force I would love to work through the solution of Schrodinger's equation for parabolic coordinates for both the bound state problem and the scattering problem which for the coulomb potential is exact even if it does involve functions which go by the ridiculous name of the Confluent Hypergeometric Function.

Still fair enough for the type of course this is meant to be. On a slightly down side it does push the standard line that particles in set ups as the two slit experiment must be travelling down two slits at once and treats wave function collapse as a physical process (although as is usual the mechanism is never given). No mention is made of the epistemic view of the wave function that the wave function is a mathematical representation of the probabilities associated with a quantum system and that when an event occurs one of the possibilites is realised thats all.

2) Quantum Mechanics and It's interpretation. This gives a good overview of the Dirac Formalism and it's application to angular momentum and spin. However it stops short of Clebsch Gordan coefficients which describe the coupling of many particles.

http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients

It then goes on to discuss the violation of the Bell inequalities and how they rule out local hidden variable theories. However they don't seem to accept that there are interpretations such as Bohm's theory which get round the problem by invoking Non-local Hidden variables or even the many worlds theory I would have thought some mention of these alternative views might have been mentioned. Conversely the text seems to want to have it's cake and eat it. Having dismissed the hidden variable point of view it still maintains that non-locality is an essential feature of quantum mechanics but the alternatives are Non local hidden variable theories (or realist theories) or local non hidden variable theories or non realist theories.

To expand slightly for pairs of properties subject to the uncertainty principle such as spin components or postion and momentum it is impossible to assign definite values to each pair of values. Thus these are undetermined or cannot be measured simultaneously. This introduces a degree of Non realism into the debate and those who don't subscribe to Hidden variable theories claim that the uncertainty principle is fundamental and precludes any hidden variable for these pairs of propeties. However it must be stressed that not all properties of quantum systems are subject to the uncertainty principle these include charge mass and intrinsic spin. So rejection of Hidden variables or realism vis a vis pairs of properties subject to the uncertainty principle is not the same as rejection of realism per se.

Bell showed that the combination of locality and Hidden variables vis a vis pairs of properties subject to the uncertainty principle was in contradiction with the predictions of quantum mechanics but if you reject Hidden variables as the course text does you are not commited to non locality it is perfectly acceptable to have non realistic local theories. However the course text speaks about signals between two particles travelling faster than the speed of light. Oh really what is the mechanism for this. Anyway despite this caveat it gives a reasonable over-view of the mathematics involved. It then goes on in the final chapter to discuss applications of entanglement to cryptography and so called tele-portation.

The final book Quantum mechanics of matter, discusses some of the standard applications of quantum mechanics to the hydrogen atom, molecule, solid state and lasers. Again the maths is a bit sketchy enough is given for those dedicated to fill in the gaps but the emphasis seems to be on using the wave functions to predict various quantites rather than the derivations of the wave functions themselves. Given the limitations this is a reasonable overview, but the subjects touched are far deeper and this can only provide an brief sketch of the applications of quantum mechanics.

In general this promises to be quite a good course, however it is a shame that this is the only course on quantum physics that the OU offers. There should be follow up courses on the applications of Group theory to quantum mechanics, A course on statistical physics, A course on Solid state physics, a course on quantum optics, quantum information theory and computing, quantum scattering theory and ideally a course on particle and nuclear physics. I doubt this will ever happen and so people who do this course will be left wanting more. Unfortunately there seems to be a dire lack of on line courses offering this so one is left to ones own devices.

As a final point the method of assessment for physics courses seems different to that for maths courses in that there are 6 so called ICMAs (Interactive Computer Marked assignments) which you can have as many attempts as you like and 4 TMA's But none of these count towards your final assessment provided you get an average of 30% or more. This means that far more emphasis is placed on the exam so I need to refine my exam technique in order to be sure of a good grade.

As a footnote I managed to catch a lecture given by the ubiquitous Brian Cox on quantum mechanics given to celebreties. On the whole quite a good overview, but one thing slightly puzzled me he made the claim that the Pauli Exclusion principle applied to all electrons in the universe so that when one electron changed it's energy all the other electrons in the universe did so as well. I find this a bit bizarre surely one can speak of isolated systems so that the energy levels of a hydrogen atom in Manchester will be the same as the energy levels of a hydrogen atom in Edinburgh. If they were not then how would spectroscopy work. So I confess to not understanding the claim if anyone knows of a reference then I would be grateful

1) Wave Mechanics, this sets the background to the development of Schrodinger's equation and solves it for simple problems such as the square well potential, transmission and reflection at a barrier and the simple harmonic oscillator which it solves by annhilation and creation operators as well as giving a brief overview of the solution by series using Hermite's polynomials. As this is a physics course, not one addressed to mathematicians, then it does tend to skip over some mathematical details. Certainly one is not likley to be asked to solve Schrodinger's equation in parabolic coordinates or solve many complicated problems. For that one would need a course based on a book such as Landau and Lifshitz volume 3

http://www.amazon.co.uk/Quantum-Mechanics-non-relativistic-theory-Theoretical/dp/0750635398/ref=sr_1_1?ie=UTF8&qid=1360429881&sr=8-1

If I get time as a tour de-force I would love to work through the solution of Schrodinger's equation for parabolic coordinates for both the bound state problem and the scattering problem which for the coulomb potential is exact even if it does involve functions which go by the ridiculous name of the Confluent Hypergeometric Function.

Still fair enough for the type of course this is meant to be. On a slightly down side it does push the standard line that particles in set ups as the two slit experiment must be travelling down two slits at once and treats wave function collapse as a physical process (although as is usual the mechanism is never given). No mention is made of the epistemic view of the wave function that the wave function is a mathematical representation of the probabilities associated with a quantum system and that when an event occurs one of the possibilites is realised thats all.

2) Quantum Mechanics and It's interpretation. This gives a good overview of the Dirac Formalism and it's application to angular momentum and spin. However it stops short of Clebsch Gordan coefficients which describe the coupling of many particles.

http://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients

It then goes on to discuss the violation of the Bell inequalities and how they rule out local hidden variable theories. However they don't seem to accept that there are interpretations such as Bohm's theory which get round the problem by invoking Non-local Hidden variables or even the many worlds theory I would have thought some mention of these alternative views might have been mentioned. Conversely the text seems to want to have it's cake and eat it. Having dismissed the hidden variable point of view it still maintains that non-locality is an essential feature of quantum mechanics but the alternatives are Non local hidden variable theories (or realist theories) or local non hidden variable theories or non realist theories.

To expand slightly for pairs of properties subject to the uncertainty principle such as spin components or postion and momentum it is impossible to assign definite values to each pair of values. Thus these are undetermined or cannot be measured simultaneously. This introduces a degree of Non realism into the debate and those who don't subscribe to Hidden variable theories claim that the uncertainty principle is fundamental and precludes any hidden variable for these pairs of propeties. However it must be stressed that not all properties of quantum systems are subject to the uncertainty principle these include charge mass and intrinsic spin. So rejection of Hidden variables or realism vis a vis pairs of properties subject to the uncertainty principle is not the same as rejection of realism per se.

Bell showed that the combination of locality and Hidden variables vis a vis pairs of properties subject to the uncertainty principle was in contradiction with the predictions of quantum mechanics but if you reject Hidden variables as the course text does you are not commited to non locality it is perfectly acceptable to have non realistic local theories. However the course text speaks about signals between two particles travelling faster than the speed of light. Oh really what is the mechanism for this. Anyway despite this caveat it gives a reasonable over-view of the mathematics involved. It then goes on in the final chapter to discuss applications of entanglement to cryptography and so called tele-portation.

The final book Quantum mechanics of matter, discusses some of the standard applications of quantum mechanics to the hydrogen atom, molecule, solid state and lasers. Again the maths is a bit sketchy enough is given for those dedicated to fill in the gaps but the emphasis seems to be on using the wave functions to predict various quantites rather than the derivations of the wave functions themselves. Given the limitations this is a reasonable overview, but the subjects touched are far deeper and this can only provide an brief sketch of the applications of quantum mechanics.

In general this promises to be quite a good course, however it is a shame that this is the only course on quantum physics that the OU offers. There should be follow up courses on the applications of Group theory to quantum mechanics, A course on statistical physics, A course on Solid state physics, a course on quantum optics, quantum information theory and computing, quantum scattering theory and ideally a course on particle and nuclear physics. I doubt this will ever happen and so people who do this course will be left wanting more. Unfortunately there seems to be a dire lack of on line courses offering this so one is left to ones own devices.

As a final point the method of assessment for physics courses seems different to that for maths courses in that there are 6 so called ICMAs (Interactive Computer Marked assignments) which you can have as many attempts as you like and 4 TMA's But none of these count towards your final assessment provided you get an average of 30% or more. This means that far more emphasis is placed on the exam so I need to refine my exam technique in order to be sure of a good grade.

As a footnote I managed to catch a lecture given by the ubiquitous Brian Cox on quantum mechanics given to celebreties. On the whole quite a good overview, but one thing slightly puzzled me he made the claim that the Pauli Exclusion principle applied to all electrons in the universe so that when one electron changed it's energy all the other electrons in the universe did so as well. I find this a bit bizarre surely one can speak of isolated systems so that the energy levels of a hydrogen atom in Manchester will be the same as the energy levels of a hydrogen atom in Edinburgh. If they were not then how would spectroscopy work. So I confess to not understanding the claim if anyone knows of a reference then I would be grateful

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