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.
Saturday, 20 July 2013
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.
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