Well something quite remarkable has happened to me over the past week. I was in Leith Cash exchange where I saw a keyboard with stand for only £30. I snapped this up immediately and dug out some of my old piano tutorial books which I had previously bought when I started my OU music course over ten years ago but never really took it seriously. The main series I'm using are Fanny Waterman's Books,
http://www.amazon.co.uk/Piano-Lessons-Waterman-Harewood-Series/dp/0571500242
and I've just started lesson 3 of book 1. I would hope to do one lesson a week if not more and then practice the admittedly limited repertoire before going on to better things. The temptation is to rush these things so I have to curb my impatience to say the least.
I've been getting up in the mornings and doing up to 1/2 an hours practice before going to work, and then about an hour in two sessions when I get home from work. Out of interest I also bought the Grade 1 piano pieces and scales and am slowly beginning to get to grips with the scale of C major and the broken chords, which I find harder to do than the scales, especially going upwards. It certainly helps I think to have a good grasp of the theory of music for example in broken chords it helps to realise that one is playing the root, the first inversion and the second inversion of the chords I'm not sure what the difference between a broken chord and an Arpeggio is
If I continue I will try and invest in a proper digital piano something like a Broadway
http://www.ukpianos.co.uk/broadway-ez101.html
which you can get for about £500 before investing in something like a Yamaha Clavinova
http://www.ukpianos.co.uk/yamaha-clp430.html
But the Broadway is definitely going to be my Christmas Present
In the new year I will get some lessons with the aim of doing at least grade 1 by the middle of next year and then take it from there. Ideally up to grade 8 in 4-5 years time. An average of six months per grade with more time once one gets past grade 5. We'll see.
Needless to say if this direction continues, I would have to curtail my mathematical and philosophical interests apart from general reading and getting back to my physics calculations, which have been neglected due to the OU maths courses getting in the way. Piano lessons are not cheap £25 a time and at an average of 1 a week that would come to £1250 a year. As my budget for education is £2000 a year one can see that would only leave room for 1 60 point OU course /per year but as the composition course also costs about £1000 a year then things would be quite expensive. I will continue to do the other two maths OU courses along with the OU music course I've booked for, but I can see the piano/keyboard and compostion taking up a large element of my time over the next 4-5 years and after June of next year I can see me not doing any OU courses for a while. Also one is talking about 20 hours per week on composition and piano practice, so it wouldn't leave much time for other things anyway.
Incidentally the new grade 1 syllabuses for 2013 - 2014 contain a really interesting piece for a beginner. A fugue in A minor by Alec Rowley who produced a set of 5 minature preludes and fugues for beginners I hope to get reasonably competent in these pieces by Christmas
http://www.amazon.co.uk/Five-Miniature-Preludes-Fugues-Rowley/dp/0711928096/ref=sr_1_2?s=books&ie=UTF8&qid=1343483502&sr=1-2
Saturday, 28 July 2012
Tuesday, 17 July 2012
Current Plans
Well despite the debacle of the third TMA for MS324 I'm pressing ahead with my plans. I decided to register for the new OU music course in preparation for my serious attempt at composition via the Open college of Arts which I hope to start in June/July 2013 (depending on Funds)
As far as maths is concerned I'll be doing M381 (Number theory and logic) and MST326 (Fluids and mathematical methods). That will then finish the maths at undergraduate level in preparation for the MSc. I will complete my second open degree in October 2013 by doing the OU third level philosophy course AA308 in its last incarnation. That will set me up for an MA in philosophy by distance learning via St Davids University
http://www.trinitysaintdavid.ac.uk/en/courses/postgraduatecourses/maeuropeanphilosophy/
Ok so the focus is Continental Philosophy and not analytic philosophy but I'm not aiming to become an academic philosopher and I feel that analytic philosophy suffers to some extent from 'Science Envy' in that it is trying to do science or mathematics without actually engaging with the subject. If I want to learn about maths or science I'll do maths and science and not philosophise about it. I'm tempted to paraphrase Bernard Shaw and say that those who can do maths and science do it those who cant philosophise about it. Yes it's important to understand why some of the problems associated with the interpretation of quantum mechanics cant be resolved, but having done that why would anyone think they have a magic key which is going to solve all the problems that other people have missed,
On the other hand Continental philosophy engages with real issues. Whilst not many people have heard of Michael Dummett, or Quine, plenty have heard of Marx, Nietszche, Schopenhauer, Sartre Foucault etc.
Also to some extent one has to make use of the opportunities available. Were I to go down the route of doing say the London external BA there would be no follow up available whereas St Davids university offers at least an MA pathway by distance learning and the opportunities to go onto a PhD.
Were London University to abandon their snobbish attitude to distance learning at Postgraduate level and open up their MA to distance learners then I might be tempted. However as that is not possible at present I'll stick with St David's.
So next year is about finishing my undergraduate maths courses, doing the new OU music course in preparation for embarking on serious composition. The years after will hopefully see me complete my compostion training, my MA in European philosophy and most if not all of the MSc in maths. Hopefully all this can be achieved in 5 years time when I'll be 60.
As far as maths is concerned I'll be doing M381 (Number theory and logic) and MST326 (Fluids and mathematical methods). That will then finish the maths at undergraduate level in preparation for the MSc. I will complete my second open degree in October 2013 by doing the OU third level philosophy course AA308 in its last incarnation. That will set me up for an MA in philosophy by distance learning via St Davids University
http://www.trinitysaintdavid.ac.uk/en/courses/postgraduatecourses/maeuropeanphilosophy/
Ok so the focus is Continental Philosophy and not analytic philosophy but I'm not aiming to become an academic philosopher and I feel that analytic philosophy suffers to some extent from 'Science Envy' in that it is trying to do science or mathematics without actually engaging with the subject. If I want to learn about maths or science I'll do maths and science and not philosophise about it. I'm tempted to paraphrase Bernard Shaw and say that those who can do maths and science do it those who cant philosophise about it. Yes it's important to understand why some of the problems associated with the interpretation of quantum mechanics cant be resolved, but having done that why would anyone think they have a magic key which is going to solve all the problems that other people have missed,
On the other hand Continental philosophy engages with real issues. Whilst not many people have heard of Michael Dummett, or Quine, plenty have heard of Marx, Nietszche, Schopenhauer, Sartre Foucault etc.
Also to some extent one has to make use of the opportunities available. Were I to go down the route of doing say the London external BA there would be no follow up available whereas St Davids university offers at least an MA pathway by distance learning and the opportunities to go onto a PhD.
Were London University to abandon their snobbish attitude to distance learning at Postgraduate level and open up their MA to distance learners then I might be tempted. However as that is not possible at present I'll stick with St David's.
So next year is about finishing my undergraduate maths courses, doing the new OU music course in preparation for embarking on serious composition. The years after will hopefully see me complete my compostion training, my MA in European philosophy and most if not all of the MSc in maths. Hopefully all this can be achieved in 5 years time when I'll be 60.
MS324 TMA03
First sorry for not blogging for a month or so been a bit bogged down with MS324 block 2 which is actually quite interesting but unfortunately the TMA does not reflect this.
The main topics are a basic overview of probability, and random walks in sections 1 and 2, An account of the diffusion equation as applied to heat problems and the most interesting part which is not assessed namely the link between microscopic diffusion, random walks and macroscopic diffusion.
The TMA is as they say in the books straightforward but tedious
Question 1 is a problem based on successive tyre failures of a cyclist where the probability distribution is an exponential one. To solve the question one has to use integration by parts a couple of times
Question 2 is a problem calculating the statistics associated with a random process defined by a recurrence relation
Question 3 concerns heat conduction in a Nuclear core, the diffusion equation reduces to a 1 dimensional form and is relatively easy to solve. Still must confess I couldn't see how to do the last part
Question 4 is a question concerning temperature waves in the earths surface again quite a straightforward question.
Overall then the TMA is quite straightforward but as there are a number of numerical calculations rather tedious. There does seem to be a disconnect between the TMA questions and the course content.
However as I was struggling to motivate myself I decided to cut my losses with only about 3/4 of the TMA done. I provided more or less complete answers to questions 1 and 4 just did the first two parts of question 2 and all but the last part of question three. A bit pathetic I realise but sometimes it's just better to move on.
It would have been more interesting had they asked us to solve the diffusion equation in three dimensions for say a cube or sphere say with the top half heated at one temperature and the other one at a different temperature. For a sphere this would involve setting up the equation in Spherical coordinates separating the variables and solving the resulting differential equations by series resulting in Spherical Harmonics and Legendre polynomials all stuff which should form the core of a third level mathematics course in mathematical methods but is hardly mentioned in this course.
Still as it hasn't then I'll just have to rely on the example sheets from Cambridge to fill the gaps.
http://www.damtp.cam.ac.uk/user/examples/B8b.pdf
Hopefully Block 3 on the calculus of variations and Lagranges equations will be a bit more exciting
The main topics are a basic overview of probability, and random walks in sections 1 and 2, An account of the diffusion equation as applied to heat problems and the most interesting part which is not assessed namely the link between microscopic diffusion, random walks and macroscopic diffusion.
The TMA is as they say in the books straightforward but tedious
Question 1 is a problem based on successive tyre failures of a cyclist where the probability distribution is an exponential one. To solve the question one has to use integration by parts a couple of times
Question 2 is a problem calculating the statistics associated with a random process defined by a recurrence relation
Question 3 concerns heat conduction in a Nuclear core, the diffusion equation reduces to a 1 dimensional form and is relatively easy to solve. Still must confess I couldn't see how to do the last part
Question 4 is a question concerning temperature waves in the earths surface again quite a straightforward question.
Overall then the TMA is quite straightforward but as there are a number of numerical calculations rather tedious. There does seem to be a disconnect between the TMA questions and the course content.
However as I was struggling to motivate myself I decided to cut my losses with only about 3/4 of the TMA done. I provided more or less complete answers to questions 1 and 4 just did the first two parts of question 2 and all but the last part of question three. A bit pathetic I realise but sometimes it's just better to move on.
It would have been more interesting had they asked us to solve the diffusion equation in three dimensions for say a cube or sphere say with the top half heated at one temperature and the other one at a different temperature. For a sphere this would involve setting up the equation in Spherical coordinates separating the variables and solving the resulting differential equations by series resulting in Spherical Harmonics and Legendre polynomials all stuff which should form the core of a third level mathematics course in mathematical methods but is hardly mentioned in this course.
Still as it hasn't then I'll just have to rely on the example sheets from Cambridge to fill the gaps.
http://www.damtp.cam.ac.uk/user/examples/B8b.pdf
Hopefully Block 3 on the calculus of variations and Lagranges equations will be a bit more exciting
Sunday, 17 June 2012
M338 TMA03
Finished this today had to ask for a small extension due to work, but don't know if it's been granted as my tutor hasn't replied to my request. Anyway I'll submit it tomorrow and see what happens.
This block was a bit more straightforward than Block A as it concerns what most people consider topology to be namely the invariance of certain features of a solid under a transformation. Thus the usual example given is that that a cup with a handle is topologically equivalent to a ring doughnut but not a solid doughnut as they both have a hole and can be deformed into each other. There is also a discussion of the Mobius band.
I have to admit that my heart was sinking when I first started this block as it seemed to be all about visualisation and introducing boundaries in order to classify various combinations of holes and twists in a surface. As there seemed to be no systematic way of doing this it was all a bit vague.
However by the second half of block 2 an algebraic method of charactersing surfaces was introduced and I was much happier. There are a number of ways of systematically obtaining the characteristic properties of a solid and these are essentially algebraic if a bit fiddly. The TMA was mostly concerned with the algebraic aspects of the block. There are three main characteristics of a surface
The Euler Characteristic = V - E +F where V is the number of vertices E = the number of edges and F = is the number of faces. This is invariant for topologically equivalent surfaces.
The Boundary number which is the number of separarate pieces forming the boundary of a surface
The orientability number of a surface essentially the number of twists.
Associated with each surface is an N sided polygon with arrows along each side denoting the orientation with a number of edges of the polygon having the same name. An edge expression is obtained by labelling the sides in order for a given direction of the arrow and labelling the side by its inverse if the arrow is pointing in an opposite direction. Various identifications give rise to different solids. Given such an edge expression one can then go onto calculate the characteristic numbers of a surface. This process is quite straightforward but can be quite fiddly. Also the edge equations can be reduced to canonical form which enables the classification of the surface to be made easily both directly from the edge expression and also there is a method of classifying the solids in terms of the connected sums of constituent surfaces eg Torus's Closed disk's etc.
The final block concerned a discussion of the coloring theorems. One of the key quantities is the chromaticiy number which is related to the number of handles of a given surface and gives the minimum number of colours required to colour the parts of the surface so that no regions next to each other have the same colour. As some readers will probably know the minimum number of colours to colour a map is 4. But this wasn't actually proven until the 1970's and only by a computer so it's doubtful to old fashioned mathematicians whether it counts as a proof at all.
http://en.wikipedia.org/wiki/Four_color_theorem
Question 1 concerned identifying the edge expression of a hexagon, obtaining the edge expressions and the
characteristic numbers. This question seemed quite straightforward
Question 2 was a question on the subdivisions of a surface for a given Euler characteristic this is quite straightforward
Question 3 The bulk of the TMA with a whopping 40% of the marks concerned obtaining a single edge expression from a number of constituent ones, then deducing the characteristic numbers. Then peforming a canonical transformation and obtained the connected sum form for the surface and then using that to deduce the characteristic numbers. Fortunately the numbers seemed to tie up so I'm consisitent if not correct.
Question 4 The final question involved obtaining some inequalities for the number of handles h in terms of the chromaticity number. This seemed quite straigthforward or at least the first part. However the last part which asked us to find the range of h for a large chromaticity numbers. On the face of it this seemed quite straightforward but the sting in the tail was that dreaded phrase "justifying your answers fully" as there were 10 marks for this part. I'm sure we were supposed to do more than solve the inequality for the various values of h. But I couldn't see what.
So on the whole I think I've done reasonably well assuming my tutor accepts my late submission but that last part has me worried that I've missed something quite fundamental.
This block was a bit more straightforward than Block A as it concerns what most people consider topology to be namely the invariance of certain features of a solid under a transformation. Thus the usual example given is that that a cup with a handle is topologically equivalent to a ring doughnut but not a solid doughnut as they both have a hole and can be deformed into each other. There is also a discussion of the Mobius band.
I have to admit that my heart was sinking when I first started this block as it seemed to be all about visualisation and introducing boundaries in order to classify various combinations of holes and twists in a surface. As there seemed to be no systematic way of doing this it was all a bit vague.
However by the second half of block 2 an algebraic method of charactersing surfaces was introduced and I was much happier. There are a number of ways of systematically obtaining the characteristic properties of a solid and these are essentially algebraic if a bit fiddly. The TMA was mostly concerned with the algebraic aspects of the block. There are three main characteristics of a surface
The Euler Characteristic = V - E +F where V is the number of vertices E = the number of edges and F = is the number of faces. This is invariant for topologically equivalent surfaces.
The Boundary number which is the number of separarate pieces forming the boundary of a surface
The orientability number of a surface essentially the number of twists.
Associated with each surface is an N sided polygon with arrows along each side denoting the orientation with a number of edges of the polygon having the same name. An edge expression is obtained by labelling the sides in order for a given direction of the arrow and labelling the side by its inverse if the arrow is pointing in an opposite direction. Various identifications give rise to different solids. Given such an edge expression one can then go onto calculate the characteristic numbers of a surface. This process is quite straightforward but can be quite fiddly. Also the edge equations can be reduced to canonical form which enables the classification of the surface to be made easily both directly from the edge expression and also there is a method of classifying the solids in terms of the connected sums of constituent surfaces eg Torus's Closed disk's etc.
The final block concerned a discussion of the coloring theorems. One of the key quantities is the chromaticiy number which is related to the number of handles of a given surface and gives the minimum number of colours required to colour the parts of the surface so that no regions next to each other have the same colour. As some readers will probably know the minimum number of colours to colour a map is 4. But this wasn't actually proven until the 1970's and only by a computer so it's doubtful to old fashioned mathematicians whether it counts as a proof at all.
http://en.wikipedia.org/wiki/Four_color_theorem
Question 1 concerned identifying the edge expression of a hexagon, obtaining the edge expressions and the
characteristic numbers. This question seemed quite straightforward
Question 2 was a question on the subdivisions of a surface for a given Euler characteristic this is quite straightforward
Question 3 The bulk of the TMA with a whopping 40% of the marks concerned obtaining a single edge expression from a number of constituent ones, then deducing the characteristic numbers. Then peforming a canonical transformation and obtained the connected sum form for the surface and then using that to deduce the characteristic numbers. Fortunately the numbers seemed to tie up so I'm consisitent if not correct.
Question 4 The final question involved obtaining some inequalities for the number of handles h in terms of the chromaticity number. This seemed quite straigthforward or at least the first part. However the last part which asked us to find the range of h for a large chromaticity numbers. On the face of it this seemed quite straightforward but the sting in the tail was that dreaded phrase "justifying your answers fully" as there were 10 marks for this part. I'm sure we were supposed to do more than solve the inequality for the various values of h. But I couldn't see what.
So on the whole I think I've done reasonably well assuming my tutor accepts my late submission but that last part has me worried that I've missed something quite fundamental.
Sunday, 3 June 2012
Pilot Waves for and against part 2
So as promised here is a basic summary of the pros and cons of De-Broglies Bohm's Pilot Wave approach to quantum mechanics. A really good summary of which can be found in these lectures by Mike Towler.
http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html
So here are the pros
i) Contrary to the claim by the Copenhagen interpretation it has been shown that it is possible to define a definite trajectory for particles it's greatest success must be it's explanation of the two slit experiment. Sanity seems to be restored we no longer have to claim that a massive particle such as an electron or a bucky ball splits in two as it interacts with a screen which has a number of slits.
ii) It brings the physics back to quantum mechanics, instead of losing contact with the world of particles and their interactions as the Dirac Formalism is apt to, it does provide a causal explanation for many quantum phenomenon. Mind you the Dirac formalism is really elegant and still encapsulates the essence of many quantum mechanical problems even if it is all quite abstract.
iii) It avoids the problem of measurement, with all the inherent problems of the world around us being created by an act of measurement.
iv) It now deserves a place as an equivalent mathematical formulation of quantum mechanics (at least for Non relativistic problems). The fact that more and more papers are being published using Bohmian mechanics has to be something. I think it's fair to say that other alternative interpretations such as the Many Worlds Interpretation have not yet reached the same degree of maturity. I would argue that once a theory of physics has reached the stage where people can 'Shut up and calculate' as the Pilot Wave theory has done then it has a right to be treated as mainstream physics.
It is quite astonishing that at Solvay in 1927, when the basics of the theory was laid out by De-Broglie, that it wasn't considered at least as an alternative equivalent mathematical reformulation of quantum mechanics just as Heisenberg's matrix mechanics was. All the formulations of quantum mechanics had their problems of interpretation and it is at least arguable that the Pilot Wave theory has a well defined procedure for relating it's mathematical formalism to empirical results.
Now for the cons
i) It makes great play, that for it the wavefunction is a real field, something akin to an electric field or gravitaional one. Unfortunately this implies acceptance of the reality of 3N+1 dimensional configuration space, where N is the number of particles considered and 1 represents time. Let me explain a bit more the wave function of a manybody particle system is a function of all the coordinates of the particles considered eg for two particles with positions r1 and r2 the time independent of the wave function of the system is now a function of r1 and r2 that is we have $$\psi(x1,y1,z1,x2,y2,z2)$$ this is quite different from classical physics, for example the electric field produced by two charges, at a given point, is still a function of 3 dimensional space and not 6 dimensional space. The question then, for those who would see the wave function as a real field, is just what is the relationship between the 3N+1 dimensional space of the wave function of an N body system, (it's so called configuration space), and our 4 dimensional space time. If you claim as Towler seems to at the end of his 6th lecture, that this just a mathematical description then you cannot claim as your theory does, that the wavefunction of quantum mechanics is real, that removes one of the main motivations for the Pilot wave theory. Some clarity is required here.
ii) It seems not to be relativistically covariant, this would imply that Einstein's theory of relativity sits uneasily in this theory. I doubt whether many physicists would welcome back the introduction of a real ether and the replacement of Minkowski space time with preferred Lorentzian frames, the idea that bodies really do contract as they approach the speed of light, (rather than just being an artefact of the relative positions of two observers). See lecture 5 of the Towler lectures for more detail I for one am not convinced.
iii) As yet it seems to be difficult to extend it to relativistic particle physics especially the treatment of fermions that means for example all the current developments in particle physics are shut off in this interpretaton.
So overall I think the Pilot Wave theory, has definitely achieved quite a lot, but it still has a lot of catching up to do with the standard formulation of quautum mechanics. Does moving the problems in the interpretation of quantum mechanics to the relationship between the 3N+1 configuration space and our own 4 dimensional space time raise more questions than it answers. I don't know. Maybe these problems will be resolved at least the pilot wave theory has earnt the right to be heard as an alternative to the standard view and I for one want to learn more about it.
http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html
So here are the pros
i) Contrary to the claim by the Copenhagen interpretation it has been shown that it is possible to define a definite trajectory for particles it's greatest success must be it's explanation of the two slit experiment. Sanity seems to be restored we no longer have to claim that a massive particle such as an electron or a bucky ball splits in two as it interacts with a screen which has a number of slits.
ii) It brings the physics back to quantum mechanics, instead of losing contact with the world of particles and their interactions as the Dirac Formalism is apt to, it does provide a causal explanation for many quantum phenomenon. Mind you the Dirac formalism is really elegant and still encapsulates the essence of many quantum mechanical problems even if it is all quite abstract.
iii) It avoids the problem of measurement, with all the inherent problems of the world around us being created by an act of measurement.
iv) It now deserves a place as an equivalent mathematical formulation of quantum mechanics (at least for Non relativistic problems). The fact that more and more papers are being published using Bohmian mechanics has to be something. I think it's fair to say that other alternative interpretations such as the Many Worlds Interpretation have not yet reached the same degree of maturity. I would argue that once a theory of physics has reached the stage where people can 'Shut up and calculate' as the Pilot Wave theory has done then it has a right to be treated as mainstream physics.
It is quite astonishing that at Solvay in 1927, when the basics of the theory was laid out by De-Broglie, that it wasn't considered at least as an alternative equivalent mathematical reformulation of quantum mechanics just as Heisenberg's matrix mechanics was. All the formulations of quantum mechanics had their problems of interpretation and it is at least arguable that the Pilot Wave theory has a well defined procedure for relating it's mathematical formalism to empirical results.
Now for the cons
i) It makes great play, that for it the wavefunction is a real field, something akin to an electric field or gravitaional one. Unfortunately this implies acceptance of the reality of 3N+1 dimensional configuration space, where N is the number of particles considered and 1 represents time. Let me explain a bit more the wave function of a manybody particle system is a function of all the coordinates of the particles considered eg for two particles with positions r1 and r2 the time independent of the wave function of the system is now a function of r1 and r2 that is we have $$\psi(x1,y1,z1,x2,y2,z2)$$ this is quite different from classical physics, for example the electric field produced by two charges, at a given point, is still a function of 3 dimensional space and not 6 dimensional space. The question then, for those who would see the wave function as a real field, is just what is the relationship between the 3N+1 dimensional space of the wave function of an N body system, (it's so called configuration space), and our 4 dimensional space time. If you claim as Towler seems to at the end of his 6th lecture, that this just a mathematical description then you cannot claim as your theory does, that the wavefunction of quantum mechanics is real, that removes one of the main motivations for the Pilot wave theory. Some clarity is required here.
ii) It seems not to be relativistically covariant, this would imply that Einstein's theory of relativity sits uneasily in this theory. I doubt whether many physicists would welcome back the introduction of a real ether and the replacement of Minkowski space time with preferred Lorentzian frames, the idea that bodies really do contract as they approach the speed of light, (rather than just being an artefact of the relative positions of two observers). See lecture 5 of the Towler lectures for more detail I for one am not convinced.
iii) As yet it seems to be difficult to extend it to relativistic particle physics especially the treatment of fermions that means for example all the current developments in particle physics are shut off in this interpretaton.
So overall I think the Pilot Wave theory, has definitely achieved quite a lot, but it still has a lot of catching up to do with the standard formulation of quautum mechanics. Does moving the problems in the interpretation of quantum mechanics to the relationship between the 3N+1 configuration space and our own 4 dimensional space time raise more questions than it answers. I don't know. Maybe these problems will be resolved at least the pilot wave theory has earnt the right to be heard as an alternative to the standard view and I for one want to learn more about it.
Tuesday, 29 May 2012
Pilot Waves for and against Part 1
Well the debate has been going on the science and physics OU Fora about the meaning or not of quantum physics as all such debates we do seem to going around in circles those of my fellow OU students reading this blog may find part 1 of the discussion here
http://learn.open.ac.uk/mod/forumng/discuss.php?d=1016527
and part 2 here
http://learn.open.ac.uk/mod/forumng/discuss.php?d=1029089
You will see that I have come close to losing my temper some times with a certain person, Still I'm not the only one who finds his attitude slightly infuriating which is reassuring.
Any way as promised here is a brief summary of the Pilot wave approach. This was intially thought of by De-Broglie and the basic idea is that each particle is accompanied by a Pilot wave. (Not that a particle is a wave or vice-versa), this Pilot wave guides the particles in situations such as the two slit experiment giving rise to the characteristic interference pattern on the screen, but a particle only passes through one slit at time. It was proposed at a conference at Solvay in 1927 but wasn't really taken up as most physicists were more impressed with the views of Bohr/Heisenberg and Schrodinger. The rest is history as they say, physicists got on with the business of using quantum physics to calculate the properties of molecules, atoms and solids etc and the pilot wave theory got quietly dropped.
That was until David Bohm rediscovered it and published two papers in the early 1950's these weren't taken seriously most notoriously because David Bohm claimed that the theory required the use of hidden variables to explain things like paticle states. So it was ignored until Bell came on the scene, as is well known he devised an experimental test which could distinguish between hidden variables and the standard predictions of quantum mechanics, it was shown that the predictions of quantum were vindicated and hidden variables were ruled out. You might have thought that would be the end of the story physicists could get on with the real business of developing the applciations of quantum mechanics to ever more and more complex problems. However there was a get out clause (there always is) In deriving his contradiction Bell made two assumptions
i) There were no hidden variables dictating spin components
ii) There were no non local interactions affecting the measurement of 1 particle a long distance away from another one.
As both were required it was perfectly possible for Bohmians to reject ii) and keep i). Most physicists were until quite recently prepared to accept i) and reject ii). However i) has its own problems if taken literally it implies that properties are created by an act of measurement against our notions of common sense.
Of course all that is a bit of an exaggeration, if on the statistical interpretation the wave function is simply a means to generate probabilities then measuring something does not create a property of a particle. All that happens as say when one throws a dice is that one of the possiblities is realised. But then that means that quantum mechanics is no more than an algorithm for correlating the mathematics of quantum mechanics with probabilities and doesn't really explain anything. Well I think I still hold that view, but nevertheless I'm slowly being persuaded, that there is more to the De-Broglie Pilot wave theory than I first would expect.
There has been quite an industry actually using the De-Broglie Bohm theory to perform calculations. Most impressive is it's explanation of the two slit experiment, It shows how the two slit experiment can be explained with each particle taking a definite trajectory and passing through a single slit, there is none of the usual problems associated with believing that an electron splits in two then magically reforms when it is detected. Neither is there any notion of wave packet collapse occuring as a result of measurement.
A really good introduction to the De-Broglie Bohm pilot wave theory is given here by these lectures by Mike Towler.
http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html
Best to start with the popular lecture
http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/towler_pilot_waves.pdf
I still think there are major problems with it which I will expound on in another post. Anyway I;ll leave you to enjoy the lectures and make your own minds up. The fact that something has moved from being a speculative tool to one where real calculations can be made has to be something.
On another note to do with real waves I was slightly disappointed with the results of my last TMA from MS324 in the low 80's. I missed a key point in one of the questions about the boundary conditions and in my favourite question number 2 I dropped a few marks because I missed out the first term in the series still a reasonable score but not as high as I hoped.
Bye for now
http://learn.open.ac.uk/mod/forumng/discuss.php?d=1016527
and part 2 here
http://learn.open.ac.uk/mod/forumng/discuss.php?d=1029089
You will see that I have come close to losing my temper some times with a certain person, Still I'm not the only one who finds his attitude slightly infuriating which is reassuring.
Any way as promised here is a brief summary of the Pilot wave approach. This was intially thought of by De-Broglie and the basic idea is that each particle is accompanied by a Pilot wave. (Not that a particle is a wave or vice-versa), this Pilot wave guides the particles in situations such as the two slit experiment giving rise to the characteristic interference pattern on the screen, but a particle only passes through one slit at time. It was proposed at a conference at Solvay in 1927 but wasn't really taken up as most physicists were more impressed with the views of Bohr/Heisenberg and Schrodinger. The rest is history as they say, physicists got on with the business of using quantum physics to calculate the properties of molecules, atoms and solids etc and the pilot wave theory got quietly dropped.
That was until David Bohm rediscovered it and published two papers in the early 1950's these weren't taken seriously most notoriously because David Bohm claimed that the theory required the use of hidden variables to explain things like paticle states. So it was ignored until Bell came on the scene, as is well known he devised an experimental test which could distinguish between hidden variables and the standard predictions of quantum mechanics, it was shown that the predictions of quantum were vindicated and hidden variables were ruled out. You might have thought that would be the end of the story physicists could get on with the real business of developing the applciations of quantum mechanics to ever more and more complex problems. However there was a get out clause (there always is) In deriving his contradiction Bell made two assumptions
i) There were no hidden variables dictating spin components
ii) There were no non local interactions affecting the measurement of 1 particle a long distance away from another one.
As both were required it was perfectly possible for Bohmians to reject ii) and keep i). Most physicists were until quite recently prepared to accept i) and reject ii). However i) has its own problems if taken literally it implies that properties are created by an act of measurement against our notions of common sense.
Of course all that is a bit of an exaggeration, if on the statistical interpretation the wave function is simply a means to generate probabilities then measuring something does not create a property of a particle. All that happens as say when one throws a dice is that one of the possiblities is realised. But then that means that quantum mechanics is no more than an algorithm for correlating the mathematics of quantum mechanics with probabilities and doesn't really explain anything. Well I think I still hold that view, but nevertheless I'm slowly being persuaded, that there is more to the De-Broglie Pilot wave theory than I first would expect.
There has been quite an industry actually using the De-Broglie Bohm theory to perform calculations. Most impressive is it's explanation of the two slit experiment, It shows how the two slit experiment can be explained with each particle taking a definite trajectory and passing through a single slit, there is none of the usual problems associated with believing that an electron splits in two then magically reforms when it is detected. Neither is there any notion of wave packet collapse occuring as a result of measurement.
A really good introduction to the De-Broglie Bohm pilot wave theory is given here by these lectures by Mike Towler.
http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html
Best to start with the popular lecture
http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/towler_pilot_waves.pdf
I still think there are major problems with it which I will expound on in another post. Anyway I;ll leave you to enjoy the lectures and make your own minds up. The fact that something has moved from being a speculative tool to one where real calculations can be made has to be something.
On another note to do with real waves I was slightly disappointed with the results of my last TMA from MS324 in the low 80's. I missed a key point in one of the questions about the boundary conditions and in my favourite question number 2 I dropped a few marks because I missed out the first term in the series still a reasonable score but not as high as I hoped.
Bye for now
Sunday, 20 May 2012
Interpretation of quantum mechanics part I (Again)
Hi been having a heated discussion on the OU science fora about the meaning or not of quantum mechanics
Those who follow this blog will know that I tend to be quite sceptical about any attempts to go beyond the current formalism as
a) No new physics will come out of it (Or none that we can distinuguish between experimentally)
b) Any attempt to see the wavefunction as somehow real leads to all sorts of problems
i) The idea of superposition being something physical until observed seems to imply that we create reality
by an act of measurement
ii) In the two slit experiment the idea that an electron or large particle somehow splits in two and then magically reforms (if taken literally) at the detector seems totally incredible (where does the energy come from etc) why bother with CERN if we can split electrons in two simply by passing them through slits.
iii) If a particle really is a wave how come the pattern only emerges after several impacts on the screen rather than all at once. It is only after a statisitically significant number of events have occurred that anything like a pattern interpreable as a wave function can occur. So that the 'wave aspects' are esssentially statistical the usual fuss about the pattern still occurring even though there is only one particle in the intererometer being irrelevant (or just as relevant as the throw of a single dice).
For these reasons I prefer the statistical interpretation of quantum mechanics, which says that the 'wavefunction' is essentially a probability amplitude whose modulus squared gives the probability of certain events happening. This implies that the wave function is not a property of a single system but more a mathematical device for generating probabilities, it differs from that of classical probability in the sense that to account for the quantum mechanical viewpoint we have to use complex numbers. I then went on to show how you could account for the sinusoidal dependence of the probabilites on the phase factor for a two state system. Also how it was quite striking how classical probabilty could be recast in the language of quantum mechanics specifically the Dirac formalism. For a recap see these two posts
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/quantum-mechanics-of-two-state-systems.html
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/mathematics-of-two-state-systems-2.html
I also gave a reference to a paper by Marcella which gave an explanation of how the typical form of the two slit interference pattern can be interpreted as a single particle build up of many events, where the particle does pass through a single slit, which acts as a measuring device the uncertainty in the particles position being due to not being able to know precisely the position of the particle and being responsible for the wave like appearance.
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf
In all fairness I should point out that a subsequent paper has been written criticising the above paper
http://arxiv.org/abs/1009.2408
And this was presented as a falsification of the Marcella paper, by one of the contributers to the forum. I beg to differ, all it shows is that Marcella has hidden some of his assumptions and that the use of free particle eigenstates is equivalent to a classical wave theory. It would be possible to adapt the Marcella paper to make it more accurate eg by the use of a superposition of free particle eigenstates (often referred to as a Gaussian wavepacket) and removing the assumption that it was equally likely for the particle to emerge from one of the slits. All this would be a distraction from the main point, namely that it is possible to give a particle like interpretation of the two slit experiment using the formalism of quantum mechanics. Indeed as that is what physically happens namely particles really do appear individually and only eventually is a wave like pattern revealed it would seem bizzare to attribute wave like properties to individual electrons, neutrons or buckyballs. Obviously collectively the wave like properties are manifest so wave particle duality is simply when considered as a single entity quantons (for want of a better word) behave like particles but when considered collectively they behave like waves.
The statistical interpretation to my mind is the bottom line, it makes the least number of metaphysical interpretations, one can avoid all the usual problems, real wave collapse, and so forth. It by definition is consistent with the formalism so physicists can get on with the real job, namely developing and applying the formalism to predict and understand the properties of solids, stars, quantum fluids, elementary particles, lasers etc. Or in a word physicists can 'shut up and calculate' and leave the 'interpretational stuff' to other people.
For more information on the statistical interpretation this web site gives a good introduction and overview
http://statintquant.net/siq/siq.html
However as a consequence of the dialogue on the OU science forum, I have become a bit more interested in the so called Pilot wave theory initiated by De-Broglie and subsequently developed by David Bohm. I'll discuss more about it in another blog, giving my reasons as to why out of all the myriad interpretations of quantum mechanics which seek to go beyond the statistics, this is the one I consider most promising. One of the striking things is that the motivations behind the pilot wave theory, seem very similar to the motivations behind the statistical ensemble theory. I will point out the comparisons between the two in a later post. Also there has recently been an experiment showing that classical systems can show wave particle duality.Something that hitherto has never happened before. I'll leave you to ponder about the significance of this experiment for now.
http://phys.org/news78650511.html
Those who follow this blog will know that I tend to be quite sceptical about any attempts to go beyond the current formalism as
a) No new physics will come out of it (Or none that we can distinuguish between experimentally)
b) Any attempt to see the wavefunction as somehow real leads to all sorts of problems
i) The idea of superposition being something physical until observed seems to imply that we create reality
by an act of measurement
ii) In the two slit experiment the idea that an electron or large particle somehow splits in two and then magically reforms (if taken literally) at the detector seems totally incredible (where does the energy come from etc) why bother with CERN if we can split electrons in two simply by passing them through slits.
iii) If a particle really is a wave how come the pattern only emerges after several impacts on the screen rather than all at once. It is only after a statisitically significant number of events have occurred that anything like a pattern interpreable as a wave function can occur. So that the 'wave aspects' are esssentially statistical the usual fuss about the pattern still occurring even though there is only one particle in the intererometer being irrelevant (or just as relevant as the throw of a single dice).
For these reasons I prefer the statistical interpretation of quantum mechanics, which says that the 'wavefunction' is essentially a probability amplitude whose modulus squared gives the probability of certain events happening. This implies that the wave function is not a property of a single system but more a mathematical device for generating probabilities, it differs from that of classical probability in the sense that to account for the quantum mechanical viewpoint we have to use complex numbers. I then went on to show how you could account for the sinusoidal dependence of the probabilites on the phase factor for a two state system. Also how it was quite striking how classical probabilty could be recast in the language of quantum mechanics specifically the Dirac formalism. For a recap see these two posts
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/quantum-mechanics-of-two-state-systems.html
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/mathematics-of-two-state-systems-2.html
I also gave a reference to a paper by Marcella which gave an explanation of how the typical form of the two slit interference pattern can be interpreted as a single particle build up of many events, where the particle does pass through a single slit, which acts as a measuring device the uncertainty in the particles position being due to not being able to know precisely the position of the particle and being responsible for the wave like appearance.
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf
In all fairness I should point out that a subsequent paper has been written criticising the above paper
http://arxiv.org/abs/1009.2408
And this was presented as a falsification of the Marcella paper, by one of the contributers to the forum. I beg to differ, all it shows is that Marcella has hidden some of his assumptions and that the use of free particle eigenstates is equivalent to a classical wave theory. It would be possible to adapt the Marcella paper to make it more accurate eg by the use of a superposition of free particle eigenstates (often referred to as a Gaussian wavepacket) and removing the assumption that it was equally likely for the particle to emerge from one of the slits. All this would be a distraction from the main point, namely that it is possible to give a particle like interpretation of the two slit experiment using the formalism of quantum mechanics. Indeed as that is what physically happens namely particles really do appear individually and only eventually is a wave like pattern revealed it would seem bizzare to attribute wave like properties to individual electrons, neutrons or buckyballs. Obviously collectively the wave like properties are manifest so wave particle duality is simply when considered as a single entity quantons (for want of a better word) behave like particles but when considered collectively they behave like waves.
The statistical interpretation to my mind is the bottom line, it makes the least number of metaphysical interpretations, one can avoid all the usual problems, real wave collapse, and so forth. It by definition is consistent with the formalism so physicists can get on with the real job, namely developing and applying the formalism to predict and understand the properties of solids, stars, quantum fluids, elementary particles, lasers etc. Or in a word physicists can 'shut up and calculate' and leave the 'interpretational stuff' to other people.
For more information on the statistical interpretation this web site gives a good introduction and overview
http://statintquant.net/siq/siq.html
However as a consequence of the dialogue on the OU science forum, I have become a bit more interested in the so called Pilot wave theory initiated by De-Broglie and subsequently developed by David Bohm. I'll discuss more about it in another blog, giving my reasons as to why out of all the myriad interpretations of quantum mechanics which seek to go beyond the statistics, this is the one I consider most promising. One of the striking things is that the motivations behind the pilot wave theory, seem very similar to the motivations behind the statistical ensemble theory. I will point out the comparisons between the two in a later post. Also there has recently been an experiment showing that classical systems can show wave particle duality.Something that hitherto has never happened before. I'll leave you to ponder about the significance of this experiment for now.
http://phys.org/news78650511.html
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