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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Bye for now

## Tuesday, 23 April 2013

## Tuesday, 2 April 2013

### SM358 Quantum Physics TMA01

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

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

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

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

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

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

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

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

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

1) Silverman Quantum Superposition

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

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

2) Ballentine Quantum Mechanics A Modern development

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1) Silverman Quantum Superposition

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

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

2) Ballentine Quantum Mechanics A Modern development

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

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

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

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

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

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

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