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


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


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.


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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