Saturday, 9 February 2013

SM358 Quantum Mechanics First Impressions

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

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

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

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

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

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

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

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

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

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

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

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

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