Saturday, 20 July 2013

SM358 quantum mechanics TMA03

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.

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