Wednesday, 26 December 2012

Quantum Probability

Have being having yet another debate (or rather a continuation of one) about the meaning or otherwise of quantum mechanics on the physics fora. Those who have been reading this blog, know, that I think the simplest way to understand quantum mechanics is to see it as an extension of classical  probability, to embrace the concept of a probability amplitude which is a complex number, the modulus squared of which gives rise to a probability density function.  This is to take the Born Interpretation to it's logical conclusion, the standard view point is a curious mix of probability and the implications of this to a single particle or system. It is this switch between an essentially statistical idea and the claim that we can predict what will happen to one particle that I believe to leads what Karl Popper calls the great quantum muddle.

In a couple of previous posts, I have indicated how such a perspective clears up the interpretation of the two slit experiment. To recap the so called wave properties are essentially statistical applying to a population or ensemble. Thus questions which are essentially statistical tend to be confused with what we can say about individual particles. The claim being for example that a particle splits into two as it passes through the slits that magically recombines to form a single particle at the screen. The screen being alleged to 'collapse the wavefunction' so that it appears as a single point on the screen. Of course empirically what happens is that it is only after a significantly number of events have occurred that anything like a wave pattern can be discerned. This would imply that the 'wave properties' are related to the probabilistic aspects associated with an ensemble of photons and not to an individual particle. I show how this can be explained by the concept of complex probability amplitudes and the same applies to beam splitters etc in these posts

http://chrisfmathsphysicsmusic.blogspot.co.uk/search?q=Two+state

Anyway I am gratified to learn that there is a whole branch of mathematics called quantum probability which includes classical probability. On this view, the so called collapse of the wave packet is a mapping from the complex number space C^n to the real numbers. There is nothing physical about it all. Indeed those who did M338 will be familiar with the concept of a projection from a higher dimensional space to the real numbers.

Anyway a good introduction to the whole subject is given here

http://igitur-archive.library.uu.nl/math/2001-0713-122250/km.pdf

What is apparent is that it is clear that classical probability theory cannot account for many quantum phenomenon. The paper also makes it clear, that in so called Aspect experiments, where a correlation has been measured between the spins of two photons emitted from the same source and sent in opposite directions, The attempt to understand it's implications which has caused so much fuss over the past 20 -30 years, measurement of the spin of one particle does not affect the spin of the other. This is counter to so much of the prevailing wisdom (or at least that which dominates the debate in the media and the popular literature) that I really think this approach deserves further attention. What is gratifying about the quantum probabilistic approach is the way in which the correlation between two particles emitted from the same source just pops out of the formalism. I will expand on this in later posts.

The Open University physics project course includes one on quantum entanglement and I'm seriously contemplating doing this within the next year or so based on the above paper.  I'm swithering between doing the OU course SM358 in October, which I'm not sure I would need, or just diving in straight away.

Monday, 17 December 2012

MST326 Fluids TMA01

Well I've just finished the first TMA for MST326 Fluids although the course is on Fluid dynamics much of Block 1 is devoted to developing mathematical techniques

The first unit does cover fluids essentially hydrostatics with an interesting application of how lock gates work and also development of simple atmospheric models. The first question of this TMA covered this I confess I couldn't quite see how to do the later half of the first part which involved working out stability conditions for an inverted half open cylinder immersed in a liquidso left it, The second part was based on developing a simple atmospheric model of the pressure in the mesosphere. This I found tedious but quite straightforward.

The second unit covers techniques for solving second order differential equations specifically for those differential equations which have polynomial coefficeints in front of the derivatives not just constants . These included the Cauchy Euler method, the method of variation of parameters which enables you to generate a second solution to a differential equation if you know the first solution, order reduction another technique for generating the second solution to the differential equation from the first, Finally solution by series essentially you substitute a trial solution in the form of a polynomial into the differential equation and then generate recurrence relations for the coefficients. This method is probably the most powerful method for solving second order differential equations with polynomial coefficients. It leads to the so called special functions such as Bessel functions Legendre polynomials etc associated with the classic differential equations which crop up all the time in physics.

The question covered all these techniques and I think I got most of it out apart from the last part which was quite tricky algebraically and I couldn't get the equation for the reduction of order to resemble anything like
the equation in the hint so I gave up

The third unit is on techniques for solving first order partial differential equations this involves the method of characteristics. The question on this I found quite straightforward and am reasonably confident of getting a good mark for this one.

Finally the fourth unit on block 4 is Vector calculus, whilst this will be familiar to those who have done MST209 this is covered in more depth in this unit and the question in the TMA was correspondingly a lot more tricky. One part in particular involved calculating the curl of a vector field in spherical polar coordinates to show that it was zero. In order to do this involved quite a bit of playing around with trig identities. I was gratified to get it out, as also the last part which involved verifying Gauss's divergence theorem by calculating both a volume integral and a surface integral I was really pleased with myself when I showed that the volume integral was indeed the same as the surface integral

So overall two full questions more or less correct, and 4/5th s of another question and 3/5ths of the first question should be enough for a respectable grade 2 on this one.

In general the TMA is quite challenging and not just a repetition of exercises with slight modifications in the book. I enjoyed the intensity of it all even though slightly frustrated in parts. I hope by June to have got in a lot more practice on the past exam papers than I did for MS324 and realistically I can expect grade 2. But I need to spread the load a bit rather than let it all pile up in the last few days before the deadline.

Best wishes to you all Chris

Sunday, 9 December 2012

A224 Music TMA02 Pointless

Well I've just sent off what must have been the most pointless TMA I've ever done for an OU course.
This is for my music course. The exercise consists of listening to three short songs and trying to write some coherent sentences about them. The first was OK as it had the music and we were asked to identify it's phrase structure, key and other things including whether or not the word setting was syllabic or melisismatic.

However the other two did not and we had to identify the role various instruments played, the texture and so forth. It's really not clear (or at least to me) what the point of the exercise was. I mean apart from saying things like the flute plays some twiddly bits at 1"03 it gets louder at 2"43 what was the point. I'm afraid after having listened to the second song endless times and putting some half coherent sentences together I gave up. I really feel that this has been a total waste of time.

It's not as if we haven't done any thing which could have been tested such as some counterpoint exercises or basic harmony. I appreciate that say 1 third of the TMA could have been devoted to the exercise but why not also test the theoretical aspects as well. That is certainly what happend in A214.

The next TMA will involve setting a verse to music so hopefully I will find that a lot more interesting but really asking people to say something about music without the score in front so you can analyse it must be one of the most futile things I've ever been asked to do.

On a much brighter note at the invitatiion of a friend I heard the Scottish Ensemble play at Greyfriars Kirk a safe haven from the Christmas Bling of princes Street. The main piece was an arrangement of Bach's Goldberg variations for strings the concert was one of the most uplifting I've been to for a long time.

Next post I'll talk about the first TMA for the fluids course

(By  the way this is now post 100 so I'm just under 1 a week on average thanks to all my readers).