Interpretation of quantum mechanics part I (Again)

Hi been having a heated discussion on the OU science fora about the meaning or not of quantum mechanics
Those who follow this blog will know that I tend to be quite sceptical about any attempts to go beyond the current formalism as

a) No new physics will come out of it (Or none that we can distinuguish between experimentally)

b) Any attempt to see the wavefunction as somehow real leads to all sorts of problems

   i) The idea of superposition being something physical until observed seems to imply that we create reality
     by an act of measurement

  ii) In the two slit experiment the idea that an electron or large particle somehow splits in two and then magically reforms (if taken literally) at the detector seems totally incredible (where does the energy come from etc) why bother with CERN if we can split electrons in two simply by passing them through slits.

 iii) If a particle really is a wave how come the pattern only emerges after several impacts on the screen rather than all at once. It is only after a statisitically significant number of events have occurred that anything like a pattern interpreable as a wave function can occur. So that the 'wave aspects' are esssentially statistical the usual fuss about the pattern still occurring even though there is only one particle in the intererometer being irrelevant (or just as relevant as the throw of a single dice).

For these reasons I prefer the statistical interpretation of quantum mechanics, which says that the 'wavefunction' is essentially a probability amplitude whose modulus squared gives the probability of certain events happening. This implies that the wave function is not a property of a single system but more a mathematical device for generating probabilities, it differs from that of classical probability in the sense that to account for the quantum mechanical viewpoint we have to use complex numbers. I then went on to show how you could account for the sinusoidal dependence of the probabilites on the phase factor for a two state system. Also how it was quite striking how classical probabilty could be recast in the language of quantum mechanics specifically the Dirac formalism.  For a recap see these two posts


http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/quantum-mechanics-of-two-state-systems.html


http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/mathematics-of-two-state-systems-2.html

I also gave  a reference to a paper by Marcella which gave an explanation of how the typical form of the two slit interference pattern can be interpreted as  a single particle build up of many events, where the particle does pass through a single slit, which acts as a measuring device the uncertainty in the particles position being due to not being able to know precisely the position of the particle and being responsible for the wave like appearance.

http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf 

In all fairness I should point out that a subsequent paper has been written criticising the above paper

http://arxiv.org/abs/1009.2408

And this was presented as a falsification of the Marcella paper, by one of the contributers to the forum. I beg to differ, all it shows is that Marcella has hidden some of his assumptions and that the use of free particle eigenstates is equivalent to a classical wave theory. It would be possible to adapt the Marcella paper to make it more accurate eg by the use of a superposition of free particle eigenstates (often referred to as a Gaussian wavepacket) and removing the assumption that it was equally likely for the particle to emerge from one of the slits. All this would be a distraction from the main point, namely that it is possible to give a particle like interpretation of the two slit experiment using the formalism of quantum mechanics. Indeed as that is what physically happens namely particles really do appear individually and only eventually is a wave like pattern revealed it would seem bizzare to attribute wave like properties to individual electrons, neutrons or buckyballs. Obviously collectively the wave like properties are manifest so wave particle duality is simply when considered as a single entity quantons (for want of a better word) behave like particles but when considered collectively they behave like waves.

The statistical interpretation to my mind is the bottom line, it makes the least number of metaphysical interpretations, one can avoid all the usual problems, real wave collapse, and so forth. It by definition is consistent with the formalism so physicists can get on with the real job, namely developing and applying the formalism to predict and understand the properties of solids, stars, quantum fluids, elementary particles, lasers  etc. Or in a word physicists can 'shut up and calculate' and leave the 'interpretational stuff' to other people.

For more information on the statistical interpretation this web site gives a good introduction and overview

http://statintquant.net/siq/siq.html  

However as a consequence of the dialogue on the OU science forum, I have become a bit more interested in the so called Pilot wave theory initiated by De-Broglie and subsequently developed by David Bohm. I'll discuss more about it in another blog, giving my reasons as to why out of all the myriad interpretations of quantum mechanics which seek to go beyond the statistics, this is the one I consider most promising. One of the striking things is that the motivations behind the pilot wave theory, seem very similar to the motivations behind the statistical ensemble theory. I will point out the comparisons between the two in a later post. Also there has recently been an experiment showing that classical systems can show wave particle duality.Something that hitherto has never happened before.  I'll leave you to ponder about the significance of this experiment for now.   

http://phys.org/news78650511.html

The Importance of Pedantry

Hi as a break from Topology and Waves I have been working on my long term ambition to understand the Big Bang line by line. I'm currently trying to understand the derivation of the Friedmann equations from General relativity. This exercise is somewhat tedious and heartbreaking to say the least. However reading some text books eg Lasenby I have noticed at best a  misleading error in the calculations of the connection coefficients and this has led me to all sorts of confusion. A connection coefficient is a three indexed quanitity which can be calculated from the metric tensor g  as follows
$$\Gamma^{i}_{jk}= \frac{1}{2}g^{il}(\partial_{k}g_{li}+\partial_{j}g_{lk}-\partial_{l}g_{jk})$$
Where the indices run from 0-3 0 corresponding to t and 1,2,3 x,y,z or whatever coodinate system is required. As there are potentially 128 different terms to caclulate one can see the temptation to take short cuts. However if we interchange the indices we get
$$\Gamma^{i}_{kj}= \frac{1}{2}g^{il}(\partial_{j}g_{li}+\partial_{k}g_{lj}-\partial_{l}g_{kj})$$
Which for cases where g is a diagonal metric ie the only non zero terms are
$$g_{00},g_{11},g_{22},g_{33}$$ or $$g_{ii}$$
as in the case of the Robertson Walker, the first equation  reduces to
$$\Gamma^{i}_{jk}= \frac{1}{2}g^{ii}(\partial_{j}g_{ii}+\partial_{k}g_{ii}-\partial_{i}g_{kj})$$
and the second equation reduces to
$$\Gamma^{i}_{kj}= \frac{1}{2}g^{ii}(\partial_{k}g_{ii}+\partial_{j}g_{ii}-\partial_{i}g_{jk})$$
But as addition is commutative  this is just the first equation with the first two terms interchanged and the last term in both equations will be zero for the off diagonal terms (ie when j does not equal k) . So for diagonal metrics  then we must have
$$\Gamma^{i}_{jk} = \Gamma^{i}_{kj}$$
However this symmetry is often missed by standard textbook accounts (eg Lasenby et al page 377) and the claim is made that the only non zero off diagonal connection coefficients for the Robertson Walker metric are
$$\Gamma^{1}_{01}, \Gamma^{2}_{02},\Gamma^{2}_{12},\Gamma^{3}_{03},
\Gamma^{3}_{13},\Gamma^{3}_{23}$$
When they should also include the other off diagonal terms:
$$\Gamma^{1}_{10}, \Gamma^{2}_{20},\Gamma^{2}_{21},\Gamma^{3}_{30},
\Gamma^{3}_{31},\Gamma^{3}_{32}$$
The reason why this is important is because a key quantity in general relativity is the Ricci Tensor which can be calculated from the connection coeffients as
$$R_{ij}=\partial_{j}\Gamma^{k}_{ik}-\partial_{k}\Gamma^{k}_{ij}+
\Gamma^{l}_{ik}\Gamma^{k}_{l}{j}-\Gamma^{l}_{ij}\Gamma^{k}_{lk}$$
So if some gullible reader like myself neglects the other off diagonal terms then you will not get the correct answer for the Ricci Tensor and will spend many hours of frustration wondering why you can't get the answers quoted in the text books. Of the many books I have which give results for the connection coefficients for the Robertson Walker metric at least three namely

Narlikar " Introduction to Cosmology" page 106
http://www.amazon.co.uk/An-Introduction-Cosmology-J-Narlikar/dp/0521793769/ref=sr_1_1?ie=UTF8&qid=1336329189&sr=8-1

Lasenby "General Relativity An Introduction for physicists" page  377
http://www.amazon.co.uk/General-Relativity-An-Introduction-Physicists/dp/0521829518/ref=sr_1_1?s=books&ie=UTF8&qid=1336329247&sr=1-1

and
Collins, Martin and Squires "Particle physics and Cosmology" page 373

http://www.amazon.co.uk/Particle-Physics-Cosmology-P-Collins/dp/0471600881/ref=sr_1_2?s=books&ie=UTF8&qid=1336329321&sr=1-2

All neglect the other off diagonal connection coefficients for the Robertson Walker metric in their tables of connection coefficients.

All this goes to show is that one should not just naively take on trust results quoted in text books I can only agree with my colleague Duncan in his last post about the importance of pedantry when trying to understand maths or physics books

http://matrices-reloaded.blogspot.co.uk/2012/04/last-book.html


On another note I have decided not to embark on the composition course immediately as my finances couldn't really cope with it and also I want to spend more time on my extra curricular physics.

M338 TMA02 Back

So got this back reasonably pleased top end of grade 2. Made a silly mistake in the last question and as predicted didn't do all that well in question 2. My tutor says I should be pleased as it was a hard TMA anyway going to take a week off from OU maths till Monday of next week hopefully giving it a bit of a rest will help my subconscious digest it all.

MST324 TMA02 (Nearly finished my pudding)

Well after finally eating my meat for M338 I've nearly finished my pudding for MST324. What a joy (says he optimistically) The questions were on the wave equation and Fourier transforms.

Question 1 Involved using D'Alembert's solution to the wave equation to solve a particular boundary problem.

Question 2 (The best)  involved solving the inhomogeneous damped wave equation for a particular function using seperation of variables. This is a showcase question consolidating all the techniques we have covered so far. The steps are

1) Write the function u(x,t)  as a product  of two functions f(x)g(t)
2) Substitute these into the partial differential equation and this gives two separate ordinary differential equations in both f(x) and g(t) in terms of a constant .
3) Solve the homeogeneous  differential equations for f(x) and g(t) using the appropriate boundary conditons
4) Use the inhomogeneous function on the RHS of the partial differential equation to obtain particular integrals for f(x) and g(t).
5) The resulting solution is then u(x,t) = f(x)g(t)

This is maths at it's best, some of the steps might be a bit tricky but I think it's really cool how it all hangs together.

Question 3 Another question on a solution of a partial differential equation by separation of variables. This was another nice question and a bit simpler than question 2

Question 4 Testing your knowledge of the properties of Fourier transforms based around a question involving the recurrence relationships of Hermite Polynomials. This was OK but a bit tricky in parts and I still have to do the last bit.

So overall I'm quite pleased and this TMA has been a joy to do, there was none of the head scratching associated with M338, but I can see the last bit of ice cream in the tub and it will soon be back to struggling with the many definitions of continuity for Topological spaces and so forth. Anyway it looks like I'm not the only one to have struggled with the TMA for M338. According to my friend Duncan who is doing M208 this year, Alan my tutor for M338 has said that it is taking him 2 hours per question per person to mark the scripts. So if the tutors find it difficult it's not surprising that we do.

M338 TMA2 away

Well for better or worse (probably worse) TMA02 is away. I have no real confidence that I understand what is going on. However for interest here is a summary of the questions and my response. Part 1 had to calculate d(x,y) a so cakked distance function  for various x y

Question 1 a distance function  is defined and we have to show that it is a metric ie we have to show

M1 d(x,y) >= 0 with equality only if x = y

M2 d(x,y) = d(y,x) ie d(x,y) is symmetric

M3  Finally show that d(x,z) >= d(x,y) + d(y,x) (ie, the triangle inquality)

The metric had 2 forms depending on whether or not given two points in R^2 (x1,y1) (x2,y2) y2 = x2 or not

First part just asked  you to show  that you can apply the definition correctly think I did OK on this.

Second part show that the metric satisfies M1 and M2 of the definition of a metric

M1 is simply that d(x,y) >= to 0 with equality only if x = y. A bear trap here is to neglect to show that if

d(x,y) = 0 x = y think I managed to negotiate this sucessfully. Again this seemed relativiely sttaightforward

(In what follows e means element of )

Then one had to show that given a,b e R^2,  that |a1-b1|+ |a2-b2| < d(a,b) again this seemed straightforward

But this was supposed to be a hint to be used in proving the triangle inequality ie to prove that d satisfied M3

I seemed to prove that M3 was satisified directly without using the hint so there is a nagging doubt that I missed some cases.

Then one had to sketch 2 open balls of d for various d(x,y) an open ball is such that d(a,r) < r for a e R

where a is the centre of the open ball, This is a generalisation of the definition of an open interval

This was relatively straightforward although conterintuitive for the second ball. I came to the conclusion that there was no open Ball centered on (1,1) with radius 1/2 which satisfied the second condition of the metric,

logic tells me I'm correct but my instinct doesn't. Antyway I couldn;t fault my logic so it stands for better or worse.

It was then desired to show that d(x,y) was not metrically equivalent to the Euclidean metricd2(x,y)  The euclidean metric is simply the disance between two points Here I totally failed. The definition of two metrics d1 and d2 being equivalent is that it is impossible to find two real numbers m and M such that

                       md1(x,y) < d2(x,y) < M d1(x,y)

for all x,y e R^2 and we were supposed to use a solution to the first part as a hint to show this. Unfortunately all I could demonstrate was that given d2 and d1 it was indeed possible to find two number m and M such that the above inequality was satisfied. I'm looking forward to Alan's solution to put me right (watch this space).

So for question 1 reckon I've got about 75% of the full marks at least I've explained my confusion

The first part of Question 2 introduced a definition of some open sets and we had to show that they formed a topology on R the first part.

A topology T on a set X is a collection of subsets of X satisfying  the following axioms

T1 the topology must include the empty set and the set  X

T2 the topology T must be such that any two intersections of the subsets of T must also be an element of T

T3 the topology T must be such that any unions of subsets of T must also be an element of T

For finite toplogies it is sufficient to demonstrate this for any two subssets of X which are elements of T. 

The elements of T are called open sets of X, their complements are closed sets of X. 

At first sight the definition of an open set of X may seem totally arbitrary but T2 and T3 guarantee a form of closure. 

 I reckon I got this correct. The second part introduced a new definition of left continuity and we had to show that a specific function was left continous. I think I got most of these parts correct,

Finally we had to show that given a function is both left continuous and monotonically increasing that the funcition was indeed continuous on R.

The background to this is that, so far the purpose of M338 has been to generalise the M208 definition of continuity to reformulate it in terms of whether or not the inverse of f(a) gives rise to an open set of X. As my mastery of the various terminology is still a bit hazy I'm not convinced I got anything resembling a perfect answer to this part of the question.

Question 3 tested whether or not you understand basic definitions such as closure, boundary, exterior interorior or the 'density' or 'non density' of certain finite sets given a topology T defined  on those sets

As there were no explicit examples (unlike M208) this involved a) understanding some basic definitions and b) how to apply them. I think I got there in the end but was a bit of a struggle to say the least. Also when defining a map from a set X to X if the function exclude a certain element a then the inverse of f(a) = 0 the empty set, But this is still part of the topology of T. So the ambiguity when testing for continuity is whether or not f(a) has an inverse mapping onto the topology, I decided that it did as 0 is still an open set of T but I can see arguments for the other point of view. So again confident that I got about 75% of this question correct.

Question 4 involved sketching certain sets finding their exterior and interior and boundary and deciding whether or not they were open or closed or neither. I think I got most of this correct.

So overall I might be pushing the boundary between grade 2 and grade 1 for this one but if I get above grade 2 that will only be due to the genoristiy or otherwise of Alan's marking. In terms of really understanding the subject I'm still only on block A2. I neeed over the next 4 weeks to get to grips with A3 and A4 before moving onto block B which admittedly looks a bit more straightforward.  So yes I'm being pushed out of my comfort zone, but as my mate Neil said I wanted a 'brain fuck' and I think I've got this in spades. Obviously my confidence has been quite shattered and as a caution for those who think this is the ultimate in pure maths, it should be noted that a course in Topology, along the lines of M338 usally forms a second year course in most brick universities. Anyway I've eaten most of my main course so for the next two weeks, I'll be having my pudding of MST324 I can't wait.

Obviously I will have to try and consolidate my extremely limited understanding of the first part of M338 as well so it will be like eating porridge with salt as opposed to sugar. Still KBO (keep buggering on ) as Churchill is alleged to have said.

Hand being Forced

Sometimes events just force your hand. As some of my followers know I had planned eventually to study composistion via the Open College of Arts.

http://www.oca-uk.com/distance-learning/music

 However as I have been getting bogged down in Topology I had intended to postpone this till October, which would have given me time to pay all fees upfront and also get Topology out of the way. However they have just announced that their fees will be increasing by 60% for students who register after June 1st 2012. However they will keep a TA arrangement for those who register before then. So it looks like I'll be registering for the course in the next month or two,  taking advantage of their installment plan. I have enough savings for their deposit, it also means less beer for  the next 6 months but that's probably a good thing.

In a way I'm quite excited, but it means something is going to have to give over the next few years, I really can't study Maths, Philosophy and Music to anything like the depth I want to simultaneously. As my maths is important, especially as I have a long term ambiton to become an OU maths tutor, and I want to do some more undergraduate courses, especially M347 mathematical statistics and M336 groups,  before I embark on the MSc.  I think philosophy will have to take a back seat,for now. I will continue to do one or two courses via Edinburgh University every year and there are also Geoffrey Klempner's Pathway modules

 http://www.philosophypathways.com/programs/pack.html

but it looks like I will have to abandon my plan of doing AA308 philosophy of the mind and wait till the replacement course comes in within the next couple of years.

So Provisional Plan is

                        Maths                       Music                      Philosophy
  Current          M338, MS324         Comp 1 (June)         Philosophy of Arts (E/U)
  Oct 2012       MST326 M381
  June 2013                                      Comp 2                    TBD
  Oct 2013      M347  M336
  June 2014                                      Composition of
                                                        Extended                  TBD
                                                        Piece (Comp 3)
  Oct 2014      Start MSc                                                  OU 3rd level course

So if all goes well I could have composed my first symphony by June 2015. Not that it will ever get performed.

I had let my music Studies slide so it's good that I'm being forced to bring them back Still need to finish TMA02 for topology before doing anything else.

Added 2 pm
Have found a blog of someone who has been through the OCA music experience
http://www.fionacoulter.com/blog/

Schubert 1797 - 1828

As a relief from all the angst about Pure maths and Topology which seems to be getting quite a few of us down. Radio 3 has decided to spend the whole of this week playing continuous Schubert.

http://www.bbc.co.uk/radio3/

 Schubert is amongst my favourite composers. Some years ago I wrote this little biography and I share it with my friends


http://dl.dropbox.com/u/16049029/Schubert_bio_cdf.pdf

For those who don't know Schubert then you don't know what you've been missing. I would recommend starting with the last 3 piano sonatas preferably played by Alfred Brendel.

http://www.amazon.co.uk/Schubert-Piano-Works-Franz/dp/B00000417C/ref=sr_1_1?ie=UTF8&qid=1332792862&sr=8-1

The last quartets and String quintet by the Lindsay quartet

http://www.amazon.co.uk/Schubert-Late-String-Quartets-Quintet/dp/B0001Z2RSW/ref=sr_1_1?s=music&ie=UTF8&qid=1332792942&sr=1-1

Finally the three great song cycles with Fischer Dieskau and Gerald Moore.

http://www.amazon.co.uk/s/ref=nb_sb_noss?url=search-alias%3Dpopular&field-keywords=Fischer+Dieskau+schubert+song+cycles+

Anyway I hope you enjoy exploring Schubert as much as I have. Take the opportunity to listen to Radio 3 this week especially the main evening concerts. I've just finished hearing a performance by Paul Lewis, one of Alfred Brendel's pupil's of the two A major sonata's.

It's quite surprising just how quickly time flies I can't believe it's almost 10 years since I wrote the short biography. Hopefully it wont be another 10 years before I write another one.