As promised (or threatened ) whatever way you want to take it, Here is another post on trying to debunk the alleged mysterious nature of quantum mechanics. This post will try and set the framework for more detailed analysis of the alleged paradoxes. To do this we have to look at the basic mathematical structure of quantum mechanics. I can do no better than quote from Richard Feynman who in the first chapter of his book on the subject gives the following summary of the essential features these are
(1) The probability of an event in an ideal experiment is given by the absolute value of a complex number a which is called the probability amplitude
P = probability
a = probability amplitude
P = |a|^2
(2) When an event can occur in several alternative ways the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference
a = a1 + a2
P = |a1 + a2 |^2 note this is not the same as |a1|^2 + |a2|^2
This is crucial, all the stuff about interference etc which bedevils much of the agonising about say the two slit experiment boils down to the fact that in order to get the total probability for an event to occur we must add the sum of the pobability amplitudes together and then take the square of the total sum rather than simply adding the sums together.
(3) On the other hand if it is possible to determine whether or not one event has actually occurred then the probability of the event is the sum of the probabilities for each alternative.
The crucial point to recognise is that we are simply dealing with probability amplitudes when I take the modulus squared of the probability amplitude taking into account all possibilities this gives me the probability distribution associated with the situation of interest. This approach was first taken by Dirac in his founding book 'The principles of quantum mechanics'. He showed that the mathematical structure of quantum mechanics was that of a linear vector space over the field of complex numbers. This was extended by Von Neumann in his book The mathematical structure of quantum mechanics which tidied up some of the loose ends when extending the dimensions of the vector space to infinite dimensions. Note these dimensions are essentially the dimensions of the probability state space associated with a given system they are not real fields in an infinite dimensional space. For those who want more detail on the underlying mathematical structure of quantum mechanics I refer to the freely available notes by David Mermin
http://people.ccmr.cornell.edu/~mermin/qcomp/CS483.html
In this set of notes he spells out how the formalism of quantum mechanics can be seen as a generalisation of probability applied to classical systems by making real probablilty amplitudes complex numbers. It's interesting to note that Dirac's approach is quite different to the standard approach based on solving Schrodinger's wave equation. The point is that the solutions to the wave equation i.e Schrodinger's wave function are only one way (albeit a convenient one) of representing the function space associated with quantum systems. Because Schrodinger's wave equation is usually solved in coordinate space the impression is given to many people that the wave function is some how a real wave in a multi-dimensional space time. This is unfortunate and misleading I would contend that the abstract formalism initiated by Dirac gives a much clearer picture as it shows that we are not tied to one particular representation. Indeed for complicated problems one can abstract the essential features by listing the possibilities. It's interesting a lot of the current problems in quantum mechanics to do with the relatiionship between information theory and the possibility of quantum computiation generally speaking do not need to solve Schrodinger's equation at all. It may well be that a new generation of physicists bought up on the Dirac approach might be more readily inclined to accept the abstract nature of quantum mechanics than an obssession with whether or Schrodinger's wave function is a real wave or as I would argue simply the complex square root of a probability distribution function. It is my contention that all the paradoxes etc beloved of the popular literature stem from seeing Schrodinger's wave function as a real field in a multi dimensional space time. I would urge people to have a look at the Mermin lectures.
Sunday 23 January 2011
Back in the Swing
Well the websites for my three maths courses opened on Monday and now I can access the TMA's I'm back in the swing of things. As a warm up I'll be concentrating on the first TMA for M208 over the next week or so. Sunday afternoons and Saturday mornings are usually the time I devote to writing TMA's whilst reading the relevant course material on the bus to work or in the evenings. The first deadline is middle of Feb for a part TMA for M208 this (as I predicted) was essentially about curve sketching, There is a systematic method outlined in the course material and I spent this afternoon applying it to the task in hand which was a rational function of two quadratics, The other question in the first part tests your ability to transform certain basic curves and draw a hybrid graph taking care to specify the intervals properly. So a warm up for the real stuff ahead but at least I feel I've started.
M208 has seven TMA's. M346 4 TMA's and M337 also 4 TMA's but only half of them have been published, In an ideal world I would like to do an average of two TMA's for M208 per month. I TMA for M337 and M346 which would mean I finish all the material by about June and leave the rest for consolidation and revision and looking at my other maths interests. Effectively it works out as 1 TMA question per week for M337 and M346 and 2 or 3 TMA questions for M208.
I had a little spat with some people on the MS221 forum in which some people were boasting they had completed half the material and it is true that M208 has a lot of 'optional material' which isn't examined mainly proofs of certain theorems so it is quite possible to zip through the course material and ignore a good half of it and then revise exam technique to get a good grade. This can be done relatively straightforwardly and quickly. On the other hand I'm doing these courses to cross the mental block I've had since I first came across say the epsilon delta definition of continuity when I started my physics degree. So at least for the analysis and group theory blocks of M208 and for all of M337 I really do want to make an effort to try and understand the proofs. I accept that this level of detail can't be done for all aspects of the course so it's a question of concentrating on what is important and I cannot see the educational value of just zipping through a course, Conversely I will be tempted to just zip through M346 but as it doesn't really go into the mathematics behind the statisitical modelling I don't feel to guilty. Of course whether such high minded intentions stay the course once the pressure of TMA deadlines loom I don't know.
M208 has seven TMA's. M346 4 TMA's and M337 also 4 TMA's but only half of them have been published, In an ideal world I would like to do an average of two TMA's for M208 per month. I TMA for M337 and M346 which would mean I finish all the material by about June and leave the rest for consolidation and revision and looking at my other maths interests. Effectively it works out as 1 TMA question per week for M337 and M346 and 2 or 3 TMA questions for M208.
I had a little spat with some people on the MS221 forum in which some people were boasting they had completed half the material and it is true that M208 has a lot of 'optional material' which isn't examined mainly proofs of certain theorems so it is quite possible to zip through the course material and ignore a good half of it and then revise exam technique to get a good grade. This can be done relatively straightforwardly and quickly. On the other hand I'm doing these courses to cross the mental block I've had since I first came across say the epsilon delta definition of continuity when I started my physics degree. So at least for the analysis and group theory blocks of M208 and for all of M337 I really do want to make an effort to try and understand the proofs. I accept that this level of detail can't be done for all aspects of the course so it's a question of concentrating on what is important and I cannot see the educational value of just zipping through a course, Conversely I will be tempted to just zip through M346 but as it doesn't really go into the mathematics behind the statisitical modelling I don't feel to guilty. Of course whether such high minded intentions stay the course once the pressure of TMA deadlines loom I don't know.
Wednesday 19 January 2011
Two Slit paper
Ok as promised here is what I think is the definitive paper on the two slit experiment
http://arxiv.org/abs/quant-ph/0703126
It shows how the quantum formalism works. Most text books as Marcella points out never show a consistent explanation from the principles of quantum mechanics. The analogy is made with classical optics one never sees a quantum explanation. I really wish this paper were known more widely it's a gem.
http://arxiv.org/abs/quant-ph/0703126
It shows how the quantum formalism works. Most text books as Marcella points out never show a consistent explanation from the principles of quantum mechanics. The analogy is made with classical optics one never sees a quantum explanation. I really wish this paper were known more widely it's a gem.
Tuesday 18 January 2011
Demystifying quantum mechanics prologue
Ok it was bound to happen sooner or later I would be provoked by an article or TV program on quantum mechanics claiming that it's all mysterious, no one can understand it, it is totally counterintuitive and so forth. I'm afraid I find all this talk at misleading and at worse really no more than absolute tosh. The culprit in question was last nights Horizon programme shown on BBC2 last night. I can only sketch my objections into what I imagine will be a big theme of these blogs. For now I just want to point out some of the claims that I object to and I will try and provide appropriate references for each of the claims in a series of posts.
1) The two slit experiment implies that particles split in two and then reform before impinging on a screen.
2) Particles after separating from a common source still stay in communication with each other and send signals faster than the speed of light
3) Every time a measurement is made a parallel universe is made
4) Objects are created simply by an act of measurement
and so forth,
Ok to give some clues to more detailed posts, the interpretation I favour, if it can be called an interpretation is the minimal statistical model. It seems to me that much of the so called mystique surrounding the interpretation of quantum mechanics stems from the fact that people refuse to take the probabilistic aspects of quantum mechanics seriously. I will expound more in a later post but let's just take the two slit experiment for example. The facts are that it is only possible to see the classic interference pattern emerging as a result of a cumulative number of events. Indeed it is only after a statistical significant events have occured that anything like wave like properties can be attributed to a photon or electron. Conversely it makes sense to speak of certain attributes of particles such as their mass or charge as belonging to individual particles. Rather than as some people would have it that particles split in two and then magically reform why not simply admit that the so called interference pattern is no different from say the way in which other probability distributions emerge from a collection of events eg the binomial distribution. OK the interesting question is why the probability distribution in events like the two slit experiment are not the same as would be classically expected. But please do not try and interpret what is primarily a statistical phenomenon on what is alleged to happen to a single particle. It has just as much significance as a single throw of the dice does or a single roll of a roulette wheel. If particles really did split into when they pass through slits why are we bothering with the Large Hadron collider ?
I will provide a link to a paper which shows how the formalism of quantum mechanics taken for what it is can explain the essential features of the two slit experiment without having to invoke wave particle duality or the idea that particles split in two when passing through slits. The essential feature is that the slits can be seen as measuring devices the uncertainty in position measurement allows for the uncertainty in momentum arising in an interference pattern.
1) The two slit experiment implies that particles split in two and then reform before impinging on a screen.
2) Particles after separating from a common source still stay in communication with each other and send signals faster than the speed of light
3) Every time a measurement is made a parallel universe is made
4) Objects are created simply by an act of measurement
and so forth,
Ok to give some clues to more detailed posts, the interpretation I favour, if it can be called an interpretation is the minimal statistical model. It seems to me that much of the so called mystique surrounding the interpretation of quantum mechanics stems from the fact that people refuse to take the probabilistic aspects of quantum mechanics seriously. I will expound more in a later post but let's just take the two slit experiment for example. The facts are that it is only possible to see the classic interference pattern emerging as a result of a cumulative number of events. Indeed it is only after a statistical significant events have occured that anything like wave like properties can be attributed to a photon or electron. Conversely it makes sense to speak of certain attributes of particles such as their mass or charge as belonging to individual particles. Rather than as some people would have it that particles split in two and then magically reform why not simply admit that the so called interference pattern is no different from say the way in which other probability distributions emerge from a collection of events eg the binomial distribution. OK the interesting question is why the probability distribution in events like the two slit experiment are not the same as would be classically expected. But please do not try and interpret what is primarily a statistical phenomenon on what is alleged to happen to a single particle. It has just as much significance as a single throw of the dice does or a single roll of a roulette wheel. If particles really did split into when they pass through slits why are we bothering with the Large Hadron collider ?
I will provide a link to a paper which shows how the formalism of quantum mechanics taken for what it is can explain the essential features of the two slit experiment without having to invoke wave particle duality or the idea that particles split in two when passing through slits. The essential feature is that the slits can be seen as measuring devices the uncertainty in position measurement allows for the uncertainty in momentum arising in an interference pattern.
Saturday 8 January 2011
Review of A211 Philosophy and the Human Condition Part 1
As some people seem to be in a reflective mood (Well Nilo anyway) I thought as I'm still between courses I would finally post some initial thoughts on A211. Philosophy and the Human Condition. This year is the last time the OU will be presenting this course and in October a New course will begin. A211 tries to achieve two quite separate aims one is an introduction to critical thinking namely learning how to follow a persons argument and if need be translate it in to premise conclusion form, this is a precursor to formal logic and can be quite useful in teasing out hidden assumptions. The course material covered 6 different areas and whilst some were what I would expect in an introduction to philosophy I have to say others were really quite idiosyncratic choices.
The units were as follows
Unit 1 Arguments for Freedom. This unit looks at the concept of Freedom starting with an account of Isaiah Berlin's distinction between negative freedom and positive freedom. Negative freedom is the freedom to pursue whatever one wants without restriction whereas positive freedom is the claim that we often make bad choices and these restrict us from fulfilling our full potential. Positive freedom is fine, if it's just us becoming aware of how the choices we make (eg watching telly instead of doing a TMA ) interfere with our long term goals. However often in the hands of the state it can become coercive people such as Rousseau or the Catholic church claim to know what is the general good and seek to impose this on other people who don't share their aims. After a brief discussion of Lockes essay on tolerance the core of the unit was devoted to a discussion of Mill's harm principle which is essentially the idea that people should be free to do say or believe whatever they want provided it doesn't cause physical harm to others. To my mind (despite the recent railings of the Pope against liberal secularism) this is a key value of liberal secularism. Liberalism It is not a collapse into moral relativism as religious critics seem to think, For example to take an example which seems to cause liberals much agonising. Many cultural practices such as female circumcison, discrimimation against women, ethnic minorities and people of different sexual orientation are justified in the name of cultural tradition. However these are all clear violations of the Harm principle, it is not cultural imperialism to criticise these as some people would claim but a sense of fighting against unjust and discriminatory practices. When the Pope criticises liberal secularists for not having values he is really saying that he wants his institution to carry out his discriminatory practices free from legitimate criticism. When liberals criticise the Pope's condemnation of the use of condoms this is done as a genuine concern for the harm that turning women into essentially breeding machines in the third world has done to both the women's lives and the dangers of overpopulation. Thus it is not that liberals have no values but there is a real clash of values.
Another key point of Mill is that offence is no harm. If I don't like a play or book because it criticises or mocks my beliefs I have no right to force it to be closed or shut down or cause violence in the streets . I do have a right to criticise it if I don't like it by writing letters to the press, making a peaceful protest etc. On this basis the increasingly successful campaigns to ban certain plays or books such as the Jerry Springer show, the burning of the Satanic verses and the shutting down of a play for fear of mob violence criticising the Seikhs in Birmingham a few years ago has no justification. I'm afraid I have no sympathy with the ease at which some religious fundamentalists be they Christian, Muslim. Hindu or otherwise take offence at certain publications.
I feel that every politician, religious leader and so forth should take on board the insights of Mill here. I found this unit quite thought provoking and relevant to today. Mill's Harm principle seems to me to be an almost universal ethical principle enabling people of different beliefs and life styles to live with each other and the ability to take on board legitimate criticism and toleration of different lifestyles provided no physical harm to other people is caused seems to me to be a necessary part of growing up.
I'll just briefly mention the two units which I didn't find all that inspiring namely the one on animal rights and the one on environmental ethics. I'll talk about the other units in later posts.
The units were as follows
Unit 1 Arguments for Freedom. This unit looks at the concept of Freedom starting with an account of Isaiah Berlin's distinction between negative freedom and positive freedom. Negative freedom is the freedom to pursue whatever one wants without restriction whereas positive freedom is the claim that we often make bad choices and these restrict us from fulfilling our full potential. Positive freedom is fine, if it's just us becoming aware of how the choices we make (eg watching telly instead of doing a TMA ) interfere with our long term goals. However often in the hands of the state it can become coercive people such as Rousseau or the Catholic church claim to know what is the general good and seek to impose this on other people who don't share their aims. After a brief discussion of Lockes essay on tolerance the core of the unit was devoted to a discussion of Mill's harm principle which is essentially the idea that people should be free to do say or believe whatever they want provided it doesn't cause physical harm to others. To my mind (despite the recent railings of the Pope against liberal secularism) this is a key value of liberal secularism. Liberalism It is not a collapse into moral relativism as religious critics seem to think, For example to take an example which seems to cause liberals much agonising. Many cultural practices such as female circumcison, discrimimation against women, ethnic minorities and people of different sexual orientation are justified in the name of cultural tradition. However these are all clear violations of the Harm principle, it is not cultural imperialism to criticise these as some people would claim but a sense of fighting against unjust and discriminatory practices. When the Pope criticises liberal secularists for not having values he is really saying that he wants his institution to carry out his discriminatory practices free from legitimate criticism. When liberals criticise the Pope's condemnation of the use of condoms this is done as a genuine concern for the harm that turning women into essentially breeding machines in the third world has done to both the women's lives and the dangers of overpopulation. Thus it is not that liberals have no values but there is a real clash of values.
Another key point of Mill is that offence is no harm. If I don't like a play or book because it criticises or mocks my beliefs I have no right to force it to be closed or shut down or cause violence in the streets . I do have a right to criticise it if I don't like it by writing letters to the press, making a peaceful protest etc. On this basis the increasingly successful campaigns to ban certain plays or books such as the Jerry Springer show, the burning of the Satanic verses and the shutting down of a play for fear of mob violence criticising the Seikhs in Birmingham a few years ago has no justification. I'm afraid I have no sympathy with the ease at which some religious fundamentalists be they Christian, Muslim. Hindu or otherwise take offence at certain publications.
I feel that every politician, religious leader and so forth should take on board the insights of Mill here. I found this unit quite thought provoking and relevant to today. Mill's Harm principle seems to me to be an almost universal ethical principle enabling people of different beliefs and life styles to live with each other and the ability to take on board legitimate criticism and toleration of different lifestyles provided no physical harm to other people is caused seems to me to be a necessary part of growing up.
I'll just briefly mention the two units which I didn't find all that inspiring namely the one on animal rights and the one on environmental ethics. I'll talk about the other units in later posts.
Wednesday 5 January 2011
Happy new Year
Well another new year starts doesn't feel like it in gloomy Edinburgh going to work in the dark and coming home back in the dark is guaranteed to make me grumpy and also lethargic. I managed to stick to my time table for watching the Ring cycle but my other ambitions took a back burner and I basically vegetated for most of the Christmas break. The materials for M208 have arrived but as I already had a copy no great surprises. The unfortunate thing is that this year the Open University have decided to issue the TMA's on line only. As a consequence despite reading Brannan's book I feel I haven't really started. However as the Introductory units seem to be a basic revision of Unit D of MST221 (without the tedious RSA coding thank goodness) it should be fairly straightforward. The first unit is on curve sketching so I suspect the first question will be a 'Tedious but Straightforward' exercise of systematically finding the asymptotes, stationary points etc of a function which will probably be a rational polynomial of some form or other with at least two zero's. The rest of the Introductory units cover Complex Numbers, proof by Induction (fine if you don't have to do it under exam pressure) and other related proof stuff. Also a first look at the number system in preparation for the analysis units. Block 1 covers the first part of group theory and goes further than MST221. Having used quite a lot of group theory in my physics studies it will be interesting to compare the 'Pure maths approach' with the physicist's approach.
Physicist's are usually given a quick run through of the main definitions then its straight into representation theory (basically reducing all group operations to matrices) with the aim of applying it first to things like molecular vibrations, then when particle physics is studied its continuous group theory with an intuitive approach to Lie groups.
Then a somewhat tenuous link (IMHO) is made to grouping particles by their various quantum numbers. It starts off OK given that the mass of the neutron and proton are roughly the same so it makes sense to see them as identical apart from their charge. This symmetry is called isospin. Then when it comes to grouping various mesons and baryons together there is quite a convincing decuplet and octet which Gell Mann found could be considered simplified using a basis state of three particles known as quarks. Mesons essentially being seen as a combination of a quark and an antiquark and baryons being a combination of 2 quarks with one antiquark. If you postulate as Gell Mann did that quarks have a charge of either 1/3 the charge of an electron or two thirds all the particle states can in principle be constructed from these basic building blocks. This led to the postulation of SU(3) symmetry. However amongst these islands of connection with reality there are a vast combination of possible quark states which do not correspond to any particle.
Furthermore since Gell Mann's day we now know there are 6 quarks instead of the original three. In the standard model of particle physics these are grouped in pairs allegedly of similar mass. Whilst this is OK for the so called first generation, the mass discrepancy for the second generation is quite significant, and by the time you get to the third generation it really seems incredible to group the 5th and 6th quarks together on the basis of their mass. The current masses along with their ratios are as follows
1 u (up) 6 MeV
2 d (down) 10 MeV ratio d/u = 1.67
3 s (Strange) 0.25 GeV
4 c (Charm) 1.2 GeV ratio c/s = 4.8
5 b (bottom) 4.3 GeV
6 t (top) 180 GeV ratio t/b = 180/4.3 = 41.8
It may well be that this current grouping is incorrect, hopefully when data starts coming from the Large Hadron Collider that the top quark is seen as the lower mass of another grouping of quarks, in which case
much of the underlying symmetry of the current Standard model will have to be rewritten. I really hope something new will come along to shake us out of our current complacency.
Hm I seem to have digressed. For those who want a physicists approach to group theory I refer to the excellent notes given by Peter Osborne for Cambridge Part III students
http://www.damtp.cam.ac.uk/user/examples/indexP3.html
The notes and example sheets are for the course 3P2 Particles and Symmetries. I hope to make a reasonable attempt to read these notes and make a decent stab at some of the example sheets over the next year to complement the Pure maths aspects of group theory that I will be getting from M208.
For those who don't have the background in quantum mechanics then sections 1 and section 5 of the undergraduate course on the principles of quantum mechanics gives a quick overview.
http://www.damtp.cam.ac.uk/user/examples/
Some of you reading this blog will be aware than Duncan and I have tried one or two of the more basic undergraduate problems from the dynamics sheets to do with rotational motion and I hope to continue to work my way through some of the other sheets. Particularly Mathematical Methods, Differential equations and Complex Methods. However I also want to reach for the sky and try my hand at the particle symmetries course.
Physicist's are usually given a quick run through of the main definitions then its straight into representation theory (basically reducing all group operations to matrices) with the aim of applying it first to things like molecular vibrations, then when particle physics is studied its continuous group theory with an intuitive approach to Lie groups.
Then a somewhat tenuous link (IMHO) is made to grouping particles by their various quantum numbers. It starts off OK given that the mass of the neutron and proton are roughly the same so it makes sense to see them as identical apart from their charge. This symmetry is called isospin. Then when it comes to grouping various mesons and baryons together there is quite a convincing decuplet and octet which Gell Mann found could be considered simplified using a basis state of three particles known as quarks. Mesons essentially being seen as a combination of a quark and an antiquark and baryons being a combination of 2 quarks with one antiquark. If you postulate as Gell Mann did that quarks have a charge of either 1/3 the charge of an electron or two thirds all the particle states can in principle be constructed from these basic building blocks. This led to the postulation of SU(3) symmetry. However amongst these islands of connection with reality there are a vast combination of possible quark states which do not correspond to any particle.
Furthermore since Gell Mann's day we now know there are 6 quarks instead of the original three. In the standard model of particle physics these are grouped in pairs allegedly of similar mass. Whilst this is OK for the so called first generation, the mass discrepancy for the second generation is quite significant, and by the time you get to the third generation it really seems incredible to group the 5th and 6th quarks together on the basis of their mass. The current masses along with their ratios are as follows
1 u (up) 6 MeV
2 d (down) 10 MeV ratio d/u = 1.67
3 s (Strange) 0.25 GeV
4 c (Charm) 1.2 GeV ratio c/s = 4.8
5 b (bottom) 4.3 GeV
6 t (top) 180 GeV ratio t/b = 180/4.3 = 41.8
It may well be that this current grouping is incorrect, hopefully when data starts coming from the Large Hadron Collider that the top quark is seen as the lower mass of another grouping of quarks, in which case
much of the underlying symmetry of the current Standard model will have to be rewritten. I really hope something new will come along to shake us out of our current complacency.
Hm I seem to have digressed. For those who want a physicists approach to group theory I refer to the excellent notes given by Peter Osborne for Cambridge Part III students
http://www.damtp.cam.ac.uk/user/examples/indexP3.html
The notes and example sheets are for the course 3P2 Particles and Symmetries. I hope to make a reasonable attempt to read these notes and make a decent stab at some of the example sheets over the next year to complement the Pure maths aspects of group theory that I will be getting from M208.
For those who don't have the background in quantum mechanics then sections 1 and section 5 of the undergraduate course on the principles of quantum mechanics gives a quick overview.
http://www.damtp.cam.ac.uk/user/examples/
Some of you reading this blog will be aware than Duncan and I have tried one or two of the more basic undergraduate problems from the dynamics sheets to do with rotational motion and I hope to continue to work my way through some of the other sheets. Particularly Mathematical Methods, Differential equations and Complex Methods. However I also want to reach for the sky and try my hand at the particle symmetries course.
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