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I Clarification of the postulates of QM

  1. Jun 5, 2016 #1
    So I'm taking a course in QM right now and would like some clarifications on the postulates of QM, mainly I'm looking for physical intuition and may be for someone to clear some misconceptions if I have any, so without further ado let's begin:

    first I'd like to state the postulates as I'm familiar of them for the sake of clearance

    1. all the information about a system is represented by ## |\psi\rangle## a member of Hilbert space.

    2. the possible measurement results of the system are the eigenvalues of hermitian operators in the Hilbert space, after measurement the state of the system is ##|\phi\rangle## which is the eigenvector corresponding to the measurement result.

    3. the probability of measuring a result ##\lambda## is ##\langle\phi|\psi\rangle^2## where ##\langle\phi|## is the complex conjugate of the eigenvector corresponding to the eigenvalue ##\lambda## (this is assuming normalization, if not the vectors should be normalized)

    4. the time evolution of the system is governed by the Schroedinger equation.

    after we got that out of the way here are my questions:

    1. in hindsight, representing the information of a system by some general mathematical object instead of forcing it to be the path of a particle (as is the case in classical mechanics) seems like the more logical route, what I don't get is why should the group of these objects form a Hilbert space, what physical intuition is there to support this notion?

    2. why the requirement for the results to be eigenvalues of operators? more importantly what do the operators represent, is it the actual act of measurement?

    3. again what is the intuition behind the measurement being probabilistic, I mean how from the interference of electrons do you come to the conclusion that the measurement should be a probabilistic result?

    lastly, this is not a question but more of my point of view on QM and I would be delighted if someone could give me some direction to whether this point of view is sensible or whether is contradicts some aspect of QM, so from my point of view QM seems to be a theory about what information is available to us more so than being a description of the physical system under consideration, that is it seems to state ( in postulates 1+2) that there is the info I have on the system, I measure it and then the info I have of the system is the measurement result, as such for the purposes of further calculations/predictions the system is now in the state representing the information I've obtained, this does not imply anything about what happens "under the hood" or what physically changed in the system if anything, it's just the information I have is updated.

    I know this is a long post with some questions that might seem stupid to some of the people with more familiarity with QM than I, so I'd like thank anyone who bothered to read this far and would be truly grateful to anyone who takes the time to reply.
     
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  3. Jun 5, 2016 #2

    mfb

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    Only if your eigenvalues are not degenerate.

    The theory was formulated to match observations. It is possible to find different theories, but those don't describe quantum mechanics in our universe.
    Originally this was a necessary requirement to make the theory work, but with the Bell inequality there is now also experimental support that we need something that looks probabilistic (there are deterministic interpretations, but none of them correspond to a classical deterministic universe where everything is predictable).

    Those measurements are interactions as well, and it is possible to describe them quantum-mechanically. At that point you quickly run into the field of interpretations of quantum theory. Keep the Schroedinger equation everywhere and you get many worlds, stop the quantum-mechanical treatment somewhere and select one result at random and you get collapse interpretations, argue that the wavefunction was never physical and you get yet other interpretations, and so on.
     
  4. Jun 5, 2016 #3
    I think that this is a very useful point of view. Too often one falls into a trap of naively associating realities to various elements of quantum theory and ends up with paradoxes. Search for "delayed-choice quantum eraser." (I'm not saying that one can't attach any reality to elements of quantum theory, but one must be very careful in doing so to evade demonstrably false conclusions.)

    These postulates given in an introductory course are half-truths. (I personally know someone who experiences a culture shock when he learns Nielsen & Chuang-styled QM for the first time that makes him questions his knowledge of QM that he has learned up to that point.)

    My favorite way of understanding most, but not all, of the postulates is the following:

    I. Start with the assumption that QM is a generalized probability theory with complex vector spaces replacing the set of vectors that have positive entries (in some fixed basis) in the classical probability theory, with mutually exclusive outcomes corresponding to orthogonal subspaces. (Actually complex vectors corresponds to only pure states. The most general statistical description uses density operators: https://www.physicsforums.com/threads/pure-and-mixed-states.873400/#post-5485669)

    In this way of thinking, your first question has no answer. The vector space structure is the mystery. But the rest are consequences of this mystery.

    II. From I. the Kochen-Specker argument forbids the assignment of values of outcomes before measuring, so the prediction has to be statistical. The closely related Gleason's theorem also gives from I. the rule to calculate probabilities. (For experts, both of these can be done in 2 dimensions using POVMs.)

    III. To preserve the probability in II. (for pure states) the time evolution must be governed by the Schrödinger equation.

    Postulate 2 in your list is generally false. The standard example is after a photon counting measurement, the photon is absorbed and destroyed. A more general description of measurements and states after the measurement uses POVMs (positive-operator valued-measure or measurement) and CP (completely-positive) maps respectively. Eigenvalues of Hermitian operators play no role in the postulate for POVMs.
     
    Last edited: Jun 6, 2016
  5. Jun 5, 2016 #4

    bhobba

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    Last edited by a moderator: May 7, 2017
  6. Jun 6, 2016 #5

    atyy

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    But if you think about it carefully, updating the information does not imply that nothing has happened under the hood. In the orthodox interpretation, one is agnostic about such a possibility.
     
  7. Jun 6, 2016 #6
    Just out of interest : what are these two ? Unfortunately I do not immediately have access to the text you mention.
     
  8. Jun 6, 2016 #7

    atyy

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    But Nielsen & Chuang also point out that there is nothing wrong with the traditional postulates, at least for discrete variables. For discrete variables, it is a matter of taste whether one uses their postulates or the traditional ones.
     
  9. Jun 6, 2016 #8

    atyy

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    BTW, given the OP's interest in information, he may like to look at

    http://arxiv.org/abs/1011.6451
    Informational derivation of Quantum Theory
    G. Chiribella, G. M. D'Ariano, P. Perinotti
     
  10. Jun 6, 2016 #9
    I agree, because of the Stinespring and Neumark's theorems. But more general postulates are also less arbitrary, which I think may help answering the OP's question of "why the requirement for the results to be eigenvalues of operators?"?
     
  11. Jun 7, 2016 #10
    So the answer to most of these is " that's what works"? I don't find that very compelling, I mean one sees a wave like interference pattern for electrons and goes yeah they most have some sort of wave like behavior, but from that to eigenvalues and probabailistic interpretations seems like quite a jump in the thought process .

    Can you please refer me to some source where I can study this formulation of QM? a Google search yielded no meaningful results.

    As for the rest of the post, most of it flew way over my head, but I"ll sit on it some more when I get the time.

    Thanks, my university happens to have a copy of the book so I"ll be checking it out soon.

    I never said that nothing changes, what I did say is that the time evolution of state or the state itself don't necessarily correspond to direct knowledge about the physical system in question (unlike classical mechanics where you can go ahead and say things like " hey the ball is over there" and the information about the physical system is actually encoded in the path it takes.), I believe that this is the definition of agnosticism.

    I did happen to get a preview of the first edition from the university's library website and they seem to be that:
    1) to each observable there corresponds a linear operator, the eigenvalues of which are the possible observation results.
    2) to each state there corresponds a unique state operator which is the average of an observable.

    it would be interesting to see how the rest of QM stem from these.

    Thanks! from a quick look it sure does seem interesting, saved in my computer so I could tackle it after the exams, although much of it seems like it would go over my head I sure am happy to be able to try at least.
     
    Last edited by a moderator: May 7, 2017
  12. Jun 7, 2016 #11

    mfb

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    That's how physics works - we observe the universe, then we make theories that describe those observations and predict future observations. There are self-consistent theories where nothing is wrong about those theories, they just have the flaw that they do not describe our universe.
     
  13. Jun 7, 2016 #12

    atyy

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    Ballentine is a terrible book for learning about the postulates of QM, because Ballentine's formulation of QM is fundamentally flawed. Ballentine has spent his career opposing standard QM, and wrote a famous review in 1970 criticizing textbook QM. Of course, it is not QM that is wrong, but Ballentine. His book differs from his review, but it is still unorthodox.

    Rather, I second Truecrimson's suggestion to look at Nielsen and Chuang, which is a very good book.

    https://arxiv.org/abs/1110.6815 (Postulates II.1 to II.5, p9) gives the same postulates as Nielsen and Chuang.
     
  14. Jun 7, 2016 #13
    It was indeed a remarkable jump, and all the pyhysicists behind it eventually were awarded with Nobel prizes.
     
  15. Jun 7, 2016 #14

    stevendaryl

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    You do know that at least one frequent contributor to this forum has exactly the opposite opinion, don't you? Not that there is anything wrong with healthy disagreement.
     
  16. Jun 7, 2016 #15

    atyy

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    If that contributor is bhobba, it's fine because he knows where Ballentine's severe errors are. One can read Ballentine after learning the correct basic material from Nielsen and Chuang.
     
  17. Jun 7, 2016 #16

    vanhees71

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    Well, postulate Nielsen and Chuang a collapse? If so, I'd be as rude against these authors as you are against Ballentines (imho marvelous) textbook and call it flawed ;-)). Of course, on Ballentine's textbook we agree to disagree for a long time within this forum. This is so, because we disagree on interpretational issues (but of course not on physics) of QT.
     
  18. Jun 7, 2016 #17

    atyy

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    It depends on what you mean by "collapse". In your terminology, does the textbook by Cohen-Tannoudji, Diu and Laloe postulate a collapse?
     
  19. Jun 7, 2016 #18
    The approach of states (density operators) and effects (POVMs) is popular in quantum information theory. I think historically it started from G. Ludwig's brand of axiomatic quantum theory in the 80's. Many practitioners such as Holevo, Werner, Keyl, Busch and Lahti, use it in their papers and books. Quantum Computation and Quantum Information by Nielsen and Chuang is regarded by many as the bible of quantum information (and computation) theory and is usually used as a graduate-level textbook for the subject.

    Here are some free online notes.
    Guide to mathematical concepts of quantum theory (This was later published as a book.)
    https://arxiv.org/abs/0810.3536

    Algebraic approach to quantum theory: A finite-dimensional guide
    http://arxiv.org/abs/1505.03106

    But most of these don't mention Kochen-Specker and Gleason's theorem. You have to hunt around on arXiv to find modern proofs of them. I don't know from the top of my head a source that develops quantum theory exactly in the way that I outlined. But I will keep looking for it.
     
  20. Jun 8, 2016 #19

    bhobba

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    That is VERY much a minority view - in fact Atty is the only one I know that holds it.

    Many around here, including me, think its the best out there.

    Thanks
    Bill
     
  21. Jun 8, 2016 #20

    atyy

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    While Ballentine may be considered by many at PF to be a good book, one can simply compare his version of quantum mechanics against the following books to see whether Ballentine's distorted view of QM is mainstream.

    Dirac https://www.amazon.com/Principles-Quantum-Mechanics-International-Monographs/dp/0198520115
    Landau and Lifhistz https://www.amazon.com/Quantum-Mechanics-Third-Non-Relativistic-Theory/dp/0750635398
    Messiah https://www.amazon.com/Quantum-Mechanics-2-Volumes-1/dp/0486409244
    Cohen-Tannoudji, Diu and Laloe https://www.amazon.com/Quantum-Mechanics-2-vol-set/dp/0471569526
    Weinberg https://www.amazon.com/Lectures-Quantum-Mechanics-Steven-Weinberg/dp/1107028728
    Le Bellac https://www.amazon.com/Quantum-Physics-Michel-Bellac/dp/1107602769
    Sakurai and Napolitano https://www.amazon.com/Modern-Quantum-Mechanics-2nd-Sakurai/dp/0805382917
    Nielsen and Chuang https://www.amazon.com/Quantum-Computation-Information-10th-Anniversary/dp/1107002176
    Holevo https://www.amazon.com/Statistical-Structure-Quantum-Lecture-Monographs/dp/3540420827
    Busch and Lahti https://www.amazon.com/Operational-Quantum-Physics-Lecture-Monographs/dp/3662140349
     
    Last edited by a moderator: May 7, 2017
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