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B After the 'Theoretical minimum' series, what is essential to know about QM?

  1. May 13, 2016 #1
    The adagium of most quantumphysics-afficionado's is: "Shut up and calculate" - 'learn the formalism'. So I started with Leonard Susskind's 'Theoretical minimum' textbooks.

    So now I know a little (very little) about the formalism, I started to wonder to which extent I have to go to educate myself in order to understand what I need to know. Is what you learn ever enough? And if not, why start with quantummechanics at all? Is it at all satisfying to study QM? Or is it that you learn more precisely what you don't know?

    So my question is: after the 'Theoretical minumum' series, what is essential to know about QM? I have planned "Mathematical Methods in the Physical Sciences" by Mary Boas, follow by "An Introduction to Quantummechanics" by David Griffiths. This is quite a lifelong planning for me it seems to me. So, do I know anything more than I did when I've read all this? Is it worth it to read all this?

    Can anyone elaborate on this? Much appreciated.
     
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  3. May 13, 2016 #2

    A. Neumaier

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    This depends on your ultimate goals. Why do you learn at all? What do you want to understand/do/achieve, and at which level? In science, learning never ends, as long as one is motivated to learn.
     
  4. May 13, 2016 #3
    Is there a level at which one could say you know 'enough' to 'understand' QM? And if not, does that mean I will never understand it? And if that is the case, what do I learn from studying QM?
     
  5. May 13, 2016 #4

    A. Neumaier

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    Your questions don't have an answer without specifying the context - in this case your values, desires, and goals.
    Do you understand life? yourself? your girl friend? Will you ever understand?

    You learn some quantum mechanics by studying quantum mechanics, and you learn something about how it relates to other subjects. It is a huge subject - after almost 30 years of learning I still don't know enough to understand it at the deepest level. But I understand it well enough to explain most things about quantum mechanics that interest me to others (lay people and students) in an intelligible way.
     
    Last edited: May 13, 2016
  6. May 13, 2016 #5
    Could you illuminate me on whether reading 'The theoretical minimum' can be self-contained as a text? I feel that now I understand pure and mixed states a bit better (I have to re-peruse it though) that I know pretty much enough. So, what do I lack then?

    I feel I would be happy if I understood the uncertainty principle, entanglement, the double slit experiment and the delayed quantum eraser. :smile: (and their relation)
     
    Last edited: May 13, 2016
  7. May 13, 2016 #6

    A. Neumaier

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    There are levels of understanding. In some sense you understand something if you can answer to your satisfaction the questions you have about it. You understand better if you can answer them to the satisfaction of someone else. On a deeper level, you understand something if you can make sense of what others write about it, can discriminate between whether they write nonsense, or something meaningful. This includes noticing when they were sloppy (e.g., an indexing mistake). Reaching this level takes significantly longer. At this stage you should also be able to solve exercises related to the subject. Answering questions of others about the subject is another level that takes more practice, though of a different kind. You reached a really deep level of understanding if you can assess the subject for yourself and arrange the material in a personally motivated and sound way, ready for others to understand. This may take days or years of thinking about the subject, depending on what it is.

    You'll probably always lack a lot unless you studied quantum mechanics for a whole life. So don't assess your growth by what you lack but by what you gained. A simple check would be how much of the wikipedia pages on the subject you can read with understanding before you get lost....
     
  8. May 13, 2016 #7

    jtbell

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    Thirty-four years after finishing a Ph.D. in physics, I'm still learning new things about QM on this forum. :cool:
     
  9. May 13, 2016 #8
    Thank you for the elaborated reply! So, how 'deep' can I go by self-study?? (at home)
     
  10. May 13, 2016 #9

    Demystifier

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    It depends only on you (and on the literature you have access to).
     
  11. May 13, 2016 #10

    atyy

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    But a very important point is that in a sense, the first few lectures are enough. There are a couple of mathematical details, but they are not that important. Even quantum field theory doesn't go beyond elementary quantum mechanics. For example, section 2.1 of http://www.theory.caltech.edu/people/preskill/ph229/notes/chap2.pdf is in an important sense all of quantum mechanics, and all of quantum field theory, and all of string theory.

    The Theoretical Minimum is a decent text and will teach you all of quantum mechanics. It is a little idiosyncratic, but no more so than Landau and Lifshitz or Weinberg, which are both great texts. However, it is a bit chatty, so it may not be clear that all of quantum mechanics is very simple.
     
  12. May 13, 2016 #11

    A. Neumaier

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    Everything you need (with the exceptions mentioned below) is free on the web, usually in many variants. So if you don't get (physically or mentally) one account try another one. Until you are at research level you can safely ignore all articles and books that are behind a paywall.

    Except for your time, concentration and determination - that must be contributed by yourself. You may be interested in reading Chapter C4: How to learn theoretical physics of my theoretical physics FAQ where I responded to others who self-study. Some of the other stuff in the FAQ might also be interesting for you, either now or at some later stage.
     
    Last edited: May 13, 2016
  13. May 13, 2016 #12
    Thank you all for the sources and the replies! What I'm going to say may sound odd: I learned from the theoretical minimum how states work (more or less :wink: ). A state represents properties (what is 'known') of a physical element like a particle, if you want to call it that. However, a state requires a new concept to be representable: a state space. The state space works perfectly well as a mathematical representation, but it has (I was told) no physical significance! Nonetheless states are mathematically very well defined (it seems to me). So, is the principle of defining a new space for each new concept you want to describe the guiding principle in QM? Then, all concepts would share a common similarity, and the core of QM could be seen as (abstract!) vectors, functions and spaces in a mathematical setting, every new space built on the previous one. Does that make any sense? I hope the question is clear.
     
    Last edited: May 13, 2016
  14. May 13, 2016 #13

    A. Neumaier

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    There are spaces for each kind of objects that one may possibly want to study in a geometric way. This is one of the guiding principles in mathematics. And much of understanding quantum mechanics is understanding its mathematical language.
     
  15. May 13, 2016 #14

    A. Neumaier

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    That's not quite true. It is like saying space has no physical significance since one cannot measure it, though it is the arena in which all motions occur.

    Similarly, the state space has no direct physical significance but it is the arena where all the dynamics happens, and hence is eminently physical. For example, with an appropriate state space you can shrink a complex molecule in 3 dimensions to a single point in high dimension - and this is everywhere made use of.
     
  16. May 13, 2016 #15
    Last edited by a moderator: May 7, 2017
  17. May 13, 2016 #16
    Classical probability theory also has states and state spaces even though we don't usually call them that. (But https://www.amazon.com/Quantum-Measurement-Control-Howard-Wiseman/dp/0521804426 did. I recommend the first chapter if you want to see the similarity.) Only the shapes are different. They are simplices (generalization of a triangle) in the classical theory. The shape of quantum state space is spherical for a qubit but more complicated in higher dimensions. This innovation by quantum mechanics is the source of all kinds of counter-intuitive features like entanglement and Bell violation.

    You might enjoy how Scott Aaronson "tweaks" classical probability theory a little to get quantum mechanics: http://www.scottaaronson.com/democritus/lec9.html
     
    Last edited by a moderator: May 7, 2017
  18. May 13, 2016 #17

    A. Neumaier

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    But later you have to unlearn the weird negative probability stuff presented there and replace it by mathematically more respectable notions.
     
  19. May 13, 2016 #18
    I see Aaronson either talks about negative amplitudes or negative "probabilities" always in scare quotes and he does not forget to say that probabilities are always non-negative. So I don't see a problem with it. Unless you mean something else by "mathematically more respectable notions."
     
  20. May 13, 2016 #19

    A. Neumaier

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    Actually he is just confusing the reader with mentioning negative probabilities at all. Crossing out the corresponding parts of sentences and headings doesn't change anything and would be less confusing. Apart from this (and a similar sloppiness where he talks without need about p-norms with real nonnegative p and complex entries), it is indeed ok.
    Thus he is just too wordy and emphasizes in his headings some irrelevant nonsense.
     
  21. May 13, 2016 #20

    atyy

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    It is important to understand why we consider such a notion to be suspect. The reason is that in quantum mechanics, we need a "classical" observer to say when a measurement is performed. We don't have any easy way to make the observer fully quantum by incorporating him into a larger Hilbert space. If we do so, we seem to need yet another classical observer to observer the larger Hilbert space.

    Since we cannot easily have a wave function of the universe and nothing else, we consider quantum mechanics to be an operational theory. The observer makes a subjective division of the world into a classical measuring apparatus, and a part described by a Hilbert space. Only the measurement outcomes are real (their distribution is described by expectation of observables, including correlation functions). The Hilbert space is not necessarily real, and just a fiction help us calculate the distribution of measurement outcomes.

    If we wish to have a wave function of the universe and nothing else, one approach is the Many-Worlds Interpretation. However, it is unclear whether such an interpretation is truly coherent.
     
  22. May 14, 2016 #21

    A. Neumaier

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    This is also needed in classical mechanics since the Hamiltonian formalism doesn't tell it.
    Nothing forbids to have this as easily in quantum mechanics as one can have a state of the universe in classical mechanics. The only difficulty in both cases is specifying exactly which state the universe is in. One doesn't need many worlds for it, neither in classical nor in quantum mechanics. One world is enough, and it features already all we know and ever will know.
    In classical mechanics, the observer also makes a subjective division of the world into a classical measuring apparatus and the system to be measured. And there is the problem of how to define the measurement result (a property of the measurement device) and how to relate it to the measured system (which is coupled through the dynamics of the universe, hence there is no simple bijection between what one reads off the device and a property of the measured system).

    Thus one has in quantum mechanics no additional difficulties compared to the classical situation. But in quantum mechanics people turn it into a big philosophical problem while in classical mechanics everyone always adhered to shut-up-and-calculate.

    You'll go deep only if you concentrate mainly on the formal side and mostly ignore the details of the many incompatible interpretations.

    Quantum mechanics thrives because of the predictions from the formal structure, not from endless discussions about the interpretation.
     
    Last edited: May 14, 2016
  23. May 14, 2016 #22
    While what you say about classical mechanics in the first paragraph is true I would say there are clearly additional difficulties in quantum mechanics that lie in the fact that quantum mechanics has a dependence on the classical theory that is obvious in the fact that the observables are ultimately classical and all measurements are classical in that sense. So there would be no difficulties if QM had not this foundational dependence on classical physics(wich would also make easy to consider QM as a fundamental theory, a hard thing as long as this dependency remains). Of course you can always refer to the mathematical formalism disconnected with the physical measurements, but then it is just a mathematical theory, not a physical theory, it couldn't make any prediction.
     
  24. May 14, 2016 #23

    A. Neumaier

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    Only some interpretations have it; it is not really needed - just over and over repeated for historical reasons.

    In practice, classical = macroscopic limit of quantum mechanics, describes by statistical mechanics, so the classical is an intrinsic limiting part of the quantum. Observables are macroscopic, hence classical in this sense.
     
  25. May 14, 2016 #24
    Um, do you know any interpretation without measurements(if only for predictions to be possible in the first place)? If so that interpretation is pure math, not a physical theory.
    The identity you use between classical and macroscopic limit of QM followed by declaring all observables macroscopic(therefore classical, and the observables are the connection between the formalism and the physical measurements) is another way to see this dependence on the classical theory but with the limiting part understood the other way around.
    This follows from simple logic, no theory that has an explicit dependence on other can be considered its generalization. Or expressed in different words: a theory cannot depend on its special case.
    Again if you are only referring to the mathematical formalism without reference to measurements and predictions what I'm saying doesn't apply, but then you are not dealing with a physical theory.

    Edit:I see you edited the last sentence to avoid the logical issue. So are observables macroscopic?
     
  26. May 14, 2016 #25

    A. Neumaier

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    Of course, else a human cannot observe them.

    Classical measurements also need the same specification of being macroscopic - so there is again no difference to the quantum case.
    The math is in both cases free of measurement issues.
     
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