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Why is the Hilbertspace complex?

  1. Nov 6, 2009 #1
    Hi there,

    is there a reason, why we choose a complex Hilbertspace in quantum mechanics? If we take a real field, some commutators won't make sense. Are there oher reasons?

    Thanks
    Sponti
     
    Last edited: Nov 6, 2009
  2. jcsd
  3. Nov 6, 2009 #2
    Hi,

    that's because the vectors of that space (ie. wave functions )must be complex for physical reasons.
     
  4. Nov 6, 2009 #3
    Ok and what are these physical reasons? And what do you mean by complex vectors? And thanks for the answer.
     
    Last edited: Nov 6, 2009
  5. Nov 6, 2009 #4
    by complex vector I mean the wave function of quantum mechanics.

    I don't know your background in QM so I don't really know what to say, basically the wave function is the mathematical tool that describes the system.
     
  6. Nov 6, 2009 #5
    I read the book "Modern Quantum Mechanics" by Sakurai. So I think we can speak on an advanced level. I'm familiar with the abstract formulation of quantum mechanics and actually never heard for an explanation or at least motiviation for the complex field in quantum mechanics. However, if we talk about an abstract Hilbertspace, I don't see, why we should speak about complex vectors. There are simple vectors. So please can you explain the physical reasons. Don't care about my backgrounds.

    Sponti
     
  7. Nov 6, 2009 #6
  8. Nov 6, 2009 #7
    Mathematically a vector space is generated by .. vectors (its building blocks). It is real if the vectors are real and complex if they are complex.
    In the case of QM, the vectors must be complex to account for the numerous experiments (Sakurai gives the example of Stern-Gerlach, in the first chapter I think). that's what I meant by 'physical reasons'.
    It follows naturally that the vector space should be complex too.
    I advise you to re-read the first chapter of Sakurai's.
     
  9. Nov 6, 2009 #8
    The results wouldn't agree with the experiments if we would chose a real space. Apparently the complex vector space is the simplest description that agrees with the experiments.

    Can somebody give some experiment example that wouldn't agree?

    -- Dmtr
     
  10. Nov 6, 2009 #9
    Just to clean thinks up: It seems to be a missunderstanding. So I'm sorry for that. I talk about a complex vector space, if the defining field (in the mathematical sense) is complex and a real vector space, if the field is real. So I should have asked: Why is the field of the Hilbertspace complex in Quantum Mechanics?

    As far as I can see (after rereading) Sakurai only uses an analogy to the polarization and don't "prove" it in a sense, but maybe you could correct me. I will appreciate to see experiments contradicting a real field. Thank you for that

    @Gerenuk: I read the threads and the articles. Actually I think that Quantum Logic restricts the possibilities to real numbers, complex numbers and quaternions. So we only need experiments. Another auspicious way is the representation of rotations with 2x2 matrices. Then we need complex numbers in the Hilbertspace, because spin creates a two dimensional Hilbertspace (also verified by experiment). So we have to represent rotations with 2x2 matrices. This would be a proof for the complex numbers.
     
  11. Nov 6, 2009 #10
    For you is important

    [tex]|\psi|^2[/tex]

    From theory of complex numbers you know that [tex]z^*z=|z|^2[/tex]

    Solution of this equation

    [tex]i\hbar\frac{\partial \psi}{\partial t}=\hat{H}\psi[/tex] is complex function!

    If solutions of Srodinger equation are complex states and these are elements of complex vector space you need complex vector space!
     
  12. Nov 7, 2009 #11

    Fra

    User Avatar

    In the conventional presentation I've seen there is no other "reason" except that it's a model that is so far successful as a predictive framework, at least for atomic and particle physics.

    As for possible deeper a priori reasons to suspec this, that's I think an open question and is inseparable from several other interpretational and conceptual problems of QM.

    I think one can convince oneself if you spend some time analysing it, that an intrinsic theory of measurement, that is supposed to define some structure representing information, and operations on information in a way that is define operationallly in terms of a finite observer interacting with an unknown environment, can in the general case not be as simple as boolean logic or follow standard probability. Mainly beceause there is not physical backup of the definition of timeless observerindependent "sets" of distinguishable events.

    Standard formulations contains identifiable assumptions. In an intrinsic operational perspective, I think assumptions are the basis of further actions, nothing more.

    So to speak for myself I feel quite clear on why boolean logic / standard probability and boolean logic is not sufficiently fit here. However why quantum logic / quantum statistics is the right choice, and even IF it is the right choice (or just "good enough for now") is something that is open.

    I haven't yet seen an argument yet that is better than the standard "because it works".

    (The born rule and the complex numbers are related. I have seen papers infere on from the other, but it proves nothing except an internal connection between quantum statistics and complex numbers, but it's not that surprising that the representation structure and the logical operations are related. - but it still doesn't make an independent argument for why)

    /Fredrik
     
  13. Nov 7, 2009 #12
    There is no deeper explanation for the complex nature, only that it is indeed necessary. The "experiments" where it comes into play are many. Consider wavepackets of the form A(x)+i B(x) corresponding to single particles that are reasonably localized in some region. Let two particles scatter. Then the "observed probability distributions" after they are well-separated will only result from two wavepackets of the form A+iB and C+iD interfering with each other, and after scattering, becoming A'+iB' and C'+iD'. You cannot reproduce the resulting interference effects with generally real waveforms; squaring real prob amplitudes don't reproduce what you can get with squaring complex amplitudes. You can find details about scattering in quantum mechanics in the texts you're using, and try imagine real waveforms in place of the imaginary ones; you should see how using real waveforms would introduce different mixed terms when you obtain squared probability distributions.
     
  14. Nov 9, 2009 #13
    I think one cannot really be convinced by mathematical or physical analogies for that. The fact is that it just works ! and we don't know 'why'.

    Now to see any logic in this is for me an a posteriori reasoning, which is not really deep. The analogy with scattering of waves and interference doesn't certainly explain why the wavefunction (in the case of this 'wavy' behaviour) is interpreted as a distribution function !
    In classical mechanics, the same happens with water or EM waves but the interpretation is quite different.


    I agree. The form of the probability function itself is 'mystery' (why psi^2 ??). This reminds me the work of A. Valentini on non-equilibrium quantum mechanics.. he also gives a bohmian explanation for 'why psi^2?'.
    You can find his papers on spires, and his thesis at the SISSA server.
     
    Last edited: Nov 9, 2009
  15. Nov 9, 2009 #14

    Fra

    User Avatar

    Thanks for the name, I haven't heard of hime before.

    I noted he has done some talks at perimeter, and he seems to be a bohmian?

    I am very far from the hidden variable ideas, however I see a connection between the notion of "hidden information" and "subjective information". The phrasing hidden information, gives the impression that although it's hidden, it's in a certain sense well defined (just hidden). That's where I differ. Although there are similarities, I don't seek to restore realism, on the contrary do I think QM still has too much realism.

    I pretty much replace the "hidden variable realism" with the opposite view of evolving subjective views, and ultimately evolving law. In a way one can argue that the subjective views are in a sense "hidden information", but I disagree that this can be characterized as hidden information in the sense of an objective information measure. I'm think that even the information measure is subjective.

    /Fredrik
     
  16. Nov 9, 2009 #15
    valentini has a maverick career.. because of his non-main-stream interests (foundatioons of QM) he could not get any position after his phd (although his advisor was Sciama) but after some 10 years of being outside academia Smolin gave him a postdoc at perimeter.
    He's now 'somewhere'..

    His theory is interesting.. take a look at his paper about its predicted astrophysical signatures.

    what do you mean by : evolving subjective views ?
    I'm sure you already explained this numerous times here...but bear with me, I cannot read everything.
     
  17. Nov 13, 2009 #16
    Are the vectors complex because of a result of the mathematics? Like when you solve some differential equations you get complex eigenvalues and thus complex eigenvectors? Is it basically an artifact of the mathematics or does it have any kind of physical interpretation?

    That's what I love about QM. The math is so interesting but it's more fun to just think about it and what it implies/might imply.
     
  18. Nov 13, 2009 #17
    Someone was talking about the de Broglie-Bohm/Valentini point of view. Let me expand on this.

    In that theory, particles have trajectories guided by the wave.

    The Hilbert space is complex because the Schroedinger equation is a complex equation and its solutions are complex functions.

    Why is the Schroedinger equation complex? Because we are using a standard mathematical trick to write two real equations as one complex equation. The two real ones are the continuity equation (which keeps the probability distribution of the particles normalized over time) and the quantum Hamilton-Jacobi equation (which describes the dynamics). You can replace the latter with the ordinary classical Hamilton-Jacobi equation if you want to describe classical mechanics with a Schroedinger-like equation.

    So there is nothing fundamental about complex Hilbert spaces, or indeed in the use of a Hilbert space at all. It just turns out to be precisely the right object to provide a compact summary of the statistics of particle trajectories in a de Broglie universe.

    So there you go - a complete answer, no?
     
    Last edited: Nov 13, 2009
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