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Composite system, rigged Hilbert space, bounded unbounded operator, CSCO, domain

  1. Jan 31, 2013 #1
    Is something wrong in my assertions below?

    Suppose we have two quantum systems [itex]N[/itex] and [itex]X[/itex]. Let [itex]N[/itex] is described by discrete observable [itex]\hat{n}[/itex] (bounded s.a. operator with discrete infinite spectrum) with eigenvectors [itex]|n\rangle[/itex]. Let [itex]X[/itex] is described by continuous observable [itex]\hat{x}[/itex] (unbounded s.a. operator with continuous spectrum) with generalized eigenvectors [itex]|x\rangle[/itex]. Then:
    1. physical states of [itex]N[/itex] lie in Hilbert space [itex]H_{N}[/itex];
    2. [itex]H_{N}[/itex] is spanned by [itex]|n\rangle[/itex];
    3. [itex]|n\rangle[/itex] lie [itex]H_{N}[/itex];
    4. basis set [itex]|n\rangle[/itex] has cardinality aleph-null (countable);
    5. system [itex]X[/itex] is considered in rigged Hilbert space [itex]Ω_{X}\subset H_{X}\subset Ω^{\times}_{X}[/itex];
    6. physical states of [itex]X[/itex] lie in [itex]Ω_{X}[/itex];
    7. [itex]Ω_{X}[/itex] is spanned by [itex]|x\rangle[/itex];
    8. basis set [itex]|x\rangle[/itex] has cardinality aleph-one (uncountable);
    9. [itex]|x\rangle[/itex] lie in [itex]Ω^{\times}_{X}\backslash H_{X}[/itex];
    10. the complete set of commuting observables (CSCO) for composite system [itex]NX[/itex] is [itex]\hat{n}[/itex], [itex]\hat{x}[/itex];
    11. composite system [itex]NX[/itex] is considered in rigged Hilbert space [itex]Ω_{NX}\subset H_{NX}\subset Ω^{\times}_{NX}[/itex];
    12. [itex]H_{NX}=H_{N}\otimes H_{X}[/itex] (tensor product);
    13. physical states of [itex]NX[/itex] lie in [itex]Ω_{NX}[/itex];
    14. [itex]Ω_{NX}[/itex] is spanned by [itex]|n,x\rangle[/itex] ( [itex]|n,x\rangle = |n\rangle\otimes|x\rangle[/itex] );
    15. basis set [itex]|n,x\rangle[/itex] has cardinality aleph-one (uncountable);
    16. [itex]|n,x\rangle[/itex] lie in [itex]Ω^{\times}_{NX}\backslash H_{NX}[/itex];
    17. operator [itex]\hat{X}=\hat{1}\otimes\hat{x}[/itex] is unbounded;
    18. [itex]\hat{X}[/itex] has domain [itex]Ω_{NX}[/itex] and maps [itex]Ω_{NX}[/itex] into [itex]Ω_{NX}[/itex];
    19. operator [itex]\hat{N}=\hat{n}\otimes\hat{1}[/itex] is bounded;
    20. [itex]\hat{N}[/itex] has domain [itex]H_{NX}[/itex] and maps [itex]H_{NX}[/itex] into [itex]H_{NX}[/itex];
    21. Suppose [itex]NX[/itex] is in the some state [itex]ψ\inΩ_{NX}[/itex]. One has measured observables [itex]\hat{X}[/itex] and/or [itex]\hat{N}[/itex] in state [itex]ψ[/itex]. After this procedure [itex]ψ[/itex] collapses to vector from [itex]|n,x\rangle[/itex] set, this vector [itex]\notinΩ_{NX}[/itex]. It implies that 6 and 13 must be reformulated: physical states lie in [itex]Ω_{NX}[/itex] and in some subspace of [itex]Ω^{\times}_{NX}\backslash H_{NX}[/itex] ( subspace of generalized eigenvectors).
    22. And what about [itex]H_{NX}\backslash Ω_{NX}[/itex] ? I can’t apply [itex]\hat{X}[/itex] to vector [itex]\varphi[/itex] from [itex]H_{NX}\backslash Ω_{NX}[/itex], because [itex]||\hat{X}\varphi||\rightarrow∞[/itex] and [itex]\hat{X}\varphi\inΩ^{\times}_{NX}\backslash H_{NX}[/itex] , i.e. I can’t measure observable [itex]\hat{X}[/itex] in states [itex]H_{NX}\backslash Ω_{NX}[/itex]. But I can apply [itex]\hat{N}[/itex] to [itex]\varphi[/itex], because [itex]\hat{N}\varphi\in H_{NX}[/itex] and [itex]||\hat{N}\varphi||<∞[/itex] , i.e. I can measure observable [itex]\hat{N}[/itex] in states [itex]H_{NX}\backslash Ω_{NX}[/itex]. But [itex]\hat{N}[/itex] and [itex]\hat{X}[/itex] form CSCO and they can be measured simultaneously. What can I do in this case? I can decrease domain of [itex]\hat{N}[/itex] from [itex]H_{NX}[/itex] to [itex]Ω_{NX}[/itex]. Thus [itex]\hat{N}[/itex] and [itex]\hat{X}[/itex] will have common domain, but for [itex]\hat{N}[/itex] this domain is not invariant, because in general case [itex]\hat{N}[/itex] maps [itex]Ω_{NX}[/itex] into [itex]H_{NX}[/itex].
     
    Last edited: Jan 31, 2013
  2. jcsd
  3. Jan 31, 2013 #2

    dextercioby

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    My comments are markes with caps lock in the quote.

    21 & 22: My comment. I don't believe in von Neumann's measurement => collapse postulate. I don't know of a formulation of von Neumann's postulate for distribution spaces.
     
    Last edited: Feb 1, 2013
  4. Jan 31, 2013 #3

    rubi

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    8, 15: This is actually neither true nor false. It's equivalent to the continnuum hypothesis, which is undecidable in ZFC. A true statement would be that the cardinality is beth-one.
    9: This is true in my opinion.

    21: There are two ways out:
    a) You never really measure an exact value for a continuous variable and thus the state will not collapse to [itex]|x\rangle[/itex], but rather to something like [itex]\int e^{-\frac{(y-x)^2}{\sigma^2}}|y\rangle dy[/itex].
    b) You are only in a distributional state for an infinitesimal amount of time, since the diffusive nature of the Schrödinger equation will quickly evolve the state into a physical state. This might not apply for every possible Hamiltonian however.

    22: The states in [itex]H_{NX}\backslash\Omega_{NX}[/itex] are not physical. They correspond to things like infinite energy. A physical state must always give you finite values for all measurements. The fact that you can apply operators to a state doesn't imply that this state can be realized physically. The situation you described corresponds to something like a particle with infinite energy and spin up. You can easily write down a wavefunction for this situation, but that doesn't mean that this is realized in nature.
     
  5. Feb 1, 2013 #4

    dextercioby

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    9 is true, of course. I've corrected the point above.
     
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