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

[Petro z sela]

Is something wrong in my assertions below?

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

 

[dextercioby]

Is something wrong in my assertions below?

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

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.

 

[rubi]

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 |x\rangle, but rather to something like \int e^{-\frac{(y-x)^2}{\sigma^2}}|y\rangle dy.
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 H_{NX}\backslash\Omega_{NX} 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.

 

[dextercioby]

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