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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].

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

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