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

• Petro z sela
In summary, the conversation discusses the properties of quantum systems N and X, where N is described by a discrete observable and X by a continuous observable. The physical states of N and X lie in their respective Hilbert spaces and are spanned by their respective basis sets. The composite system NX is considered in a rigged Hilbert space, with physical states lying in this space and being spanned by a basis set of composite states. The operators for NX form a complete set of commuting observables, but there is uncertainty around the exact cardinality of the basis set for the composite states. The conversation then delves into a discussion about the application of these operators to states in the rigged Hilbert space, with considerations for the collapsing of states and the
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 $H_{NX}$.

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Petro z sela said:
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.

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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 $|x\rangle$, 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.

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

## 1. What is a composite system in physics?

A composite system in physics refers to a physical system that is composed of multiple subsystems. These subsystems can interact with each other and form a larger, more complex system. Composite systems are often used to model real-world phenomena and are studied in fields such as quantum mechanics and thermodynamics.

## 2. What is a rigged Hilbert space?

A rigged Hilbert space, also known as a Gelfand triplet, is a mathematical concept used to describe a space of functions that are not necessarily square-integrable, but can be paired with square-integrable functions in a way that preserves the inner product. It is commonly used in quantum mechanics to study states that are not in the Hilbert space but can be approximated by states in the Hilbert space.

## 3. What is a bounded and unbounded operator?

A bounded operator is a linear transformation between two vector spaces that does not "grow too fast". In other words, the operator maps finite values to finite values. On the other hand, an unbounded operator is a linear transformation that can map finite values to infinite values. This concept is important in functional analysis and is used to study the properties of operators in various mathematical models.

## 4. What is a CSCO in quantum mechanics?

CSCO stands for "complete set of commuting observables" and refers to a set of operators that commute with each other, meaning that they can be measured simultaneously without affecting each other's results. In quantum mechanics, these observables correspond to physical quantities such as position, momentum, and energy. They are used to fully describe the state of a quantum system and predict the outcomes of measurements.

## 5. What is the domain of an operator?

The domain of an operator refers to the set of all vectors in a vector space on which the operator is defined and can act. In other words, it is the set of all possible inputs for the operator. In functional analysis, the domain of an operator is often restricted to a specific subset of the vector space to ensure that the operator is well-defined and has certain properties, such as being bounded.

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