The discussion revolves around the representation of spin-1/2 quantum states within the framework of quantum mechanics, specifically focusing on the relationship between two-dimensional Hilbert spaces for spin and infinite-dimensional Hilbert spaces for position and momentum. Participants explore the implications of tensor products of these spaces and the nature of wave functions in this context.
Participants generally agree on the existence of different Hilbert spaces for spin and position/momentum, but there is disagreement regarding the implications of these spaces for representing quantum states, particularly in cases of entanglement. The discussion remains unresolved regarding the specific nature of bases in the spin-1/2 Hilbert space.
Participants express uncertainty about the definitions and implications of separability in quantum states, as well as the nature of basis types in different Hilbert spaces. There are unresolved questions about the completeness of wave functions in the presence of spin and entanglement.
Yes.cianfa72 said:It should mean that $$(I \otimes \sigma_z)(\vec{x} \otimes I) \ket{\psi} = (\vec{x} \otimes I) (I \otimes \sigma_z) \ket{\psi}, \forall \ket{\psi}$$
Yes.cianfa72 said:We can check that it is true using the definition of tensor product operators acting on a generic tensor product of type ##\ket{\psi} = \ket{\alpha} \otimes \ket {\beta}##.
Yes.cianfa72 said:Since both ##I \otimes \sigma_z## and ##\vec{x} \otimes I## are hermitian and commute each other, they have at least a common eigenbasis -- see also Commuting Operators Have the Same Eigenvectors.
Yes.cianfa72 said:However any eigenbasis of the first operator is an eigenbasis for the second (and the other way around) if and only if the commutating hermitian operators have both nondegerate eigenvalues -- see Common eigenfunctions of commuting operators.