In Shankar's Principle of Quantum Mechanics, Section 10.1, part The Direct Product Revisited (he calls tensor products direct products), he attempts to show that a two-particle state space is the tensor product of two one-particle state spaces. He begins by letting Ω(1)1 be an operator on the state space V1 of a particle in one dimension, whose nondegenerate eigenfunctions ψω1(x1) form a complete basis, and similarly, letting ψω2(x2) form a basis for the state space of a second particle. He then states that if a function ψ(x1,x2) that represents an abstract vector |ψ⟩ from the state space V1⊗2 of a system consisting of both particles has x1 fixed at some value ˉx1, then ψ becomes a function of x2 alone and may be expanded as ψ(ˉx1,x2)=∑ω2Cω2(ˉx1)ψω2(x2)
where
Cω2(ˉx1)=∑ω1Cω1ω2ψω1(ˉx1)
Substituting Equation
(2) into Equation
(1) and dropping the bar on
ˉx1, he states that the resulting expansion
ψ(x1,x2)=∑ω1∑ω2Cω1ω2ψω1(x1)ψω2(x2)
imply that
V1⊗2=V1⊗V2 for
ψω1(x1)×ψω2(x2) is the same as the inner product between
|x1⟩⊗|x2⟩ (
|x1⟩ and
|x2⟩ are the position basis vectors of
V1 and
V2 respectively) and
|ω1⟩⊗|ω2⟩ (
|ω1⟩ and
|ω2⟩ are the basis eigenvectors of
Ω1 on
V1 and
Ω2 on
V2 respectively).
Question: How does Equation (1) follow from fixing x1 in ψ(x1,x2) as ˉx1? Using the simultaneous eigenbasis |ω1ω2⟩ of the operators Ω1 and Ω2 on V1⊗2, ψ(ˉx1,x2)=⟨ˉx1x2|ψ⟩=∑ω1∑ω2⟨ω1ω2|ψ⟩⟨ˉx1x2|ω1ω2⟩=∑ω1∑ω2⟨ω1ω2|ψ⟩ψω1ω2(ˉx1,x2).
If the author intends
Cω1ω2 to mean
⟨ω1ω2|ψ⟩, what is the reason (besides the tensor product since he is trying to show that
V1⊗2=V1⊗V2) for
ψω1ω2(ˉx1,x2)=ψω1(ˉx1)ψω2(x2)?