- #1

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- Homework Statement
- Consider a Hilbert space H that consists of all functions ψ(x) such that

##\int_{-\infty}^{\infty} |\psi(x)|^2, dx##

is finite. Show that there are functions in H for which ˆxψ(x) ≡ xψ(x) is not in H.

That is to say, there are functions in H that are taken out of H when acted upon by

the position operator (or equivalently, the position operator does not preserve H)

- Relevant Equations
- ##\int_-\infty^\infty |ψ(x)|^2\, dx##

ˆxψ(x) ≡ xψ(x)

The attempt

##\int_{-\infty}^{\infty} |ψ^*(x)\, \hat x\,\psi(x)|\, dxˆ##

Using ˆxψ(x) ≡ xψ(x)

=##\int_{-\infty}^{\infty} |ψ^*(x)\,x\,\psi(x)|\, dxˆ##

=##\int_{-\infty}^{\infty} |ψ^*(x)\,\psi(x)\,x|\, dxˆ##

=##\int_{-\infty}^{\infty} |x\,ψ^2(x)|\, dxˆ##

I'm pretty sure this is not the correct approach. ψ(x) is a stationary right? Is it safe to attempt this using the definition of a stationary state? I apologize if I'm completely wrong, I've read the text and lecture notes but am still having a lot of trouble understanding this

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