 #1
Moolisa
 20
 5
 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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