Hey guys Im a little confused with the concept of plane waves and how to perform an expectation value. Let me show you by an example. Suppose you have a wave function of the form
ψp0(x)=f(p0)eiℏp0⋅x
where p0=(0,0,p0) and suppose you want to perform an expectation value of the position of the particle, that is
$
wich I think is nonsense. But if you define an arbitrary momentum vector p′=(p′1,p′2,p′3), and perform the transition probability
⟨ψp′(x)|x|ψp0(x)⟩=f(p′)f(p0)∫d3xxe−iℏ(p′−p0)⋅x=f(p′)f(p0)∫d3x(iℏ∂∂p′x)e−iℏ(p′−p0)⋅x==iℏf(p′)f(p0)∂∂p′xδ3(p′−p0)=−iℏδ3(p′−p0)∂∂p′x(f(p′))f(p0)
where I made use of the property f(x)δ′(x)=−f′(x)δ(x). So now with my new expresion I have a meaningful result and I can evaluate for p′=p0 and get a result that I wasnt able to get with the first method. What I'm doing wrong or the the second way is the correct way to do it? Thanks!
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