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I understand that if a given potential well, U(x), is symmetric about x = L, then the expectation value for operator [x] would be <x> = L. (I am not even entirely sure why this is, guessing that the region where x<L and x>L are equally probable)

Is it possible to draw conclusion about expectation values of other operators?

Heres one example:

Given ψ(x) = sin(nπx/2L) for a infinite potential well with barriers at x=0 and x=2L,

Using symmetry arguments or otherwise, explain why or show that the expectation value of the particle momentum in this infinite well is <p> = 0

I got <p> = 0 by applying the momentum operator [p]ψ and evaluated it at the point of symmetry x=L, which gave [p]ψ = 0 (at x=L). => <p> = ∫ψ*[p]ψ = 0

Is that correct? Could someone explain to me what really is the "symmetry argument"?

Cheers