Consider a particle which is confined in a one-dimensional box of size L, so that the position space wave function ψ(x) has to vanish at x = 0 and x = L. The energy operator is H = p2/2m + V (x), where the potential is V (x) = 0 for 0 < x < L, and V (x) = ∞ otherwise.
Find the position-space wave functions ψn(x) ≡ <x|ψn> of the energy eigenstates |ψn>. Make sure that your wave functions are normalized, so that <ψn|ψn> = 1
<x'|x|ψ> = x'<x'|ψ>
The Attempt at a Solution
I'm not 100% sure what the question is asking for but I'm guessing it's asking me to find the RHS of
ψn(x) ≡ <x|ψn>
I can write the RHS as a sum since energy eigenstates are discrete
<x|ψn> = <x|1|ψn> = ∑ψni2
I'm not sure if this is the right step but it's all I could come up with.