# Position wave function of energy eigenstates in 1D box

## Homework Statement

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 <ψnn> = 1

## Homework Equations

maybe

<x'|x|ψ> = x'<x'|ψ>
1 =

## 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.

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Chandra Prayaga
Are you familiar with the Schrodinger equation for such a system?

Are you familiar with the Schrodinger equation for such a system?
Yes, for a mass moving in 1D, the Schrodinger equation gives

n(x) + Vψn(x) = Eψn(x)

H = Hamiltonian

So if I solve the equation for a 1D box I would get something like

ψn(x) = Asin(kx)+Bsin(kx)

Do I then solve for k,A, and B?

I am unsure of what it means by position wave functions of energy eigenstates. Does it mean Solve for ψn(x) for the definite levels of energy levels?

BvU
Yes. You have the solution, now apply the boundary conditions and normalization. Only specific $k$ do the trick and they can be numbered from n=0 to infinity.