Position wave function of energy eigenstates in 1D box

[jasonchiang97]

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 ψ_n(x) ≡ <x|ψ_n> of the energy eigenstates |ψ_n>. Make sure that your wave functions are normalized, so that <ψ_n|ψ_n> = 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

ψ_n(x) ≡ <x|ψ_n>

I can write the RHS as a sum since energy eigenstates are discrete

<x|ψ_n> = <x|1|ψ_n> = ∑ψ_ni²

I'm not sure if this is the right step but it's all I could come up with.

 

[Chandra Prayaga]

Are you familiar with the Schrödinger equation for such a system?

 

[jasonchiang97]

Are you familiar with the Schrödinger equation for such a system?

Yes, for a mass moving in 1D, the Schrödinger equation gives

Hψ_n(x) + Vψ_n(x) = Eψ_n(x)

H = Hamiltonian

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

ψ_n(x) = Asin(kx)+Bsin(kx)

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

 

[jasonchiang97]

I am unsure of what it means by position wave functions of energy eigenstates. Does it mean Solve for ψ_n(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.

(I assumed you wrote A sin .. + B cos .. ?)