Solve 3D TISE in Potential Well: Eigenvals/Fns & Normalize

[obnoxiousris]

Homework Statement

A particle in a 3D cubical potential well. The walls are L_x, L_y, L_z long.
Inside the well, V(x,y,z)=0 when 0<x<L_x, 0<y<L_y, 0<z<L_z. V= ∞ elsewhere.
Solve the TISE to find the eigenfunctions and eigenvalues of this potential. And to normalise the wavefunctions.

(Hint: look for separable solutions in the form ψ(x,y,z)=X(x)Y(y)Z(z))

Homework Equations

(-hbar/2m)(∇²)ψ(x,y,z)+V(x,y,z)ψ(x,y,z)=Eψ(x,y,z)

The Attempt at a Solution

I'm confused about which equation to use for this question. I know that for a potential well, the V terms are zero, and i can write -hbar/2m and E in terms of k, so the equation reduces to ∇²ψ(x,y,z)=-k²ψ(x,y,z).

we did the one dimensional well in class. my teacher used ψ(x)=Asin(kx)+Bsin(kx) as a general solution, then subbed in the boundary conditions like when x=L, ψ(x)=0 and so on.

However, I know that ψ(x)=Ae^ikx is also a general solution to the differential equation. And this is used for free particles. I did some research online, and apparently I can treat a particle in the well as a free particle.

So i don't know which equation i should use, and what are the differences between the two equations? Some sites I've seen used the trig, some used the complex.

I tried to use the complex one, I got (after normalisation) ψ(x,y,z)=(1/sqrt3L)e^{i(k_xx+k_yy+k_zz)}. Then I tried to sub in the boundary conditions, only to realize no matter what i did i couldn't make ψ=0.
So I tried to use the trig, but the equation wasn't separable.

I really don't know what to do here, some help would be appreciated! Thanks!

 

[clamtrox]

There are two independent solutions, e^ikx and e^-ikx. You will need both in order to satisfy your boundary conditions.