- #1

- 21

- 0

## 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 ∇

^{2}ψ(x,y,z)=-k

^{2}ψ(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(kxx+kyy+kzz)}. 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!

Last edited: