Working with Three-Dimensional Rectangular Potentials

This article takes a look at a 3D potential that forms a box, as you see in the following figure. You want to get the wave functions and the energy levels here.

A box potential in 3D.
A box potential in 3D.

Inside the box, say that V(x, y, z) = 0, and outside the box, say that

image1.png

So you have the following:

image2.png

Dividing V(x, y, z) into Vx(x), Vy(y), and Vz(z) gives you

image3.png

Okay, because the potential goes to infinity at the walls of the box, the wave function,

image4.png

must go to zero at the walls, so that's your constraint. In 3D, the Schrödinger equation looks like this in three dimensions:

image5.png

Writing this out gives you the following:

image6.png

Take this dimension by dimension. Because the potential is separable, you can write

image7.png

Inside the box, the potential equals zero, so the Schrödinger equation looks like this for x, y, and z:

image8.png

The next step is to rewrite these equations in terms of the wave number, k. Because

image9.png

you can write the Schrödinger equations for x, y, and z as the following equations:

image10.png

Start by taking a look at the equation for x. Now you have something to work with — a second order differential equation,

image11.png

Here are the two independent solutions to this equation, where A and B are yet to be determined:

image12.png

So the general solution of

image13.png

is the sum of the last two equations:

image14.png
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