Applying the Spherical Bessel and Neumann Functions to a Free Particle
Applying the Radial Equation Inside the Square Well
Finding the x, y, and z Equations for Three-Dimensional Free Particle Problems

How to Work with a Cubic Potential

In quantum physics, when working with a box potential, you can make things simpler by assuming that the box is actually a cube. In other words, L = Lx = Ly = Lz. When the box is a cube, the equation for the energy becomes

image0.png

So, for example, the energy of the ground state, where nx = ny = nz = 1, is given by the following, where E111 is the ground state:

image1.png

Note that there's some degeneracy in the energies; for example, note that

image2.png

So E211 = E121 = E112, which means that the first excited state is threefold degenerate, matching the threefold equivalence in dimensions.

In general, when you have symmetry built into the physical layout (as you do when L = Lx = Ly = Lz), you have degeneracy.

The wave function for a cubic potential is also easier to manage than the wave function for a general box potential (where the sides aren't of the same length). Here's the wave function for a cubic potential:

image3.png

So, for example, here's the wave function for the ground state

image4.png
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When to Use Spherical Coordinates Instead of Rectangular Coordinates
Determining the Angular Part of a Wave Function
How to Keep a Function of r Finite as r Goes to Infinity
How to Calculate the Spin of an Electron in a Hydrogen Atom
Combine the Solutions for Small r and Large r in the Schrödinger Equation
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