Question

For a particle in a three-dimensional cubical box, what is the degeneracy (number of different quantum states with the same energy) of the energy levels (a)$\mathbf{3}{\mathbf{\pi }}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}\mathbf{/}\mathbf{2}{\mathbf{mL}}^{\mathbf{2}}$and (b)$\mathbf{9}{\mathbf{\pi }}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}\mathbf{/}\mathbf{2}{\mathbf{mL}}^{\mathbf{2}}$?

Short answer

The answer is not given in the document.

Step-by-step solution

Degeneracy

The term "degeneracy" refers to the existence of two or more distinct quantum states with the same energy.

In a cubical box with side L, the allowed energies for a particle of mass m are given by;

As a result, the degenerate states have the samevalue.

Degeneracy of the energy levels at 3π2h2/2mL2

(a) As, the given energy level;

On comparing the equations;

This is unique to the combination

Hence, this energy level has degeneracy=1.

Degeneracy of the energy levels at 9π2h2/2mL2

(b) As, the given energy level;

On comparing the equations;

This is unique to the combination

Hence, this energy level has degeneracy=3.