04 - Degeneracy

$d$-fold Degenerate Energy Level

Energy Level $E$ is $d$-fold degenerate if

Space of Solutions to Stationary State Schrödinger Equation with energy $E$ has dimension $d$
where $d>1$

Non-Degenerate Energy Level

Energy Level $E$ is non degenerate if

Space of Solutions to Stationary State Schrödinger Equation with energy $E$ has dimension $1$

Finding Degeneracy of an Energy Level

Generally is the number of different ways to pick $n_1,n_2,\cdots ,n_k$ for a specific value of $E$
where $n_1,n_2,n_k\in {\mathbb{Z}}^{>0‘}$

Particle in a Three-Dimensional Box

As

$$E_{n_1,n_2,n_3}=\frac{{\pi }^2{\hslash }^2}{2m}\left(\frac{n_1^2}{a^2}+\frac{n_2^2}{b^2}+\frac{n_3^2}{c^2}\right)$$

So for $E=\frac{6{\pi }^2{\hslash }^2}{2m{a}^2}$ then

It is $3$-fold degenerate as it has solutions

$$(n_1,n_2,n_3)=(2,1,1)\text{ or }(1,2,1)\text{ or }(1,1,2)$$

Degeneracy Energy Levels with Full Schrödinger Equation

Suppose energy level $E_n$ is $d_n$-fold degenerate (with $d_n\ge 1$) then

Full schrödinger equation can be written as

$$\Psi (\mathbf{x},t)=\sum _n\sum _{i=1}^{d_n}{c}_{n,i}{\psi }_{n,i}(\mathbf{x})e^{-i{E}_{n}t/\hslash }$$

with ${\varphi }_{n,i}(\mathbf{x})$ are linearly independent stationary states of energy $E_n,i=1,\cdots ,d_n$