What is Degeneracy of Wave Functions in Quantum Mechanics?
Definition
In quantum mechanics, degeneracy refers to the condition where two different (i.e., linearly independent) wave functions have the same eigenvalue.
Explanation
In simpler terms, when two wave functions are degenerate, it means that their energies are equal. Mathematically, this implies that the geometric multiplicity of the eigenvalue is 2 or more.
In Griffiths’ textbook, this is referred to as overlapping or overlapped states. According to the translator, before the term ‘overlapping’ was used in the 1995 terminology dictionary published by the Korean Physical Society, it was called degeneracy. However, I have never seen the term “overlapped states” being used, and since the term “overlapping” overlaps with everyday language and is not very practical for searches, I think using “degeneracy” is better.
Let the two eigenfunctions of an arbitrary operator $A$ be $\psi_{1}$ and $\psi_2$, and their eigenvalue be $a$. The expression that $\psi_{1}$ and $\psi_2$ are degenerate in formula is as follows.
$$ A\psi_{1}=a\psi_{1} $$
$$ A\psi_2=a\psi_2 $$
Here, if the operator $A$ is the Hamiltonian, the energy operator $H$, it implies that the energies (eigenvalues) of the two wave functions are the same, and based on the energy alone, it is impossible to distinguish between the two wave functions (eigenfunctions). It’s similar to being unable to distinguish two people named Kim Cheolsu just by looking at their names.
Example
If we solve the eigenvalue equation for the two wave functions $\psi_{1}=e^{ikx}$ and $\psi_2=e^{-ikx}$,
$$ \begin{align*} H\psi_{1} &= E_{1}\psi_{1} \\ &= -\frac{\hbar^2}{2m} \frac{\partial ^2}{\partial x^2} e^{ikx} \\ &= -\frac{\hbar^2}{2m}(ik)^2 e^{ikx} \\ &= \frac{\hbar^2 k^2}{2m}\psi_{1} \end{align*} $$
$$ \begin{align*} H\psi_2 &= E_2\psi_2 \\ &= -\dfrac{\hbar^2}{2m}\dfrac{\partial ^2}{\partial x^2} e^{-ikx} \\ &= -\frac{\hbar^2}{2m}(-ik)^2 e^{ikx} \\ &= \frac{\hbar^2 k^2}{2m}\psi_2 \end{align*} $$
Since the energies of the two wave functions are equal, they are degenerate. Therefore, solving the eigenvalue equation alone is not sufficient to distinguish between the two wave functions.
$$ E_{1}=\dfrac{\hbar^2 k^2}{2m}=E_2 $$