# 18.6: Wave Functions, Quantum States, Energy Levels, and Degeneracies

We approximate the wavefunction for a molecule by using a product of approximate wavefunctions, each of which models some subset of the motions that the molecule undergoes. In general, the wavefunctions that satisfy the molecule’s Schrödinger equation are degenerate; that is, two or more of these wavefunctions have the same energy. (The one-dimensional particle in a box and the one-dimensional harmonic oscillator have non-degenerate solutions. The rigid-rotor in a plane has doubly degenerate solutions; two wavefunctions have the same energy. The $$J$$-th energy level of the three-dimensional rigid rotor is $$\left(2J+1\right)$$-fold degenerate; there are $$\left(2J+1\right)$$ wavefunctions whose energy is $$E_J$$.) We use doubly subscripted symbols to represent the wavefunctions that satisfy the molecule’s Schrödinger equation. We write $${\psi }_{i,j}$$ to represent all of the molecular wavefunctions whose energy is $${\epsilon }_i$$. We let $$g_i$$ be the number of wavefunctions whose energy is $${\epsilon }_i$$. We say that the energy level $${\epsilon }_i$$ is $$g_i$$-fold degenerate. The wavefunctions

${\psi }_{i,1},\ {\psi }_{i,2},\ \dots ,\ {\psi }_{i,j},\dots ,\ {\psi }_{i,g_i}$

are all solution to the molecule’s Schrödinger equation; we have

$H_{molecule}{\psi }_{i,j}={\epsilon }_i{\psi }_{i,j}$

for $$j=1,\ 2,\ \dots ,\ g_i$$. Every energy level $${\epsilon }_i$$ is associated with $$g_i$$ quantum states. For simplicity, we can think of each of the $$g_i$$ wavefunctions, $${\psi }_{i,j}$$, as a quantum state; however, the molecule’s Schrödinger equation is also satisfied by any set of $$g_i$$ independent linear combinations of the $${\psi }_{i,j}$$. For present purposes, all that matters is that there are $$g_i$$ quantum-mechanical descriptions—quantum states—all of which have energy $${\epsilon }_i$$.