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

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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

\[{\psi }_{i,1},\ {\psi }_{i,2},\ \dots ,\ {\psi }_{i,j},\dots ,\ {\psi }_{i,g_i} \nonumber \]

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

\[H_{molecule}{\psi }_{i,j}={\epsilon }_i{\psi }_{i,j} \nonumber \]

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 \({\epsilon }_i\).