Energy Levels and Spectroscopic Notation
Wednesday, March 9, 2022
Previously, we have used three quantum numbers to describe electronic states in the hydrogen atom: , , and . Spin is another quantum number needed to fully describe a state, but it is always for the electron, so including it is not necessary.
Instead, the magnetic quantum number for spin () is needed. Therefore, we need four quantum numbers to describe a state: , , , and .
The ground state of the hydrogen atom was previously labeled as . With , this becomes or . Note: the degeneracy of the ground state is now two. And since each state that previously had degeneracy now has degeneracy due to spin.
Instead of keeping track of the components of the angular momentum vectors of the electron (which do not matter much most of the time), spectroscopic notation is used.
In this system, different letters correspond to different values. See the chart below for designations for different values.
The stands for sharp, the for principal, for diffuse, and for fundamental (these terms were used to describe atomic spectra before atomic theory).
In the ground state, , which is denoted in spectroscopic notation. Transitions between different levels can be determined using the Schrödinger equation (called transition probabilities). It is most common for the to change by during a transition. This restriction is a selection rule, and for atomic transitions the selection rule is