# Energy Levels and Spectroscopic Notation

Wednesday, March 9, 2022

## Quantum Numbers

Previously, we have used three quantum numbers to describe electronic states in the hydrogen atom: $n$, $l$, and $m_l$. Spin is another quantum number needed to fully describe a state, but it is always $1/2$ for the electron, so including it is not necessary.

Instead, the magnetic quantum number for spin ($m_s$) is needed. Therefore, we need four quantum numbers to describe a state: $n$, $l$, $m_l$, and $m_s$.

### Ground state

The ground state of the hydrogen atom was previously labeled as $\left(n,l,m_l\right)=\left(1,0,0\right)$. With $m_s$, this becomes $\left(n,l,m_l,m_s\right)=\left(1,0,0,+1/2\right)$ or $\left(n,l,m_l,m_s\right)=\left(1,0,0,-1/2\right)$. Note: the degeneracy of the ground state is now two. And since each state that previously had degeneracy $n^2$ now has degeneracy $2n^2$ due to spin.

## Spectroscopic Notation

Instead of keeping track of the $z$ 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 $l$ values. See the chart below for designations for different $l$ values.

Value of $l$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|---|

Designation | $s$ | $p$ | $d$ | $f$ | $g$ | $h$ | $i$ |

The $s$ stands for *sharp*, the $p$ for *principal*, $d$ for *diffuse*, and $f$ for *fundamental* (these terms were used to describe atomic spectra before atomic theory).

In the ground state, $n=1$, which is denoted $1s$ in spectroscopic notation. Transitions between different levels can be determined using the Schrödinger equation (called *transition probabilities*). It is most common for the $l$ to change by $1$ during a transition. This restriction is a *selection rule*, and for atomic transitions the selection rule is

$\Delta l=\pm 1$