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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

$\Delta l=\pm 1$