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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: nn, ll, and mlm_l. Spin is another quantum number needed to fully describe a state, but it is always 1/21/2 for the electron, so including it is not necessary.

Instead, the magnetic quantum number for spin (msm_s) is needed. Therefore, we need four quantum numbers to describe a state: nn, ll, mlm_l, and msm_s.

Ground state

The ground state of the hydrogen atom was previously labeled as (n,l,ml)=(1,0,0)\left(n,l,m_l\right)=\left(1,0,0\right). With msm_s, this becomes (n,l,ml,ms)=(1,0,0,+1/2)\left(n,l,m_l,m_s\right)=\left(1,0,0,+1/2\right) or (n,l,ml,ms)=(1,0,0,1/2)\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 n2n^2 now has degeneracy 2n22n^2 due to spin.

Spectroscopic Notation

Instead of keeping track of the zz 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 ll values. See the chart below for designations for different ll values.

Value of ll 00 11 22 33 44 55 66
Designation ss pp dd ff gg hh ii

The ss stands for sharp, the pp for principal, dd for diffuse, and ff for fundamental (these terms were used to describe atomic spectra before atomic theory).

In the ground state, n=1n=1, which is denoted 1s1s in spectroscopic notation. Transitions between different levels can be determined using the Schrödinger equation (called transition probabilities). It is most common for the ll to change by 11 during a transition. This restriction is a selection rule, and for atomic transitions the selection rule is

Δl=±1\Delta l=\pm 1