Consider the (eight) $\mathbf{n}\mathbf{=}\mathbf{2}\mathbf{}\mathbf{states}\mathbf{,}\mathbf{}\mathbf{|}\mathbf{2}{\mathbf{lm}}_{\mathbf{l}}{\mathbf{m}}_{\mathbf{s}}\mathbf{⟩}\mathbf{.}$Find the energy of each state, under strong-field Zeeman splitting. Express each answer as the sum of three terms: the Bohr energy, the fine-structure (proportional to${\mathbf{a}}^{\mathbf{2}}$), and the Zeeman contribution (proportional to${\mathbf{\mu }}_{\mathbf{B}}{\mathbf{B}}_{\mathbf{ext}}\mathbf{.}$). If you ignore fine structure altogether, how many distinct levels are there, and what are their degeneracies?

When radiation (such as light) originates in a magnetic field, Zeeman splitting occurs, which is the splitting of a single spectral line into two or more lines of different frequencies.

For $\mathrm{n}=2$, there are eight different states $|2{\mathrm{lm}}_{\mathrm{l}}{\mathrm{m}}_{\mathrm{s}}⟩$:

$$\begin{gathered} \mathrm{i})\mathrm{l}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{ii})\mathrm{l}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=\frac{1}{2}, \\ \mathrm{iii})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=-1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{iv})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=-1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=\frac{1}{2}, \\ \mathrm{v})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{vi})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=\frac{1}{2}, \\ \mathrm{vii})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{viii})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=1/2 \\ \mathrm{i})\mathrm{l}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{ii})\mathrm{l}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=\frac{1}{2}, \\ \mathrm{iii})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=-1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{iv})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=-1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=\frac{1}{2}, \\ \mathrm{v})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{vi})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=\frac{1}{2}, \\ \mathrm{vii})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=-\frac{1}{2}, \\ \mathrm{viii})\mathrm{l}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{l}}=1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mathrm{m}}_{\mathrm{s}}=\frac{1}{2} \end{gathered}$$

Total energy is equal to:

$\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}({\mathrm{m}}_{\mathrm{l}}+2{\mathrm{m}}_{\mathrm{s}})+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-\frac{\mathrm{l}(\mathrm{l}+1)-{\mathrm{m}}_{\mathrm{l}}{\mathrm{m}}_{\mathrm{s}}}{\mathrm{l}\left(\mathrm{l}+\frac{1}{2}\right)(\mathrm{l}+1)}\right]$

role="math" localid="1656064856692" $$\begin{gathered} \mathrm{i})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}-{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-1\right]. \\ \mathrm{ii})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-1\right]. \\ \mathrm{iii})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}-2{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-\frac{3}{2}\right]. \\ \mathrm{iv})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-\frac{5}{6}\right]. \\ \mathrm{v})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}-{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-\frac{2}{3}\right]. \\ \mathrm{vi})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-\frac{2}{3}\right]. \\ \mathrm{vii})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+2{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-\frac{5}{6}\right]. \\ \mathrm{viii})\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+\frac{13.6\mathrm{eV}}{{\mathrm{n}}^3}{\mathrm{\alpha }}^2\left[\frac{3}{4\mathrm{n}}-\frac{1}{2}\right]. \end{gathered}$$

If fine structure is ignored,$\mathrm{a}=0$.

Therefore, the following energies:

$\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}-{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}$Degree of degeneracy: 2

$\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}$Degree of degeneracy: 2

$\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}$Degree of degeneracy: 2

$\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}-2{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}$Degree of degeneracy: 1

$\mathrm{E}=-\frac{13.6\mathrm{eV}}{{\mathrm{n}}^2}+2{\mathrm{\mu }}_{\mathrm{B}}{\mathrm{B}}_{\mathrm{ext}}$Degree of degeneracy: 1

Now only there are five distinct energies instead of eight.