Question

A hydrogen atom is subjected to a magnetic field Bstrong enough to completely overwhelm the spin-orbit coupling. Into how many levels would the 2p level split, and what would be the spacing between them?

Short answer

5 levels,

And energy difference is$∆\mathrm{E}=\frac{\mathrm{e}\sout{\mathrm{h}}\mathrm{B}}{2\mathrm{m}}$

Step-by-step solution

total energy of magnetic field

In the case of a strong magnetic field Bsuch as this one, the total energy Eis given by

$\mathbf{E}\mathbf{=}\frac{\mathbf{e}\sout{\mathbf{h}}\mathbf{B}}{\mathbf{2}\mathbf{m}}\left(m_l+m_s\right)\mathbf{+}{\mathbf{E}}_{\mathbf{o}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$

where E_ois the zero-field energydata-custom-editor="chemistry" $\left(B=0\right)$.

find level splits

The possible values for m_l are$\left(\mathrm{l}=1\right)$

${\mathrm{m}}_{\mathrm{l}}=-1,0,1.............\left(2\right)$

Since m_s can have values of $\pm \frac{1}{2}$, we conclude that the 2_p level splits into $3+2=5$ levels, since there are 3 possible values for m_l and 2 possible values for m_s.

The quantity${\mathrm{m}}_{\mathrm{l}}+2{\mathrm{m}}_{\mathrm{s}}$ can have values of

role="math" localid="1658468430759" ${\mathrm{m}}_{\mathrm{l}}+2{\mathrm{m}}_{\mathrm{s}}=-2,-1,0,1,2............\left(3\right)$

For example, let us take the following two levels:${\mathrm{m}}_{\mathrm{l}}+2{\mathrm{m}}_{\mathrm{s}}=1$ and ${\mathrm{m}}_{\mathrm{l}}+2{\mathrm{m}}_{\mathrm{s}}=2$.

energy difference ∆E

The energy difference$∆\mathrm{E}$ is then (we use Eq. (l))

$$\begin{gathered} ∆\mathrm{E}=\frac{\mathrm{e}\sout{\mathrm{h}}\mathrm{B}}{2\mathrm{m}}\left(2-1\right) \\ ∆\mathrm{E}=\frac{\mathrm{e}\sout{\mathrm{h}}\mathrm{B}}{2\mathrm{m}} \end{gathered}$$