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76E

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Found in: Page 344

### Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

# A hydrogen atom is subjected to a magnetic field B strong 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?

5 levels,

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

See the step by step solution

## Step 1: total energy of magnetic field

In the case of a strong magnetic field B such as this one, the total energy E is given by

${\mathbf{E}}{\mathbf{=}}\frac{\mathbf{e}\overline{)\mathbf{h}}\mathbf{B}}{\mathbf{2}\mathbf{m}}\left({m}_{l}+{m}_{s}\right){\mathbf{+}}{{\mathbf{E}}}_{{\mathbf{o}}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}\left(1\right)$

where Eo is the zero-field energy data-custom-editor="chemistry" $\left(B=0\right)$.

## Step 2: find level splits

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

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

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

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 ${\mathrm{m}}_{\mathrm{l}}+2{\mathrm{m}}_{\mathrm{s}}=2$.

## Step 3: energy difference ∆E

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

$∆\mathrm{E}=\frac{\mathrm{e}\overline{)\mathrm{h}}\mathrm{B}}{2\mathrm{m}}\left(2-1\right)\phantom{\rule{0ex}{0ex}}∆\mathrm{E}=\frac{\mathrm{e}\overline{)\mathrm{h}}\mathrm{B}}{2\mathrm{m}}$