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76E

Expert-verifiedFound in: Page 344

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 $\xe2\u02c6\u2020\mathrm{E}=\frac{\mathrm{e}\overline{)\mathrm{h}}\mathrm{B}}{2\mathrm{m}}$

**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 E _{o}**

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 $\xc2\pm \frac{1}{2}$, 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 ${\mathrm{m}}_{\mathrm{l}}+2{\mathrm{m}}_{\mathrm{s}}=2$.

The energy difference $\xe2\u02c6\u2020\mathrm{E}$ is then (we use Eq. (l))

$\xe2\u02c6\u2020\mathrm{E}=\frac{\mathrm{e}\overline{)\mathrm{h}}\mathrm{B}}{2\mathrm{m}}\left(2-1\right)\phantom{\rule{0ex}{0ex}}\xe2\u02c6\u2020\mathrm{E}=\frac{\mathrm{e}\overline{)\mathrm{h}}\mathrm{B}}{2\mathrm{m}}$

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