91影视

The average total magnetic dipole moment${\mathbf{渭}}_{{\mathbf{j}}_{\mathbf{avg}}}$is given by

${\mathit{渭}}_{{\mathbf{j}}_{\mathbf{a}\mathbf{v}\mathbf{g}}}\mathbf{=}\frac{\mathbf{e}}{\mathbf{2}\mathbf{m}}\frac{{\mathbf{L}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{S}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\stackrel{\mathbf{鈫�}}{\mathbf{L}}\mathbf{-}\stackrel{\mathbf{鈫�}}{\mathbf{S}}}{\mathbf{J}}$ 鈥�(1)

The square of the total angular momentum${\mathbf{J}}^{\mathbf{2}}$is given by

${\mathit{J}}^{\mathbf{2}}\mathbf{=}{\mathit{L}}^{\mathbf{2}}\mathbf{+}{\mathit{S}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\stackrel{\mathbf{鈫�}}{\mathbf{L}}\mathbf{-}\stackrel{\mathbf{鈫�}}{\mathbf{S}}$ 鈥�(2)

The magnitudes of the angular momenta J,Land Sare given by

$\mathit{J}\mathbf{=}\sqrt{\mathbf{j}\mathbf{(}\mathbf{j}\mathbf{+}\mathbf{1}\mathbf{)}}\sout{\mathbf{h}}$ 鈥�(3)

$\mathit{L}\mathbf{=}\sqrt{\mathbf{l}\mathbf{(}\mathbf{l}\mathbf{+}\mathbf{1}\mathbf{)}}\sout{\mathbf{h}}$ 鈥�(4)

$\mathit{S}\mathbf{=}\sqrt{\mathit{s}\mathbf{(}\mathit{s}\mathbf{+}\mathbf{1}\mathbf{)}}\sout{\mathbf{h}}$ 鈥�(5)

From Eq. (2) we can write

$i-\stackrel{鈫�}{S}=\frac{1}{2}\left(J^2-L^2-S^2\right)$ 鈥�(6)

By plugging in Eq. (3) into Eq. (1) we get

$$\begin{gathered} {渭}_{j_{avg}}=\frac{e}{2m}\left(\frac{L^2+2S^2+{\displaystyle \frac{3}{2}}{J}^2-{\displaystyle \frac{3}{2}}{L}^2-{\displaystyle \frac{3}{2}}{S}^2}{J}\right) \\ =\frac{e}{2m}\left(\frac{3J^2-L^2+S^2}{2J}\right) \end{gathered}$$ 鈥�(7)

By plugging in Eqs. (3), (4) and (5) into Eq. (7) we get

${渭}_{j_{vg}}=\frac{e\sout{h}}{2m}\left(\frac{2j(j+1)-l(l+1)+s(s+1)}{2\sqrt{j(j+1)}}\right)$