91影视

We get the complete Fine-Structure Formula is,

$E_{fs}^1=\frac{E_n^2}{2m{c}^2}\left(3-\frac{4n}{j+1/2}\right)$

Using Equation 6:66,

$E_{fs}^1=\frac{E_2^2}{2m{c}^2}\left(3-\frac{8}{j+1/2}\right)=\frac{E_1^2}{32m{c}^2}\left(3-\frac{8}{j+1/2}\right);\frac{E_1}{m{c}^2}=-\frac{{伪}^2}{2}$,

So.

$E_{fs}^1=-\frac{E_1}{32}\left(\frac{{伪}^2}{2}\right)\left(3-\frac{8}{j+1/2}\right)=\frac{13.6eV}{64}{伪}^2\left(3-\frac{8}{j+1/2}\right)=纬\left(3-\frac{8}{j+1/2}\right).$

$E_{fs}^1=\frac{{\left(E_n\right)}^2}{2m{c}^2}\left(3-\frac{4n}{j+1/2}\right)$

For

For$j=3/2\left({蠄}_3,{蠄}_4,{蠄}_5,{蠄}_7\right)$

$$\begin{gathered} H_{fs}^1=纬\left(3-\frac{8}{2}\right) \\ {H}_{fs}^1=-纬 \end{gathered}$$

This confirms all the 纬 terms in -W (p. 281).

Meanwhile, $H_z^{\text{'}}=(e/2m)B_{ext}\left(L_z+2S_z\right),{蠄}_1,{蠄}_2,{蠄}_3,{蠄}_4$are Eigen states of $L_z\text{and}{S}_z$ for these there are only diagonal elements:

$H_Z^{\text{'}}=\frac{e}{2m}(L+2S)路B_{ext}$

This confirms the upper left corner of -W.

Finally:

$$\begin{gathered} \left.\begin{array}{r}\left(L_z+2S_z\right)\left|{蠄}_5\right.>=+\mathrm{鈩�}\sqrt{\frac{2}{3}}|10鉄�\left|\frac{1}{2}\frac{1}{2}\right.> \\ \left(L_z+2S_z\right)\left|{蠄}_6\right.>=-\mathrm{鈩�}\sqrt{\frac{1}{3}}|10鉄�\left|\frac{1}{2}\frac{1}{2}\right.> \\ \left(L_z+2S_z\right)\left|{蠄}_7\right.>=-\mathrm{鈩�}\sqrt{\frac{2}{3}}|10鉄�\left|\frac{1}{2}-\frac{1}{2}\right.> \\ \left(L_z+2S_z\right)\left|{蠄}_8\right.>=-\mathrm{鈩�}\sqrt{\frac{1}{3}}|10鉄�\left|\frac{1}{2}-\frac{1}{2}\right.>\\ \end{array}\right\}so \end{gathered}$$

which confirms the remaining elements.