# 精細結構

${\displaystyle H=H^{(0)}+H_{kinetic}+H_{so}\,\!}$

## 相對論性修正

${\displaystyle T={\frac {p^{2}}{2m}}\,\!}$

${\displaystyle T={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}-mc^{2}\,\!}$

${\displaystyle T={\frac {p^{2}}{2m}}-{\frac {p^{4}}{8m^{3}c^{2}}}+\dots \,\!}$

${\displaystyle H_{kinetic}=-{\frac {p^{4}}{8m^{3}c^{2}}}\,\!}$

${\displaystyle E_{n}^{(1)}=\langle \psi _{n}^{(0)}\vert H_{kinetic}\vert \psi _{n}^{(0)}\rangle =-{\frac {1}{8m^{3}c^{2}}}\langle \psi _{n}^{(0)}\vert p^{4}\vert \psi _{n}^{(0)}\rangle =-{\frac {1}{8m^{3}c^{2}}}\langle \psi _{n}^{(0)}\vert p^{2}p^{2}\vert \psi _{n}^{(0)}\rangle \,\!}$

${\displaystyle H^{(0)}\vert \psi _{n}^{(0)}\rangle =E_{n}^{(0)}\vert \psi _{n}^{(0)}\rangle \,\!}$

${\displaystyle \left({\frac {p^{2}}{2m}}+V\right)\vert \psi _{n}^{(0)}\rangle =E_{n}^{(0)}\vert \psi _{n}^{(0)}\rangle \,\!}$

${\displaystyle p^{2}\vert \psi _{n}^{(0)}\rangle =2m(E_{n}^{(0)}-V)\vert \psi _{n}^{(0)}\rangle \,\!}$

{\displaystyle {\begin{aligned}E_{n}^{(1)}&=-{\frac {1}{8m^{3}c^{2}}}\langle \psi _{n}^{(0)}\vert p^{2}p^{2}\vert \psi _{n}^{(0)}\rangle \\&=-{\frac {1}{8m^{3}c^{2}}}\langle \psi _{n}^{(0)}\vert (2m)^{2}(E_{n}^{(0)}-V)^{2}\vert \psi _{n}^{(0)}\rangle \\&=-{\frac {1}{2mc^{2}}}[(E_{n}^{(0)})^{2}-2E_{n}^{(0)}\langle V\rangle +\langle V^{2}\rangle ]\\\end{aligned}}\,\!}

${\displaystyle \langle V\rangle ={\frac {Z^{2}e^{2}}{4\pi \epsilon _{0}a_{0}n^{2}}}\,\!}$
${\displaystyle \langle V^{2}\rangle ={\frac {Z^{4}e^{4}}{(l+1/2)(4\pi \epsilon _{0}a_{0})^{2}n^{3}}}\,\!}$

{\displaystyle {\begin{aligned}E_{n}^{(1)}&=-{\frac {1}{2mc^{2}}}\left[(E_{n}^{(0)})^{2}-2E_{n}^{(0)}{\frac {Z^{2}e^{2}}{4\pi \epsilon _{0}a_{0}n^{2}}}+{\frac {Z^{4}e^{4}}{(l+1/2)(4\pi \epsilon _{0}a_{0})^{2}n^{3}}}\right]\\&=-{\frac {(E_{n}^{(0)})^{2}}{2mc^{2}}}\left({\frac {4n}{l+1/2}}-3\right)\\\end{aligned}}\,\!}

## 自旋-軌道修正

${\displaystyle H_{so}={\frac {Ze^{2}}{8\pi \epsilon _{0}m^{2}c^{2}r^{3}}}\,(\mathbf {L} \cdot \mathbf {S} )\,\!}$

${\displaystyle E_{n}^{(1)}={\frac {(E_{n}^{(0)})^{2}}{mc^{2}}}\ {\frac {2n[j(j+1)-l(l+1)-3/4]}{l(l+1)(2l+1)}}\,\!}$

## 總和

${\displaystyle E_{n}^{(1)}=-{\frac {(E_{n}^{(0)})^{2}}{2mc^{2}}}\left({\frac {4n}{l+1/2}}-3\right)+{\frac {(E_{n}^{(0)})^{2}}{mc^{2}}}\ {\frac {2n[j(j+1)-l(l+1)-3/4]}{l(l+1)(2l+1)}}\,\!}$

${\displaystyle j\,\!}$的這兩個數值分別代入總合方程式裏，經過一番運算，可以得到同樣的結果：

${\displaystyle E_{n}^{(1)}={\frac {(E_{n}^{(0)})^{2}}{mc^{2}}}\left({\frac {3}{2}}-{\frac {4n}{2j+1}}\right)\,\!}$

${\displaystyle E_{n}={\frac {E_{1}^{(0)}}{n^{2}}}\left(1+\left({\frac {Z\alpha }{n}}\right)^{2}\left({\frac {2n}{2j+1}}-{\frac {3}{4}}\right)\right)\,\!}$ ;

## 更精确的结果

${\displaystyle E_{n}=-mc^{2}\left[1-\left(1+\left[{\dfrac {Z\alpha }{n-j-{\frac {1}{2}}+{\sqrt {\left(j+{\frac {1}{2}}\right)^{2}-Z^{2}\alpha ^{2}}}}}\right]^{2}\right)^{-1/2}\right]}$

## 參考文獻

1. ^ Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004: pp. 266–276. ISBN 0-13-111892-7.
2. ^ Dirac Equation and Hydrogen Atom (PDF). [2014-09-10]. （原始内容存档 (PDF)于2016-03-05）.
• Liboff, Richard L. Introductory Quantum Mechanics. Addison-Wesley. 2002. ISBN 0-8053-8714-5.