# 類氫原子

（重定向自类氢原子

## 薛丁格方程式解答

${\displaystyle -{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi +V(r)\psi =E\psi }$

${\displaystyle V(r)=-{\frac {Ze^{2}}{4\pi \epsilon _{0}r}}}$

${\displaystyle -{\frac {\hbar ^{2}}{2\mu r^{2}}}\left\{{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial }{\partial r}}\right)+{\frac {1}{\sin ^{2}\theta }}\left[\sin \theta {\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)+{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right]\right\}\psi -{\frac {Ze^{2}}{4\pi \epsilon _{0}r}}\psi =E\psi }$

${\displaystyle \psi (r,\ \theta ,\ \phi )=R_{nl}(r)Y_{lm}(\theta ,\ \phi )}$

### 角部分解答

${\displaystyle -{\frac {1}{\sin ^{2}\theta }}\left[\sin \theta {\frac {\partial }{\partial \theta }}{\Big (}\sin \theta {\frac {\partial }{\partial \theta }}{\Big )}+{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right]Y_{lm}(\theta ,\phi )=l(l+1)Y_{lm}(\theta ,\phi )}$

${\displaystyle Y_{lm}(\theta ,\ \phi )=(i)^{m+|m|}{\sqrt {{(2l+1) \over 4\pi }{(l-|m|)! \over (l+|m|)!}}}\,P_{lm}(\cos {\theta })\,e^{im\phi }}$

${\displaystyle P_{lm}(x)=(1-x^{2})^{|m|/2}\ {\frac {d^{|m|}}{dx^{|m|}}}P_{l}(x)\,}$

${\displaystyle P_{l}(x)}$ ${\displaystyle l}$ 勒讓德多項式，可用羅德里格公式表示為

${\displaystyle P_{l}(x)={1 \over 2^{l}l!}{d^{l} \over dx^{l}}(x^{2}-1)^{l}}$

### 徑向部分解答

${\displaystyle \left[-{\hbar ^{2} \over 2\mu r^{2}}{d \over dr}\left(r^{2}{d \over dr}\right)+{\hbar ^{2}l(l+1) \over 2\mu r^{2}}-{\frac {Ze^{2}}{4\pi \epsilon _{0}r}}\right]R_{nl}(r)=ER_{nl}(r)}$

${\displaystyle R_{nl}(r)={\sqrt {{\left({\frac {2Z}{na_{\mu }}}\right)}^{3}{\frac {(n-l-1)!}{2n[(n+l)!]^{3}}}}}e^{-Zr/{na_{\mu }}}\left({\frac {2Zr}{na_{\mu }}}\right)^{l}L_{n-l-1}^{2l+1}\left({\frac {2Zr}{na_{\mu }}}\right)}$

${\displaystyle L_{i}^{j}(x)=(-1)^{j}\ {\frac {d^{j}}{dx^{j}}}L_{i+j}(x)}$

${\displaystyle L_{i}(x)={\frac {e^{x}}{i!}}\ {\frac {d^{i}}{dx^{i}}}(x^{i}e^{-x})}$

${\displaystyle \psi _{nlm}=R_{nl}(r)\,Y_{lm}(\theta ,\phi )}$

### 量子數

${\displaystyle n=1,\ 2,\ 3,\ 4,\ \dots }$
${\displaystyle l=0,\ 1,\ 2,\ \dots ,\ n-1}$
${\displaystyle m=-l,\ -l+1,\ \ldots ,\ 0,\ \ldots ,\ l-1,\ l}$

### 角動量

${\displaystyle {\hat {L}}^{2}Y_{lm}=\hbar ^{2}l(l+1)Y_{lm}}$

${\displaystyle {\hat {L}}_{z}Y_{lm}=\hbar mY_{lm}}$

${\displaystyle \Delta L_{x}\ \Delta L_{y}\geq \left|{\frac {\langle [{\hat {L}}_{x},\ {\hat {L}}_{y}]\rangle }{2i}}\right|={\frac {\hbar |\langle {\hat {L}}_{z}\rangle |}{2}}}$

${\displaystyle L_{x}}$  的不確定性與 ${\displaystyle L_{y}}$  的不確定性的乘積 ${\displaystyle \Delta L_{x}\ \Delta L_{y}}$  ，必定大於或等於 ${\displaystyle {\frac {\hbar |\langle L_{z}\rangle |}{2}}}$

### 精細結構

${\displaystyle E_{nj}=E_{n}\left[1+\left({\frac {Z\alpha }{n}}\right)^{2}\left({\frac {n}{j+{\frac {1}{2}}}}-{\frac {3}{4}}\right)\right]}$

## 穩定性

${\displaystyle E_{0}>-\infty }$

${\displaystyle E=T+V=\int _{\mathbb {R} ^{3}}\mathrm {d} x\left({\frac {1}{2}}|\nabla \psi (x)|^{2}-Z{\frac {|\psi (x)|^{2}}{|x|}}\right)}$

${\displaystyle E_{0}=-4Z^{2}/3\ [Ry]}$

## 註釋

1. ^ 為了方便運算，採用 ${\displaystyle \hbar ^{2}/2=1}$  、質量 ${\displaystyle m=1}$  、基本電荷 ${\displaystyle |e|=1}$  的單位制。

## 參考文獻

1. ^ French, A.P. Introduction to Quantum Physics. W.W. Norton & Company. 1978: pp. 542.
2. ^ 狄拉克方程式關於氫原子的解答 互联网档案馆存檔，存档日期2008-02-18.
3. ^ Lieb, Elliot. THE STABILITY OF MATTER:FROM ATOMS TO STARS (PDF). BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY. 1990, 22 (1) [2014-09-30]. （原始内容存档 (PDF)于2013-12-19）.
4. ^ Lieb, Elliot. The stability of matter (PDF). Review of Modern Physics. 1976, 48: 553–569 [2014-09-30]. （原始内容存档 (PDF)于2015-02-20）.