# 氫原子

（重定向自

## 歷史

1913 年，尼爾斯·玻耳在做了一些簡化的假設後，計算出氫原子的光譜頻率。這些假想，波耳模型的基石，並不是完全的正確，但是可以得到正確的能量答案。

1925/26 年，埃爾文·薛丁格應用他發明的薛丁格方程式，以嚴謹的量子力學分析，清楚地解釋了波耳答案正確的原因。氫原子的薛丁格方程式的解答是一個解析解，也可以計算氫原子的能級光譜譜線頻率。薛丁格方程式的解答比波耳模型更為精確，能夠得到許多電子量子態的波函數（軌域），也能夠解釋化學鍵各向異性

## 薛丁格方程式解答

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

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

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

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

### 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

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

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

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

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

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

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

### 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

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

## 穩定性

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

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

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

## 參閱

 相邻较轻同位素: (沒有, 最輕的) 氫原子是 氫的同位素 相邻较重同位素: 氫-2 母同位素：自由中子氦-2 氫原子的衰變鏈 衰變產物為 （穩定）

## 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. （注音：ㄆㄧㄝ；拼音：piē；客家話：piet5；粵語：pit8；英語：protium）
2. Griffiths, David J. Introduction to Quantum Mechanics. Upper Saddle River, NJ: Prentice Hall. 1995. ISBN 978-0-13-111892-8.
3. ^ French, A.P. Introduction to Quantum Physics. W.W. Norton & Company. 1978: pp. 542.
4. ^ 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.
5. ^ 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）.
6. ^ 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）.