# 類氫原子

## 薛丁格方程式解答

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

$V(r) = - \frac{Ze^2}{4 \pi \epsilon_0 r}$

$-\frac{\hbar^2}{2\mu r^2}\left \{ \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r}\right)+\frac{1}{\sin^2\theta}\left[\sin\theta\frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right)+\frac{\partial^2}{\partial \phi^2}\right]\right \}\psi - \frac{Ze^2}{4 \pi \epsilon_0 r}\psi= E\psi$

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

### 角部分解答

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

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

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

$P_l(x)$$l$勒讓德多項式，可用羅德里格公式表示為

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

### 徑向部分解答

$\left[ - {\hbar^2 \over 2\mu r^2} {d\over dr}\left(r^2{d\over dr}\right) +{\hbar^2 l(l+1)\over 2\mu r^2} - \frac{Ze^2}{4 \pi \epsilon_0 r} \right] R_{nl}(r)=ER_{nl}(r)$

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

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

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

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

### 量子數

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

### 角動量

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

$\hat{L}_z Y_{lm} = \hbar m Y_{lm}$

$\Delta L_x\ \Delta L_y \ge \left|\frac{\langle[\hat{L}_x,\ \hat{L}_y]\rangle}{2i}\right|=\frac{\hbar |\langle \hat{L}_z\rangle|}{2}$

$L_x$ 的不確定性與 $L_y$ 的不確定性的乘積 $\Delta L_x\ \Delta L_y$ ，必定大於或等於 $\frac{\hbar |\langle L_z\rangle|}{2}$

### 精細結構

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

## 穩定性

$E_0 > -\infty$

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

$E_0=-4Z^2/3\ [Ry]$

## 註釋

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

## 參考文獻

1. ^ French, A.P.. Introduction to Quantum Physics. en:W.W. Norton & Company. 1978: pp. 542.
2. ^ 狄拉克方程式關於氫原子的解答
3. ^ Lieb, Elliot. THE STABILITY OF MATTER:FROM ATOMS TO STARS. BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY. 1990, 22 (1).
4. ^ Lieb, Elliot. The stability of matter. Review of Modern Physics. 1976, 48: 553–569.
• Tipler, Paul & Ralph Llewellyn (2003). Modern Physics (4th ed.). New York: W. H. Freeman and Company. ISBN 0-7167-4345-0
• Griffiths, David J. Introduction to Quantum Mechanics. Upper Saddle River, NJ: Prentice Hall. 1995: 131–200. ISBN 0-13-111892-7.