# 旋轉不變性

## 球對稱位勢範例

### 哈密頓算符的旋轉不變性

$H= - \frac{\hbar^2}{2m}\nabla^2+V(r)$

$x'=x\cos\theta - y\sin\theta$
$y'=x\sin\theta+y\cos\theta$
$z'=z$

$\frac{\partial}{\partial x'}=\cos\theta\frac{\partial}{\partial x} - \sin\theta\frac{\partial}{\partial y}$
$\frac{\partial}{\partial y'}=\sin\theta\frac{\partial}{\partial x} +\cos\theta\frac{\partial}{\partial y}$
$\frac{\partial}{\partial z'}=\frac{\partial}{\partial z}$

$\nabla'^2=\left(\frac{\partial}{\partial x'}\right)^2+\left(\frac{\partial}{\partial y'}\right)^2+\left(\frac{\partial}{\partial z'}\right)^2=\left(\frac{\partial}{\partial x}\right)^2+\left(\frac{\partial}{\partial y}\right)^2+\left(\frac{\partial}{\partial z}\right)^2 =\nabla^2$

$r'=\sqrt{x'^2+y'^2+z'^2}=\sqrt{x^2+y^2+z^2}=r$

$H'= - \frac{\hbar^2}{2m}\nabla'^2+V(r')= - \frac{\hbar^2}{2m}\nabla^2+V(r)=H$

### 角動量守恆

$\sin\delta\theta\approx\delta\theta$
$\cos\delta\theta\approx 1$

$x'\approx x - y\delta\theta$
$y'\approx x\delta\theta+y$
$z'=z$

$R$ 作用於波函數 $\psi(x,\,y,\,z)$

$R\psi(x,\,y,\,z)=\psi(x',\,y',\,z')\approx \psi(x,\,y,\,z)+\frac{i}{\hbar}\delta\theta L_z \psi(x,\,y,\,z)$

$R=1+\frac{i}{\hbar}\delta\theta L_z$

$H\psi_E(\mathbf{r})=E\psi_E(\mathbf{r})$

$H'\psi_E(\mathbf{r}')=E\psi_E(\mathbf{r}')$

$RH\psi_E(\mathbf{r})=RE\psi_E(\mathbf{r})=ER\psi_E(\mathbf{r})=E\psi_E(\mathbf{r}')$
$HR\psi_E(\mathbf{r})=H\psi_E(\mathbf{r}')=H'\psi_E(\mathbf{r}')=E\psi_E(\mathbf{r}')$

$[R,\,H]=0$

$[L_z,\,H]=0$

$\frac{d}{dt}\langle L_z \rangle= \frac{1}{i\hbar}\langle [L_z,\,H] \rangle + \left\langle \frac{\partial L_z}{\partial t}\right\rangle$

$\frac{d}{dt}\langle L_z \rangle=\left\langle \frac{\partial L_z}{\partial t}\right\rangle$

$\frac{d}{dt}\langle L_z \rangle=0$

## 參考文獻

1. ^ 古斯, 阿蘭, The Inflationary Universe, Basic Books. 1998:  pp.340, ISBN 978-0201328400
• Gasiorowics, Stephen. Quantum Physics (3rd ed.). Wiley. 2003. ISBN 978-0471057000.
• Stenger, Victor J. (2000). Timeless Reality Symmetry, Simplicity, and Multiple Universes. Prometheus Books. 特別參考第十二章。非專科性書籍。