旋轉不變性

球對稱位勢範例

哈密頓算符的旋轉不變性

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

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

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

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

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

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

角動量守恆

${\displaystyle \sin \delta \theta \approx \delta \theta }$
${\displaystyle \cos \delta \theta \approx 1}$

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

${\displaystyle R}$  作用於波函數 ${\displaystyle \psi (x,\,y,\,z)}$

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

${\displaystyle R=1+{\frac {i}{\hbar }}\delta \theta L_{z}}$

${\displaystyle H\psi _{E}(\mathbf {r} )=E\psi _{E}(\mathbf {r} )}$

${\displaystyle H'\psi _{E}(\mathbf {r} ')=E\psi _{E}(\mathbf {r} ')}$

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

${\displaystyle [R,\,H]=0}$

${\displaystyle [L_{z},\,H]=0}$

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

${\displaystyle {\frac {d}{dt}}\langle L_{z}\rangle =\left\langle {\frac {\partial L_{z}}{\partial t}}\right\rangle }$

${\displaystyle {\frac {d}{dt}}\langle L_{z}\rangle =0}$

參考文獻

1. ^ 古斯, 阿蘭, The Inflationary Universe, Basic Books: pp.340, 1998, 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。