# 定態

${\displaystyle {\frac {d}{dt}}|\Psi (x,\,t)|^{2}=0}$

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi +V\Psi =i\hbar {\frac {\partial }{\partial t}}\Psi }$

${\displaystyle \Psi (x,\,t)=\psi (x)e^{-iEt/\hbar }}$

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi +V\psi =E\psi }$

## 機率密度與時間無關

${\displaystyle |\Psi (x,\,t)|^{2}=|\psi (x)|^{2}}$

{\displaystyle {\begin{aligned}\langle x\rangle &=\int _{-\infty }^{\infty }\Psi ^{*}(x,\,t)x\Psi (x,\,t)\,dx\\&=\int _{-\infty }^{\infty }\,x|\Psi (x,\,t)|^{2}\,dx\\&=\int _{-\infty }^{\infty }\,x|\psi (x)|^{2}\,dx\\\end{aligned}}}

{\displaystyle {\begin{aligned}\langle p\rangle &=\int _{-\infty }^{\infty }\Psi ^{*}(x,\,t){\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\Psi (x,\,t)\,dx\\&={\frac {\hbar }{i}}\int _{-\infty }^{\infty }\psi (x)e^{iEt/\hbar }{\frac {\partial }{\partial x}}(\psi (x)e^{-iEt/\hbar })\,dx\\&={\frac {\hbar }{i}}\int _{-\infty }^{\infty }\,\psi ^{*}(x){\frac {\partial }{\partial x}}\psi (x)\,dx\\\end{aligned}}}

## 參考文獻

• Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004. ISBN 0-13-111892-7.