# 自伴算子

（重定向自自共轭算符

## 量子力學

### 可觀察量

${\displaystyle \langle O\rangle =\langle O\rangle ^{*}\,\!}$

${\displaystyle \langle \psi |{\hat {O}}|\psi \rangle =\langle \psi |{\hat {O}}|\psi \rangle ^{*}\,\!}$

${\displaystyle {\hat {O}}={\hat {O}}^{\dagger }\,\!}$

${\displaystyle {\hat {H}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi \,\!}$

${\displaystyle \langle \psi |{\hat {p}}|\psi \rangle =\int _{-\infty }^{\infty }\ \psi ^{*}{\frac {\hbar }{i}}{\frac {\partial \psi }{\partial x}}\ dx=\left.{\frac {\hbar }{i}}\psi ^{*}\psi \right|_{-\infty }^{\infty }-\int _{-\infty }^{\infty }\ {\frac {\hbar }{i}}{\frac {\partial \psi ^{*}}{\partial x}}\psi \ dx=\int _{-\infty }^{\infty }\ \left({\frac {\hbar }{i}}{\frac {\partial \psi }{\partial x}}\right)^{*}\psi \ dx=\langle \psi |{\hat {p}}|\psi \rangle ^{*}=\langle \psi |{\hat {p}}^{\dagger }|\psi \rangle \,\!}$

## 參考文獻

Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004: pp. 96–106. ISBN 0-13-111892-7.