# 自伴算子

（重定向自厄米算符

${\displaystyle V}$ 是具有规范正交基的有限维复向量空间，其上自伴算子在该基下的矩阵埃尔米特矩阵——该矩阵等于自身的共轭转置。有限维的谱定理表明，对于一个算子 ${\displaystyle A}$ ，总能找到 ${\displaystyle V}$ 上的规范正交基使得 ${\displaystyle A}$ 在该基下的矩阵是一个对角矩阵，且这些对角元都是实数

## 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

• Rudin, Walter. Functional Analysis. Boston, Mass.: McGraw-Hill Science, Engineering & Mathematics. 1991. ISBN 978-0-07-054236-5.
• Narici, Lawrence; Beckenstein, Edward. Topological vector spaces. Monographs and textbooks in pure and applied mathematics 2. ed. Boca Raton: CRC Press. 2011. ISBN 978-1-58488-866-6.
• Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004: pp. 96–106. ISBN 0-13-111892-7.