# 可觀察量

（重定向自不相容可觀察量

## 數學表述

### 本徵態

${\displaystyle \langle e_{i}|e_{j}\rangle =\delta _{ij}}$

${\displaystyle |\psi \rangle =\sum _{i}\ c_{i}|e_{i}\rangle }$

### 統計詮釋

${\displaystyle |\psi \rangle =\sum _{i}\ c_{i}|e_{i}\rangle }$

${\displaystyle |\phi \rangle ={\hat {O}}|\psi \rangle =\sum _{i}\ c_{i}{\hat {O}}|e_{i}\rangle =\sum _{i}\ c_{i}O_{i}|e_{i}\rangle }$

${\displaystyle \langle \psi |\phi \rangle =\langle \psi |{\hat {O}}|\psi \rangle =\sum _{i}\ c_{i}O_{i}\langle \psi |e_{i}\rangle =\sum _{i}\ |c_{i}|^{2}O_{i}=\sum _{i}\ p_{i}O_{i}}$

${\displaystyle \langle O\rangle \ {\stackrel {def}{=}}\ \langle \psi |{\hat {O}}|\psi \rangle =\sum _{i}\ p_{i}O_{i}}$

### 厄米算符

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

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

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

## 不相容可觀察量

${\displaystyle [{\hat {A}},{\hat {B}}]\neq 0}$

${\displaystyle |\psi \rangle =\sum _{i}\ c_{i}|\alpha _{i}\rangle }$

${\displaystyle |\psi \rangle =\sum _{i}\ d_{i}|\beta _{i}\rangle }$

${\displaystyle \Delta A\ \Delta B\geq \left|{\frac {\langle [{\hat {A}},{\hat {B}}]\rangle }{2i}}\right|}$

## 實例

### 位置與動量

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

### 角動量

${\displaystyle \langle L_{x}\rangle ^{*}=\langle yp_{z}-zp_{y}\rangle ^{*}=\langle yp_{z}-zp_{y}\rangle =\langle L_{x}\rangle }$

## 註釋

1. ^ 通常這句話成立，但也存在有例外。思考氫原子角量子數為零（${\displaystyle \ell =0\ }$ ）的量子態，它是${\displaystyle L_{x}}$ ${\displaystyle L_{y}}$ ${\displaystyle L_{z}}$ 的本徵態，本徵值都為零，而這三個自伴算符都互不對易，它們對應的可觀察量彼此之間都是不相容可觀察量。[3]

## 參考文獻

1. Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, 2004, ISBN 0-13-111892-7
2. Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics 2nd, Addison-Wesley, 2010, ISBN 978-0805382914
3. ^ A. P. French, An Introduction to Quantum Phusics, W. W. Norton, Inc.: pp. 452–453, 1978, ISBN 9780748740789 （英语）