# 位置算符

$\hat{x}|x\rang=x|x\rang$

## 位置空間表現

$\Psi(x)\ \stackrel{def}{=}\ \lang x|\Psi\rang$
$\psi(x)\ \stackrel{def}{=}\ \lang x|\psi\rang$

$\hat{\mathfrak{x}}\psi(x)\ \stackrel{def}{=}\ x\psi(x)$

$|\psi \rang = \int_{-\infty}^{\infty} \mathrm{d}x'\ |x'\rang\lang x'|\psi\rang$

$\hat{x}|\psi \rang =\hat{x} \int_{-\infty}^{\infty} \mathrm{d}x'\ |x'\rang\lang x'|\psi\rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ x' |x'\rang\lang x'|\psi\rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ x' \psi(x')|x'\rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ \hat{\mathfrak{x}}\psi(x') |x'\rang$

$\lang x|\hat{x}|\psi \rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ \hat{\mathfrak{x}}\psi(x') \lang x|x'\rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ \hat{\mathfrak{x}}\psi(x') \delta(x-x') =\hat{\mathfrak{x}}\psi(x)$

$\Psi \rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ |x'\rang\lang x'|\Psi\rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ \Psi(x')|x'\rang$

$\lang x|\Psi \rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ \Psi(x')\lang x|x'\rang =\int_{-\infty}^{\infty} \mathrm{d}x'\ \Psi(x') \delta(x-x') =\Psi(x)$

$\Psi(x)=\hat{\mathfrak{x}}\psi(x)$

## 本徵函數

$\hat{\mathfrak{x}}g_q(x)=q g_q(x)$

$g_q(x)=g_0 \delta(x - q)$

$\int_{ - \infty}^{\infty}\ g_q^*(x)g_q(x)\ dx=|g_0|^2 \int_{ - \infty}^{\infty}\ \delta^2(x - q)\ dx=\mbox{?}$

$\int_{ - \infty}^{\infty}\ g_{q1}^*(x)g_{q2}(x)\ dx=\int_{ - \infty}^{\infty}\ \delta(x - q1)\delta(x - q2)\ dx=\delta(q1-q2)$

$\psi(x)=\int_{ - \infty}^{\infty}\ \psi(q)g_{q}(x)\ dq$

## 期望值

$\lang \psi_1| \psi_2 \rang = \int_{ - \infty}^{\infty}\psi_1^*(x)\psi_2(x) \, \mathrm{d}x$

$\lang x \rang\ \stackrel{def}{=}\ \lang \psi | \hat{x} |\psi \rang$

$\lang x \rang =\int_{ - \infty}^{\infty} \psi^\ast (x) \, x \, \psi(x) \, \mathrm{d}x = \int_{ - \infty}^{\infty} x \, |\psi(x)|^2 \, \mathrm{d}x$

$p(x) \mathrm{d}x = \psi^*(x)\psi(x) \mathrm{d}x$

## 三維案例

$\langle \bold{r} \rangle = \int_{\mathbb{V}} \bold{r} |\psi(\bold{r})|^2 \mathrm{d}^3 \bold{r}$

$\bold{\hat{\mathfrak{r}}}\psi=\bold{r}\psi$

## 對易關係

$[\hat{x},\ \hat{p}]\psi=(\hat{x}\hat{p} - \hat{p}\hat{x})\psi=x\frac{\hbar}{i}\frac{\partial\psi}{\partial x} - \frac{\hbar}{i}\frac{\partial (x\psi)}{\partial x}=i\hbar\psi$

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

## 參考文獻

1. ^ Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004: pp. 17, 104–109. ISBN 0-13-111892-7.
2. ^ 2.0 2.1 Sakurai, J. J.; Napolitano, Jim, Modern Quantum Mechanics. 2nd, Addison-Wesley. 2010, ISBN 978-0805382914