# 動量算符

$\hat{p}=\frac{\hbar}{i}\frac{\partial}{\partial x}\,\!$

$\langle p\rangle=\int_{ - \infty}^{\infty}\ \psi^*(x)\hat{p}\psi(x)\ dx=\int_{ - \infty}^{\infty}\ \psi^*(x)\frac{\hbar}{i}\frac{\partial}{\partial x}\psi(x)\ dx\,\!$

## 導引 1

$- \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \ \psi(x)=E \psi(x)\,\!$

$\psi_k(x)= e^{ikx}\,\!$

$p=\hbar k\,\!$

$\hat{p}\psi_k(x)=p\psi_k(x)\,\!$

$\hat{p}\psi_k(x)=\frac{\hbar}{i}\frac{\partial}{\partial x}\psi_k(x)=\frac{\hbar}{i}\frac{\partial}{\partial x} e^{ikx}=\hbar ke^{ikx}=p\psi_k(x)\,\!$

$\hat{p}=\frac{\hbar}{i}\frac{\partial}{\partial x}\,\!$

## 導引 2

$p=mv=m\frac{dx}{dt}\,\!$

$\langle p\rangle= m\frac{d}{dt}\langle x\rangle\,\!$

$\langle p\rangle= m\frac{d}{dt}\int_{ - \infty}^{\infty}\ \Psi^*(x,\,t)x\Psi(x,\,t)\ dx\,\!$

$\langle p\rangle=m \int_{ - \infty}^{\infty}\ \left( \frac{\partial \Psi^*}{\partial t}x\Psi +\Psi^*\frac{\partial x}{\partial t}\Psi+\Psi^*x\frac{\partial \Psi}{\partial t} \right) dx\,\!$

$\langle p\rangle=m \int_{ - \infty}^{\infty}\ \left( \frac{\partial \Psi^*}{\partial t}x\Psi +\Psi^*x\frac{\partial \Psi}{\partial t} \right) dx\,\!$(1)
$i\hbar\frac{\partial \Psi}{\partial t}= - \frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}+V\Psi\,\!$

$i\hbar\frac{\partial \Psi^*}{\partial t}=\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{\partial x^2} - V\Psi^*\,\!$

\begin{align}\langle p\rangle & = \frac{m}{i\hbar} \int_{ - \infty}^{\infty}\ \left( \frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}x\Psi - V\Psi^*x\Psi - \frac{\hbar^2}{2m}\Psi^*x\frac{\partial^2 \Psi}{\partial x^2}+\Psi^*xV\Psi \right) dx \\ & =\frac{\hbar}{i2} \int_{ - \infty}^{\infty}\ \left( \frac{\partial^2 \Psi^*}{\partial x^2}x\Psi - \Psi^*x\frac{\partial^2 \Psi}{\partial x^2} \right) dx \\ \end{align}\,\!

$\int_{ - \infty}^{\infty}\ \frac{\partial^2 \Psi^*}{\partial x^2}x\Psi\ dx= - \int_{ - \infty}^{\infty}\ \frac{\partial \Psi^*}{\partial x}\Psi\ dx - \int_{ - \infty}^{\infty}\ \frac{\partial \Psi^*}{\partial x}x\frac{\partial \Psi}{\partial x}\ dx \,\!$(2)
$\int_{ - \infty}^{\infty}\ \Psi^* x\frac{\partial^2 \Psi}{\partial x^2}\ dx= - \int_{ - \infty}^{\infty}\ \Psi^*\frac{\partial \Psi}{\partial x}\ dx - \int_{ - \infty}^{\infty}\ \frac{\partial \Psi^*}{\partial x}x\frac{\partial \Psi}{\partial x}\ dx \,\!$(3)

$(2) - (3)=\int_{ - \infty}^{\infty}\ \left( - \frac{\partial \Psi^*}{\partial x}\Psi+\Psi^*\frac{\partial \Psi}{\partial x} \right) dx=2\int_{ - \infty}^{\infty}\ \Psi^*\frac{\partial \Psi}{\partial x}\ dx\,\!$

$\langle p\rangle=\int_{ - \infty}^{\infty}\ \Psi^*\frac{\hbar}{i}\frac{\partial }{\partial x}\Psi\ dx\,\!$

## 厄米算符

$\langle O\rangle=\langle O\rangle^*\,\!$

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

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

\begin{align} \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}\ \left(\frac{\hbar}{i}\frac{\partial\psi^*}{\partial x}\right)\psi\ dx \\ & =\int_{ - \infty}^{\infty}\ \psi\left(\frac{\hbar}{i}\frac{\partial}{\partial x}\psi\right)^* \ dx =\langle\psi|\hat{p}|\psi\rangle^*=\langle\psi|\hat{p}^{\dagger}|\psi\rangle \\ \end{align}

## 本徵值與本徵函數

$\hat{p}f_p(x)=\frac{\hbar}{i}\frac{\partial f_p(x)}{\partial x}=p f_p(x)\,\!$

$f_p(x)=f_0 e^{ipx/\hbar}\,\!$

$\int_{ - L}^{L}\ f_p^*(x)f_p(x)\ dx=|f_0|^2 \int_{ - L}^{L}\ dx=|f_0|^2 2L=1\,\!$

$f_0\,\!$ 的值是 $1/\sqrt{2L}\,\!$ 。動量算符的本徵函數歸一化為 $f_p(x)=\frac{1}{\sqrt{2L}} e^{ipx/\hbar}\,\!$

$\int_{ - \infty}^{\infty}\ f_p^*(x)f_p(x)\ dx=|f_0|^2 \int_{ - \infty}^{\infty}\ dx=\infty\,\!$

$\int_{ - \infty}^{\infty}\ f_{p1}^*(x)f_{p2}(x)\ dx=\frac{1}{2\pi\hbar}\int_{ - \infty}^{\infty}e^{ - i(p1-p2)x/\hbar}\ dx=\delta(p1-p2)\,\!$

$\psi(x)=\int_{ - \infty}^{\infty}c(p) f_p(x)\ dp=\frac{1}{\sqrt{2\pi\hbar}}\int_{ - \infty}^{\infty}c(p) e^{ipx/\hbar}\ dp\,\!$

$c(p)=\int_{ - \infty}^{\infty}f_p^*(x)\psi(x)\ dx=\frac{1}{\sqrt{2\pi\hbar}}\int_{ - \infty}^{\infty}\psi(x) e^{ - ipx/\hbar}\ dx\,\!$

## 正則對易關係

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

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

## 參考文獻

1. ^ A. P. French, An Introduction to Quantum Phusics, W. W. Norton, Inc., pp. 443–444, 1978, ISBN 0-393-09106-0 请检查|isbn=值 (帮助) （英文）
2. ^ Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, pp. 15–18, 97–116, 2004, ISBN 0-13-111892-7