# 動量算符

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

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

## 導引 1

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

${\displaystyle \psi _{k}(x)=e^{ikx}\,\!}$

${\displaystyle p=\hbar k\,\!}$

${\displaystyle {\hat {p}}\psi _{k}(x)=p\psi _{k}(x)\,\!}$

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

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

## 導引 2

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

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

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

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

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

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

{\displaystyle {\begin{aligned}\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{aligned}}\,\!}

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

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

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

## 厄米算符

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

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

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

{\displaystyle {\begin{aligned}\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{aligned}}}

## 本徵值與本徵函數

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

${\displaystyle f_{p}(x)=f_{0}e^{ipx/\hbar }\,\!}$

${\displaystyle \int _{-L}^{L}\ f_{p}^{*}(x)f_{p}(x)\ dx=|f_{0}|^{2}\int _{-L}^{L}\ dx=|f_{0}|^{2}2L=1\,\!}$

${\displaystyle f_{0}\,\!}$ 的值是 ${\displaystyle 1/{\sqrt {2L}}\,\!}$ 。動量算符的本徵函數歸一化為 ${\displaystyle f_{p}(x)={\frac {1}{\sqrt {2L}}}e^{ipx/\hbar }\,\!}$

${\displaystyle \int _{-\infty }^{\infty }\ f_{p}^{*}(x)f_{p}(x)\ dx=|f_{0}|^{2}\int _{-\infty }^{\infty }\ dx=\infty \,\!}$

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

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

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

## 正則對易關係

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

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

## 參考文獻

1. ^ A. P. French, An Introduction to Quantum Phusics, W. W. Norton, Inc.: pp. 443–444, 1978, ISBN 978-0393091069 （英語）
2. ^ Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall: pp. 15–18, 97–116, 2004, ISBN 0-13-111892-7