# 康普頓散射

（重定向自康普頓效應

## 康普顿频移公式

${\displaystyle \lambda -\lambda _{0}={\frac {h}{mc}}\left(1-\cos \theta \right)}$

 ${\displaystyle \lambda _{0}\,}$ 撞前波长 ${\displaystyle \lambda \,}$ 撞後波长 ${\displaystyle m\,}$ 电子质量 ${\displaystyle \theta \,}$ 光子方向转动角（碰撞前後的路径夹角） ${\displaystyle h\,}$ 普朗克常数 ${\displaystyle c\,}$ 光速

 ${\displaystyle \mathbf {p} _{0}\,}$ 撞前光子動量 ${\displaystyle \mathbf {p} \,}$ 撞後光子動量 ${\displaystyle \mathbf {v} \,}$ 撞後電子速度 ${\displaystyle \gamma ={\frac {1}{\sqrt {1-\left(\mathbf {v} /c\right)^{2}}}}}$ ${\displaystyle \mathbf {p} _{0}=\mathbf {p} +\gamma m\mathbf {v} }$ 动量守恒 ${\displaystyle \left|\mathbf {p} _{0}\right|c+mc^{2}=\left|\mathbf {p} \right|c+\gamma mc^{2}}$ 能量守恒 ${\displaystyle \left|\mathbf {p} \right|={\frac {h}{\lambda }}}$ 物質波公式

${\displaystyle {\begin{array}{rcl}\mathbf {p} _{0}^{2}+\mathbf {p} ^{2}-2\left|\mathbf {p} _{0}\right|\left|\mathbf {p} \right|\cos \theta &=&\left(\mathbf {p} _{0}-\mathbf {p} \right)^{2}=\left(\gamma m\mathbf {v} \right)^{2}\\&=&\left(\gamma mc\right)^{2}-\left(mc\right)^{2}=\left(\left|\mathbf {p} _{0}\right|+mc-\left|\mathbf {p} \right|\right)^{2}-\left(mc\right)^{2}\\&=&\left(\left|\mathbf {p} _{0}\right|-\left|\mathbf {p} \right|\right)\left(\left|\mathbf {p} _{0}\right|+2mc-\left|\mathbf {p} \right|\right)\\&=&\mathbf {p} _{0}^{2}+\mathbf {p} ^{2}-2\left|\mathbf {p} _{0}\right|\left|\mathbf {p} \right|+2mc\left(\left|\mathbf {p} _{0}\right|-\left|\mathbf {p} \right|\right)\end{array}}}$

${\displaystyle {\frac {1-\cos \theta }{mc}}={\frac {\left|\mathbf {p} _{0}\right|-\left|\mathbf {p} \right|}{\left|\mathbf {p} _{0}\right|\left|\mathbf {p} \right|}}={\frac {1}{\left|\mathbf {p} \right|}}-{\frac {1}{\left|\mathbf {p} _{0}\right|}}={\frac {\lambda }{h}}-{\frac {\lambda _{0}}{h}}}$

${\displaystyle \lambda -\lambda _{0}={\frac {h}{mc}}\left(1-\cos \theta \right)}$

## 參考文獻

1. ^ George Greenstein; Arthur Zajonc. The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics. Jones & Bartlett Learning. 2006. ISBN 978-0-7637-2470-2.