# 康普頓散射

## 康普頓頻移公式

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

 $\lambda_{0}\,$ 撞前波長 $\lambda\,$ 撞後波長 $m\,$ 電子質量 $\theta\,$ 光子方向轉動角（碰撞前後的路徑夾角） $h\,$ 普朗克常數 $c\,$ 光速

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

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

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

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

## 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.