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能量守恒：
\(\frac{hc}{\lambda} + m_e c^2 = \frac{hc}{\lambda’} + \gamma m_e c^2\)
动量守恒：
x方向：\(\frac{h}{\lambda} = \frac{h}{\lambda’} \cos\theta + p_{e,x}\),
y方向：\(0 = \frac{h}{\lambda’} \sin\theta - p_{e,y}\)
\(p_e^2 =p_{e,x}^2+p_{e,y}^2\)
动量能量关系：
\(E_e^2 = (p_e c)^2 + (m_e c^2)^2\)
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出射光子波长\(\lambda’\)与入射光子波长\(\lambda\)以及电子康普顿波长\(\lambda_c\)、散射角\(\theta\)关系：
\( \lambda’-\lambda= \lambda_c (1-cos(\theta))\)
\(\nu’ = \frac{\nu}{1 + \frac{h\nu}{m_e c^2}(1 - \cos\theta)}\)
散射角为180时：
\( \lambda’ = \lambda + 2\lambda_c \)
\(E_\gamma = \frac{(\gamma - 1) + \sqrt{\gamma^2 - 1}}{2} m_e c^2 =\frac{K_e+p_e^2/c}{2}\)
\( p_\gamma = \frac{(\gamma - 1) + \sqrt{\gamma^2 - 1}}{2} m_e c =\frac{K_e/c+p_e^2/c^2}{2}\)
\( K_e = \frac{2 E_\gamma^2}{m_e c^2 + 2 E_\gamma} \)
\( p_e = \frac{2 E_\gamma (m_e c^2 + E_\gamma)}{c (m_e c^2 + 2 E_\gamma)} \)
\( E_\gamma’ = \frac{E_\gamma}{1 + \frac{2 E_\gamma}{m_e c^2}} \)
\( m_{\text{动}} = m_e + \frac{K_e}{c^2} =m_e + \frac{2 E_\gamma}{m_e + 2 E_\gamma}\frac{E_\gamma}{c^2} \)
\( \lambda_e =\frac{\lambda \lambda’}{\lambda+\lambda’}\)，\( \frac{1}{\lambda_e} =\frac{1}{\lambda}+\frac{1}{\lambda’}\)
电子反冲角：\(\phi = 0 \) 散射角为90度时：
\( \lambda’ = \lambda + \lambda_c \)
\(K_e = \frac{E_\gamma^2}{m_e c^2 + E_\gamma} \)
\(p_e = \sqrt{\left(\frac{E_\gamma}{c}\right)^2 + \left(\frac{E_\gamma}{c \left(1 + \frac{E_\gamma}{m_e c^2}\right)}\right)^2}\)
\(E_\gamma = \frac{(\gamma - 1) + \sqrt{(\gamma - 1)(\gamma + 3)}}{2} m_e c^2\)
\(E_\gamma’ = \frac{E_\gamma}{1 + \frac{E_\gamma}{m_e c^2}} \)
\( p_\gamma’ =\frac{E_\gamma}{c \left(1 + \frac{E_\gamma}{m_e c^2}\right)}\)
\( \frac{1}{\lambda_e^2} =\frac{1}{\lambda^2} + \frac{1}{\lambda’^2} \)
电子反冲角：\(\tan\phi = \frac{\lambda}{\lambda + \lambda_C}\)
其中：
\(\lambda_c=\frac{h}{m_e c}\)
\(E_\gamma\)为入射光子能量，
\(p_\gamma\)为入射光子动量，
\(K_e\)为电子动能
\(p_e\)为电子动量
\(E_\gamma’\)为散射光子能量
吸收的能量与吸收的动量的关系
\((\Delta E + E_0)^2 = (\Delta p c)^2 + E_0^2\)
=>\(\Delta E = -E_0 + \sqrt{E_0^2 + (\Delta p c)^2}\)
=>\( \Delta p =\sqrt{(\Delta E/c+E_0/c)^2-(E_c/c)^2}\)
=>\(\gamma =\frac{\Delta E + E_0}{E_0}=1+\frac{\Delta E}{E_0}\)
=> \( v=c\sqrt{1-1/\gamma^2} \)
可以看出，只有在\(E_0=0\)时，才会有\( \Delta E =\Delta p c \)，也就是光速无静止质量的情况下，这也是费米子与玻色子的差异，玻色子可以一维线性叠加聚成一个，而费米子则不行