Module 5 : MODERN PHYSICS
Lecture 25 : Derivation of Compton's Formula
  Conservation of Momentum
x-direction :
 
$\displaystyle p_i = p_f\cos\theta + p_e\cos\phi$ (1)
  y-direction :
 
$\displaystyle 0= -p_f\sin\theta + p_e\sin\phi$ (2)
  From Eqns. (1) and (2), we get
 
$\displaystyle p_e^2 = p_e^2(\cos^2\phi+\sin^2\phi)$ $\displaystyle =$ $\displaystyle (p_i-p_f\cos\theta)^2 + p_f^2\sin^2\theta$
$\displaystyle =$ $\displaystyle p_i^2 + p_f^2 -2p_ip_f\cos\theta$ (3)
  Conservation of Energy : (relativistic effect)
  If the rest mass of the electron is taken to be $ m_0$, the initial energy is $ m_0c^2$ and the final energy is $ \sqrt{m_0^2c^4 + p_e^2c^2}$. Thus
   
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