Module 5 : MODERN PHYSICS
Lecture 26 : Wave Nature of Particle - the de Broglie Hypothesis
Consider two extreme limits when the gamma ray photon is scattered by an angle $ \theta$ to the left extreme wall of the microscope. The x-component of the total momentum is the sum of the x component of the momentum of the scattered electron, $ p_x^\prime$ and the x component of the momentum of the scattered photon, which is , i.e. $ \displaystyle{p_x^\prime-\frac{h}{\lambda^\prime}\sin\theta}$ ,where $ \lambda^\prime$ is the wavelength of the scattered photon.
 
Similarly, when the photon is scattered to the extreme right, the total momentum is $ \displaystyle{p_x^{\prime\prime}+ \frac{h}{\lambda^{\prime\prime}}\sin\theta}$ ,where $ \lambda^{\prime\prime}$ is the wavelength of the scattered photon. As the angle $ \theta$ is small, the Compton shift $ \Delta\lambda=(h/mc)(1-\cos\theta)$ is small, and we may take $ \lambda^\prime\approx\lambda^{\prime\prime}\approx \lambda$. The total x-component of the momentum incases must be the same, each being equal to momentum of the incident photon.
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