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| the_compton_effect [2022/09/01 20:22] – admin | the_compton_effect [2022/10/13 17:36] (current) – admin |
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| We will use the relativistic energy-momentum relation $E^2 = p^2c^c + m_0c^4$. Photons are massless, so for the incident photon we have $E = pc$ and for the scattered photon we have $E' = p'c$. The electron is initially at rest, so $E_0 = m_e c^2$, where $m_e$ is the rest mass of the electron. For the recoiling electron, we have $E_e = \sqrt{m_e^2 c^4 + P_e^2 c^2}$ or, equivalently $E_e = \sqrt{E_0^2 + P_e^2 c^2}$. Substituting all of this into the conservation of energy formula gives | We will use the relativistic energy-momentum relation $E^2 = p^2c^c + m_0c^4$. Photons are massless, so for the incident photon we have $E = pc$ and for the scattered photon we have $E' = p'c$. The electron is initially at rest, so $E_0 = m_e c^2$, where $m_e$ is the rest mass of the electron. For the recoiling electron, we have $E_e = \sqrt{m_e^2 c^4 + P_e^2 c^2}$ or, equivalently $E_e = \sqrt{E_0^2 + P_e^2 c^2}$. Substituting all of this into the conservation of energy formula gives |
| \[pc + E_0 = p'c + \sqrt{E^2 + P_e^2 c^2},\] | \[pc + E_0 = p'c + \sqrt{E_0^2 + P_e^2 c^2},\] |
| and rearranging gives | and rearranging gives |
| \[E_0 + (p-p')c = \sqrt{E^2 + P_e^2 c^2}.\] | \[E_0 + (p-p')c = \sqrt{E_0^2 + P_e^2 c^2}.\] |
| Squaring this equation gives | Squaring this equation gives |
| \[E_0^2 + (p-p')^2c^2 + 2E_0(p-p')c = E_0^2 + P_e^2c^2,\] | \[E_0^2 + (p-p')^2c^2 + 2E_0(p-p')c = E_0^2 + P_e^2c^2,\] |