Consider the Compton scattering of a photon of momentum \(\mathbf{k}\) and energy \(E=|\mathbf{k}|=\mathrm{k}\) from an electron \(a\) t rest. Writing the four-momenta of the scattered photon and electron respectively as \(k^{\prime}\) and \(p^{\prime}\), conservation of fourmomentum is expressed as \(k+p=k^{\prime}+p^{\prime}\). Use the relation \(p^{\prime 2}=m_{e}^{2}\) to show that the energy of the scattered photon is given by $$ E^{\prime}=\frac{E}{1+\left(E / m_{e}\right)(1-\cos \theta)} $$ where \(\theta\) is the angle through which the photon is scattered.

Compton scattering involves the interaction of a photon with an electron. One of the fundamental principles governing this interaction is the conservation of four-momentum. Four-momentum includes both energy and momentum components into a single framework. In simpler terms, it's a way to combine the energy and momentum of particles in a way that respects special relativity. The conservation of four-momentum means that the total four-momentum before the scattering event is equal to the total four-momentum after the scattering event. For the initial photon and electron at rest, we have:

• The four-momentum of the photon: \[k = (E, \textbf{k})\]
• The four-momentum of the electron: \[p = (m_e, 0)\]

After interaction, for the scattered photon and electron, we have:

• The four-momentum of the scattered photon: \[k' = (E', \textbf{k'})\]
• The four-momentum of the scattered electron: \[p' = (E_{p'}, \textbf{p'}) \]

Conservation of four-momentum states: \[k + p = k' + p'\]This principle is crucial as it dictates how energy and momentum are partitioned between the photon and electron after scattering.

In the photon-electron interaction, when a photon collides with an electron, some of its energy is transferred to the electron, causing the electron to recoil. This process changes the photon's energy and direction. Initially, the photon is characterized by its energy \(E\) and momentum \(\textbf{k}\), while the electron is at rest. During the interaction, the photon's energy diminishes to \(E'\) and it gets scattered at an angle \( \theta \) relative to its original direction. Meanwhile, the electron gains energy and momentum as a result of the collision. The interaction can be described mathematically by the equation: \[k + p = k' + p'\] Here, \(k\) and \(p\) are the four-momenta of the initial photon and electron, and \(k'\) and \(p'\) are the four-momenta of the scattered photon and electron. Understanding this interaction is key to determining how the energy and momentum are redistributed post-collision.

The energy-momentum relation is central to analyzing the outcome of Compton scattering. For photons (particles of light), the energy \(E\) and momentum \(\textbf{k}\) are related by the equation: \[E = |\textbf{k}|\] For electrons (which have mass \(m_e\)), the energy and momentum are encapsulated in the relation: \[E^2 = p^2 + m_e^2\] These equations help simplify the conservation of four-momentum and understand the dynamics of the scattering process. By analyzing these equations, we can derive other important relationships that describe the scattering event. Specifically, we can use them to derive the Compton shift equation, which quantifies the change in wavelength (and hence energy) of the photon after scattering.

The scattering angle \( \theta \) is a critical parameter in Compton scattering. It describes the angle at which the photon is deflected as a result of its collision with the electron. This angle directly influences the energy of the scattered photon. To derive the energy \(E'\) of the scattered photon, we take into account the conservation of four-momentum and the energy-momentum relations. By resolving the conservation equation \[k + p = k' + p'\] and substituting the known quantities:

• \(k \cdot p = E m_e\)
• \(k' \cdot p' = E' m_e (1 - \cos \theta)\)

We get the relation: \[E' = \frac{E}{1 + \left(\frac{E}{m_e} \right) (1 - \cos \theta)}\] This formula shows how the energy of the scattered photon decreases as the scattering angle \( \theta \) increases. Larger scattering angles correlate with greater energy transfer to the electron, resulting in a lower energy photon.