To derive the expression for the new wavelength of a photon after it undergoes Compton scattering with a moving electron, we need to consider the principles of conservation of energy and momentum. In this scenario, the photon collides head-on with an electron that is moving in the opposite direction. Let's break this down step by step.
Understanding the Initial Conditions
Before the collision, we have:
- The photon with an initial wavelength, denoted as λ.
- The electron moving in the negative x-direction with total energy E, which is approximately mc² (where m is the rest mass of the electron).
Energy and Momentum Conservation
During the collision, both energy and momentum must be conserved. The energy of the photon can be expressed using the equation:
E_photon = hc/λ
where h is Planck's constant and c is the speed of light.
The total initial energy (E_initial) before the collision can be written as:
E_initial = E_photon + E_electron = hc/λ + mc²
Post-Collision Scenario
After the collision, the photon is scattered backward (180 degrees) and has a new wavelength, which we will denote as λ'. The energy of the scattered photon is:
E'_photon = hc/λ'
The electron, now moving in the negative x-direction, will have a new energy, which we can denote as E'_electron. The total energy after the collision is:
E_final = E'_photon + E'_electron = hc/λ' + E'_electron
Applying Conservation Laws
From conservation of energy, we have:
hc/λ + mc² = hc/λ' + E'_electron
From conservation of momentum, we can express the momentum of the photon and the electron. The momentum of the photon before the collision is:
p_photon = E_photon/c = hc/λc = h/λ
The momentum of the electron before the collision is:
p_electron = mv (where v is the velocity of the electron).
After the collision, the momentum of the photon is:
p'_photon = -E'_photon/c = -hc/λ'c = -h/λ'
The momentum of the electron after the collision can be expressed as:
p'_electron = mv' (where v' is the new velocity of the electron).
Setting Up the Momentum Conservation Equation
Applying conservation of momentum gives us:
h/λ + mv = -h/λ' + mv'
Solving for the New Wavelength
Now, we can rearrange the equations to isolate λ'. First, let's focus on the energy conservation equation:
hc/λ' = hc/λ + mc² - E'_electron
Next, we need to express E'_electron in terms of the new momentum and energy. For small energies (E << mc²), we can approximate the change in energy of the electron. However, for simplicity, we can assume that the electron's kinetic energy remains approximately mc² after the collision.
Now, substituting this back into the momentum equation and solving for λ' involves some algebra. After simplification, we find:
λ' - λ = \frac{h}{mc}(1 - \cos(180°))
Since cos(180°) = -1, we have:
λ' - λ = \frac{2h}{mc}
Thus, the new wavelength of the scattered photon is:
λ' = λ + \frac{2h}{mc}
Final Expression
In summary, the expression for the new wavelength of the scattered photon after a head-on collision with a moving electron is:
λ' = λ + \frac{2h}{mc}
This result illustrates how the wavelength of the photon increases after scattering, which is a key aspect of Compton scattering. The increase in wavelength corresponds to the energy transferred to the electron during the collision.
