Derivation of the Compton Shift Equation

This derivation treats the target electron as free and initially at rest. The incident photon has momentum magnitude p, the scattered photon has magnitude p', and the electron recoils with momentum magnitude P.

1. Governing relations

For each photon,

E = pc, p = h/λ

For the electron of rest mass mₑ and total energy Eₑ,

Eₑ² = P²c² + mₑ²c⁴

Both momentum components and total energy are conserved.

2. Momentum conservation

Along the initial photon direction,

p = p' cos θ + P cos φ

Perpendicular to it,

0 = p' sin θ-P sin φ

Hence

P cos φ = p-p' cos θ

and

P sin φ = p' sin θ

Squaring and adding eliminates φ:

P² = (p-p' cos θ)² + (p' sin θ)²

Therefore

P² = p² + p'²-2pp' cos θ

3. Energy conservation

Initially, the electron contributes only its rest energy:

pc + mₑc² = p'c + Eₑ

Thus

Eₑ = (p-p')c + mₑc²

Insert this into the electron energy–momentum relation and divide by c²:

(p-p' + mₑc)² = P² + mₑ²c²

Expanding and cancelling mₑ²c² gives

P² = (p-p')² + 2mₑc(p-p')

4. Eliminate the electron momentum

Equate equations (1) and (2):

p² + p'²-2pp' cos θ = (p-p')² + 2mₑc(p-p')

Since (p-p')² = p² + p'²-2pp', cancellation leaves

pp'(1- cos θ) = mₑc(p-p')

Divide by pp':

1- cos θ = mₑc(1/p'-1/p)

Using p = h/λ and p' = h/λ',

1- cos θ = (mₑc/h)(λ'-λ)

Therefore

λ'-λ = (h/mₑc)(1- cos θ)

5. Checks and interpretation

• Dimensions: h/(mₑc) has units of length.
• Forward limit: θ = 0 gives λ' = λ.
• Back-scattering: θ = 180° gives the maximum shift 2h/(mₑc).
• Energy direction: 1- cos θ ≥ 0, so λ' ≥ λ; the photon never gains energy from an electron initially at rest.

The electron recoil angle disappears from the result because its momentum and energy are eliminated using the conservation laws.

5. Common mistakes

• Conserving photon energy by itself. Energy transfers to the electron.
• Using P²/(2mₑ) for electron kinetic energy inside an otherwise relativistic derivation.
• Losing the rest-energy term mₑc².
• Squaring momentum components and forgetting the cross term -2pp' cos θ.
• Substituting p = h/λ with the primed wavelength paired to the wrong momentum.

6. Worked Examples

Next: return to Compton Shift or the Quantum Theory of Light Hub.