Close a two-dimensional photon–electron momentum ledger

Begin with evidence

Compton scattering couples photon energy and momentum. An initially stationary free electron recoils when a photon changes direction and wavelength. The outgoing photon generally has a longer wavelength and lower energy. With scattering angle θ measured from the incoming +x direction, the shift is Δλ=h(1−cosθ)/(me c). It depends on angle, not directly on the incident wavelength.

Connect the model

Incoming photon momentum is (h/λ,0), while outgoing photon momentum is (h cosθ/λout,h sinθ/λout). Conservation therefore gives electron momentum components px=h/λ−h cosθ/λout and py=−h sinθ/λout. Adding momentum magnitudes instead would discard the measured geometry. The vector diagram uses one spatial scale for both axes and labels each vector; it is not a picture of spatial trajectories.

Interpret the ledger

The electron gains kinetic energy hc(1/λ−1/λout). In the short-wavelength range used here, do not automatically substitute p²/(2me) for that recoil energy. The software’s optional relativistic consistency check uses the electron rest energy and the total momentum magnitude; the core audit closes photon energy and both momentum components. The incoming electron is free and at rest by assumption, not an arbitrary bound electron in matter.

Test the boundary

At θ=0 the photon wavelength is unchanged and recoil momentum is zero. A direction angle for that zero vector is undefined, so the interface omits it. At 180 degrees the wavelength shift is maximal. The supplied X-ray-scale quantities are numerical examples only; no ionizing source, detector or exposure procedure is part of this activity.

Worked original case

For λ=20 pm and θ=90°, λout≈22.4264 pm. Photon energy falls from≈61.9921 to 55.2849 keV, leaving≈6.70720 keV recoil energy. Electron momentum is approximately (3.31304,−2.95458)×10⁻²³ kg m/s, directed below the incoming axis.

Predict, compare and retain evidence

Can a nonzero outgoing photon y momentum coexist with zero electron y momentum in this isolated initial-rest model? Compare angles 0, 90 and 180 degrees at 20 pm. Retain wavelength, both photon components, electron components and kinetic energy. Independently sum x and y momenta; then change incident wavelength at fixed angle to test what stays invariant.

Changed-case practice

Compare the wavelength shifts at 60 and 120 degrees for the same incident wavelength. Find their ratio and state the sign of electron y momentum at both angles.

Check the reasoning after your attempt

Factors 1−cosθ are 1/2 and 3/2, so the 120-degree shift is three times the 60-degree shift. Photon y momentum is positive at both angles, so electron y momentum is negative.

Reflect and source-check

Why is a zero recoil angle different from an undefined recoil direction?

Original prose and supplied numerical evidence; checked 30 October 2026 against the current AP Physics 2 framework, fall 2026 clarifications and OpenStax. Virtual-only: no physical source, exposure or apparatus is requested. Public examples and model captures are not independent mastery or supervised laboratory participation.