Problem 3

An atom of mass

M

, initially at rest, absorbs a photon of wavelength

\lambda

. Assuming

Mc^2\gg hc/\lambda

, find its approximate recoil speed;

h

is Planck’s constant and

c

the speed of light. (a) State the assumptions and establish the governing equation for the requested quantity. (b) Solve it in terms of the given symbols, identifying the direction, sign, or physical nature of the result. (c) Check a simple limiting case and state when the model remains valid. Use SI units consistently, retain the symbols in the setup, and enforce all geometric constraints and initial conditions.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{p_\gamma=h/\lambda,\quad Mv=p_\gamma,\quad v=h/(M\lambda),\quad v\ll c}

problems.proof.analysis

The results are listed in the order requested and use one consistent sign convention. Their dimensions, initial conditions, and model constraints must all agree.