Quantum Mechanics and conservation of momentum

[ericboyer]

I need help getting started solving the following.

Show that when the recoil kinetic energy of the atom, p^2/2M, is taken into account the frequency of a photon emitted in a transition between two atomic levels of energy difference delta-E is reduced by a factor which is approximately (1 - delta E/2Mc62). (Hint: The recoil momentum is p=hv/c.) Compare the wavelength of the light emitted from a hydrogen atom in the
3-->1 transistion when the recoil is taken into account to the wavelength without accounting for recoil.

My textbook is very vague on this topic so I was wandering if anyone knows where to start answering a question like this. Any useful formulas that I can use to do this proof. If so, do you have any good links relating to this material.

What is recoil kinetic energy and momentum? I understand that a photon is emitted when an electron is reduced to a lower energy state. I just don't understand how this recoil KE fits in and how to relate everything. I need a starting point!

Thanks for any help you can provide. I am not looking for someone to answer this for me, just someone to help me through it. Thanks.

 

[Astronuc]

Start with conservation of momentum. Before the photon is emitted, the net momentum of the system is 0, therefore the momentum of atom P = Mv must equal the momentum of the photon p = E/c = h$\nu$ for the net momentum to remain 0.

So the atom then recoils with a velocity v = P/M = p/M.

Now since the atom moves, the wavelength/frequency are affected - Doppler effect.