Commutators of Operators

Please help me with this. I do not know how to do this. There is not even one example of doing these proofs in the book. The book just gives us these properties.

Momentum space representation of a wavefunction.

See attached file.

Linearly independent and normalized degenerate eigenfunctions

I don't even know where to start with this problem. If someone could show this proof I would appreciate very much.

Compton scattering

I'm looking at instructor notes for one of the problems in my textbook, the link is here: http://www.physics.smu.edu/~coan/hw_sols/5382/f02/5382_02_hw_sols_02.pdf On page (3), where it says "Part B" P'Max occurs when...., can you please explain to me how they got the equations directly below till the end of the Page (3).

Frequency shift in Compton scattering

How would one obtain the Compton shift equation if the incident photon and electron were moving in opposite directions to begin with?