3.8

a)

check that the eigenvalues of the hermitian operator in example 3.1 are real. show that the eigenfunctions are orthogonal.

Qf = if'

the eigenvalues are 0,+- 1, etc., which are obviously real.


pick two arbitrary eigenfunctions:

f = A exp(-i q phi)
g = A exp(-i q' phi)

<f|g> = A*A int[exp(i q phi) exp(-i q' phi)] dphi[0,2pi]
	= A*A int[exp(i (q - q') phi)] dphi[0,2pi]
	= A*A [i(q-q')]^-1 [exp(i(q-q') phi)]|[0,2pi]