Quantum Mechanics Solution Manual 

© Leon van Dommelen 

4.2.3.1 Solution anguca
Question:
The general wave function of a state with azimuthal quantum number and magnetic quantum number is , where is some further arbitrary function of . Show that the condition for this wave function to be normalized, so that the total probability of finding the particle integrated over all possible positions is one, is that
Answer:
You need to have 1 for the wave function to be normalized. Now the volume element is in spherical coordinates given by , so you must have
Taking this apart into two separate integrals:
The second integral is one on account of the normalization of the spherical harmonics, so you must have