\prob{28}
A system is known to be in the normalized state described by the wave function
,
where the are the spherical harmonics. The probability of finding the system in a state with azimuthal orbital quantum number m=3 is
 0
 1/15
 1/6
 1/3
 13/15

Quantum Mechanics}Probability
One doesn't actually need to know much (if anything) about spherical harmonics to solve this problem. One needs only the relation . Since the problem asks for states where , and it gives the form of spherical harmonics employed as , one can eliminate the third term after the dotproduct.
So, the given wave function gets dotproduct'ed like , as in choice (E).
