Past exam of the quantum physics course of the University of Oxford / 2016 / 8 / d / Solution

Direct integration over solid angle givesThus $|C|=\sqrt{(2l+1)!/(4\pi )}/(2^{l}l!)$. Choosing the usual alternating sign convention,Apply $L_{-}$ successively and divide by its ladder coefficient at each step. Explicitly,The normalization follows because the product of the squared ladder coefficients is ${\hslash }^{2(l-m)}(2l)!(l-m)!/(l+m)!$. This constructs every magnetic sublevel, including negative $m$.