The normalisation constant ensures that the associated Legendre polynomials are properly scaled, thereby maintaining the probabilistic interpretation of quantum states.

To determine the normalisation constant, we begin by replacing the index with in eq383 and expanding and using the binomial theorem to give:

Since and are polynomials of degrees and , respectively, only the and terms in the summations survive. Simplifying eq384 yields

###### Question

Evaluate .

###### Answer

Let and so .

To determine , consider . Let and . Then, and . Integrating by parts,

Multiplying through by and substituting eq386 yields

Changing the variable back to , where , and , gives

Eq387 is a recurrence relation, where

or equivalently |

Substituting eq388 through eq390 in eq387 results in

is the product of odd numbers of . In other words, we can express as the ratio of to the product of even numbers of . The product of even numbers of is the product of the numerators of the RHS of eq388, which is equal to . Therefore, and

Substituting eq389 in eq385 and simplifying gives

Therefore, the normalisation constant of the associated Legendre polynomials is

With reference to eq364, the normalised associated Legendre polynomials is