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 expression for the normalised associated Legendre polynomials is