2
$\begingroup$

I have a doubt. If Legendre equation has a polynomial solution, is the constant $l$ in $l(l+1)$ necessarily a integer number? Asked in another way, is possible $l(l+1)$ be a integer if $l$ is not an integer?

Thanks in advance.

1 Answers 1

1

Yes (although this is not really relevant for the Legendre polynomials): $l = \frac{-1 + \sqrt{5}}{2}$ $l(l+1) = 1$

  • 1
    It is not relevant for physics. Not considering half-integer intrinsic angular momentum (spin), the (orbital) angular momentum quantum number is always an integer. The reason that your text book says that non-integer $l$ is not relevant, is that in that case the resulting solution of the Schrodinger equation is not square-integrable - and hence the solution does not fulfill the probability postulate of quantum mechanics.2014-04-23