Any nice interpretation of differentiating $j(j+1)$ to get $2j+1$?

Question

If $j$ is a continuous variable, then differentiating the function $f(j)=j(j+1)$ with respect to $j$ gives $f'(j)=2j+1$. Of course I've chosen the letter to evoke quantum mechanical angular momentum, in which case for integer or half-integer values of $j$ we can interpret these two expressions as the eigenvalue of the squared angular momentum and the multiplicity of the angular momentum.

Is there any nice interpretation of this? As I was looking at the automatically generated list of "Questions that may already have your answer," I came across a comment that asks exactly the same question.

One reason to believe that it has no very special interpretation is that since the actual variable is discrete, the derivative $f'$ would really represent an approximation to a divided difference, and the relevant difference for a unit change in $j$ does not necessarily equal the derivative unless you evaluate the derivative at the correct place.

It seems to me that there is a second reason not to expect anything special here, which is that the correspondence doesn't seem to work except in three dimensions. For a rotor in $d$ dimensions, the eigenvalue of the squared angular momentum operator is $j(j+d-2)$. I don't know what the multiplicity of states is in general, but I suppose it's a polynomial of order $d-2$. E.g., for $d=2$, the multiplicity is 2 ($m=\pm j$), which doesn't equal the derivative of $j(j+d-2)=j^2$. On the other hand, I guess it's possible that there is a nice interpretation, and the nice interpretation tells us that there's something special about three dimensions.

Related: What is known about the hydrogen atom in $d$ spatial dimensions?

Answer 1

The number of spherical harmonics of weight $\ell$ on $S^d$ is given by $$ N(d,\ell) = \frac{d+2\ell-1}{d-1} {d+\ell-2\choose \ell} $$ Further, the eigenvalue of this spherical harmonic is $-\Delta(d,\ell) = \ell ( \ell + d - 1 )$.

In particular, note that for $d=2$ (which is the case to consider for 3 dimensional angular momenta, we find $$ N(2,\ell)= 2\ell+1~, \qquad -\Delta(2,\ell) = \ell(\ell+1)~. $$ In this special case, it is true that $-\partial_\ell \Delta(2,\ell) = N(2,\ell)$ as you have observed. I did note the following generalization to higher dimensions, $d\geq3$, $$ \partial_\ell^{d-3} N(d,\ell) = - \Delta(d,\ell) + \frac{d^2}{4} - \frac{7d}{12} + \frac{1}{2}~. $$ I do not see any real property of interest however. I believe the formula is a pure coincidence.

PS - This is one case where I would love to be proven wrong!