# How is energy linked to angular momentum for a quantum system?

I'm trying to find a relation between the energy, E, of a state and its corresponding angular momentum, $$\ell$$. I was under the impression it was loosely related by:

$$E \sim (\ell (\ell+1)) \hbar$$

Does anyone have any source that can confirm this? I've searched and have found nothing.

I only ask as I'm trying to find a source that can confirm a proportional relationship between l and E for a state within a nucleus, to prove the 1p3/2 state is more energetically favourable than the 1d5/2 state. If anyone has a source would they mind linking it in a reply?

The idea is the following. The Hamiltonian can only depend on $L^2$ (if you'd add terms like $c_x L_x + c_y L_y + c_z L_z$ you'd choose a preferential direction in space). So the simplest possible $SO(3)$-invariant Hamiltonian is

$$H = c L^2$$

for some constant $c > 0$. Next, from $SO(3)$ invariance you know that states are organized into multiplets, meaning that they are labeled as

$$|l,m \rangle$$

with $m=-l,\ldots,l$. Finally, you should remember that the Casimir $L^2$ acts on such states as

$$L^2 |l,m \rangle = l(l+1) |l,m \rangle$$

which essentially proves the relation you want to show. The coefficient $c$ depends on the microscopics of your theory and cannot be fixed by group theory alone.

More generally the Hamiltonian can be an arbitrary function $H = f(L^2)$. In particular, it's possible to have interactions like $(L^2)^2$ etc. In the field theory language, you can think of those as higher-derivative terms.