# Group-theoretical reason

#### dextercioby

Homework Helper
... for the fact that the orbital angular momentum weight is NOT a semi)integer positive number, but an integer.

Is there such a reason...? I've never seen it in some book. I know there are other reasons for which we conclude that "l" MUST be an integer, see Sakurai's thoughts attached.

However, orbital angular momentum is a type of angular momentum, the latter which, at quantum level, is the self-adjoint generator of the unitary group representations of the rotation symmetry group $SO(3)$.

So there has to be some group-theoretical reason for which "l" must be an integer and NOT a semi-integer, soe other that Sturm-Liouville theory of PDE-s, etc...(see Sakurai)

Daniel.

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#### dextercioby

Homework Helper
Here's what my QM teacher had to say.

Single-valued irreducible unitary representations of SO(3) 's covering group (i.e. SU(2) ) correspond to both single-valued and double-valued representations of SO(3). This fact is well known, it's due to the covering homomorphism which is double valued. The idea is that single-valued irreps of SO(3) correspond to proper rotations to which the orbital angular momentum is the self-adjoint generator. Since single-valued irreps of SO(3) are characterized by integer weights of angular-momentum, it thus follows that the weights of orbital angular momentum are integer. End of story.

If one sees any flaws in the argumentation above, poke me in the eye..

Daniel.

#### Haelfix

"Since single-valued irreps of SO(3) are characterized by integer weights of angular-momentum"

Yea I think thats true (off the top of my head), once thats established the rest goes through trivially.