# Group-theoretical reason

1. Mar 8, 2006

### dextercioby

... 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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2. Mar 8, 2006

### dextercioby

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.

3. Mar 8, 2006

### 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.

4. Mar 9, 2006

### dextercioby

On giving it a second thought, this part is if not false, then at least suspicious:

"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."

Daniel.

Last edited: Mar 9, 2006