Angular momentum is measured in units of hbar. If they were measured in units of 2hbar then half-integer values would be permitted. It's just a matter of convention. So the real meat of your question is (a) why is angular momentum quantized? and (b) why are half values permitted for spin? Of course, the final answer is :because nature does it that way". But, in the context of QM, it all stems from the group SU(2) which permits both integer and half-integer values. But when we project into a space-time context, only intrinsic angular momentum (of a system that has no spatial extent) can be half-integer because otherwise, spatial wave functions would be inherently two-valued (a 2pi rotation changes the sign for half-integer spin) and we would not know how to define a unique wave-function. (In the case of the spin orientation we can obtain a unique wave-function by specifying the unique rotation that takes the position (or momentum) vector into the spin vector, so the 2pi ambiguity does not arise.)

Hi Mike,

I do share your opinion, and I think single-valuedness of the wave function is indeed the reason why the orbital angular momentum must be integer-valued. How would you otherwise define a Schrödinger equation?

I just wanted to mention that this view is not universally shared though: there is an ongoing discussion dating back from Pauli's times until now. See "Biedenharn/Louck: Angular Momentum in Quantum Physics", pp 319ff. And a German article on this: http://www.itp.uni-bremen.de/~noack/orb-ang.pdf.

The way I understand the question, is why half-integer momentum numbers occur in theory and not in experiment. IMHO, the reason is that we humans can only observe the electromagnetic current, which is bilinear in fermionic operators. By laws of momentum addition, any such quantity may only bear integer momentum quantum numbers.

The rotation space group SO(3) has only single valued representations while SU(2) which is its universal covering group has both single and double valued representations. An the space statevector cannot take two different values at the same space point. So, going through a path surrounding the sphere of, e.g. circular shape, after a rotation of 360 you must have the same wavefunction and not one which differs from it for a sign. Actually, from a formal point of view, it is just requiring singlevauedness which imposes to you that the angular momentum takes only integer values.

Angular momentum describes the degree of asymmetry of an object. So, let's take the simple hydrogen atom. The mathematical convention for the state of spherical symmetry - see picture - is ℓ = 0, not ℓ > 0 (e.g. 1/2), neither ℓ < 0 (e.g  -1/2), just ℓ = 0. For the moment, it's just a convention for the case of complete angular symmetry.

In the classical physics L = Iω where I is the momentum of inertia, ω is the angular velocity, and L has some direction in space. In the case of spherical symmetry, there is no preferred direction in space. This can be ragarded as a reason for taking ℓ = 0 for an atom with spherical symmetry.

The values of the angular momentum vary only by ħ, s.t. if we begin from 0, we can have only integer multiples of ħ.

The fact that these choices for the orbital angular momentum are good choices, is further confirmed by the compatibility with the properties of the space-group SO(3) as exlained by Prof. Ghirardi.

The right answer is quite not so simple as single valuedness of WF. because there is no prescription for WF to be single valued (WF is not observable). Single valued must be only the expectation values. I read the article of Noack and it is quite informable for everyone. It turns out that this argument for WF single value is implanted in many QM textbooks (Messiah b.e.) and is erroneous of course. Noack points out that Bohr and Jordan had proved in fact the relation r.L=0 is the reason for l being integer. 

I am not convinced about Noack's line of reasoning, and I think the real topic is whether the WF is supposed to be single-valued or not.

First, if it were not, it would not be a proper function in the first place, at least not in a simply-connected domain (like R^3), and someone needs to explain to me then how to define a 2nd order partial differential equation for it.

Second: yes, the WF is not an observable of course, so it may be ambiguous. It is clear that a global phase change does not alter experimental predictions as expectation values are still unchanged. We all know that q.m. states are defined up to a complex phase, and thus transform according to ray representations of the respective symmetry group of the system under consideration. Still, it needs to be single-valued once a phase convention has been chosen, otherwise we come back to my first point.

So my question is: is the use of terminology maybe sometimes too imprecise? The WF as a solution to the Schrödinger equation may not be unique (and actually never is), but it must be single-valued by definition, and from this, integer-valuedness of the orbital angular momentum follows.

Relative single-valuedness is required because interference is observable! (E.g. double slit. Changing sign of one would convert destructive interference into constructive and vice versa.) Of course there is, as Oliver points out, an arbitrary global phase.