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PF Gold
P: 1,776
 Quote by paco_uk Thank you. I see what you mean that J can't be forced to obey a rule which spin doesn't obey but I still don't understand the difference between L and J which makes the original argument true for one and not the other. In fact, the more I think about it the less I understand the argument for integer values of ml. Why does a rotation of 2 pi have to leave you with the same state? Common sense dictates that it has to describe the same physics but couldn't this be achieved by the same state with an arbitrary phase factor in front (as is the case for J)?
Firstly a 2pi = 360deg rotation is by definition the identity transformation so by definition equivalent to doing nothing. The real question is how spinors can not be the same after a 2pi rotation if they represent a physical system. The answer of course is they do not represent the system itself but projectively represent information about the system hence the sign change for 2pi rotations of spinors has no direct physical meaning. psi and -psi correspond to the same physical mode.

As to why the orbital component of angular momentum should be integral valued, it is simply the only possibility for single valued wave-functions of an orbital particle. The angular momentum operators e.g. Sz take the form of differential operators on the wave-function psi(x,y,z).

$$L_z \propto [x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}]$$

If you then rewrite this in polar form you see that the eigen-functions are the spherical harmonics with integer eigen-values. You then see the spherical harmonics expressing the angular components of the wave-functions for a particle orbiting a central potential.

One then adds in the intrinsic angular momentum of the orbiting particle which we are surprised to discover may take on half integer values. Hence the total J is integral (orbital) plus half integral (intrinsic) which then may be half integral (3 +1/2 for example).

Ultimately though the answer to your question, "Why" is that that's how nature is observed to behave. There are possible exceptions, see for example anyons which may manifest as quasi-particles in a two dimensional material (such as thin films).