# Spin and angular momentum

1. Feb 18, 2015

ok i just learned that spin comes up when l (azimuthal quantum number) is half integer but then my book says that each elementary particle has a specific and immutable value of spin. Ok now does this mean that l (azimuth quantum number) takes two values at once - One value corresponding to spin and another to angular momentum?

2. Feb 18, 2015

### Orodruin

Staff Emeritus
There are two different concepts here, the l is related to the orbital angular momentum. Spin is something else, it is an intrinsic angular momentum of the particle itself, e.g., of an electron. The orbital angular momentum will combine with the intrinsic spin of the particle to form a total angular momentum.

As far as we have observed, there are elementary particles with spin 0, 1/2, and 1.

3. Feb 18, 2015

Ok i am getting it, but could you expand on this a little bit? So far the math i saw lead to "extrinsic" angular momentum implied spin since m when it was half integer lead to a weird result.

4. Feb 18, 2015

### Orodruin

Staff Emeritus
You never imply half integer spins exist just from the mathematics. However, the mathematics that apply to orbital angular momentum turns out to also allow representations with half integer angular momentum (although not orbital). Now, in Nature, it just so happens that there are objects that can be described by these half integer angular momentum representations.

5. Feb 18, 2015

Oh ok, so they are unrelated mathematically. But what about those half integer values of l - we discard them for orbital angular momentum?

6. Feb 18, 2015

### Khashishi

Total angular momentum, represented by J, is the vector sum of the spin angular momentum, represented by S, and orbital angular momentum, represented by L. Only spin can have half-integer values (integer values are also possible). Orbital angular momentum is always integer valued. If the spin angular momentum is half-integer, then the total angular momentum will also be half-integer. I think extrinsic angular momentum means the same thing as orbital angular momentum, which is a much more standard name.

When we say spin is 1/2, what we mean maximum projection of spin along any direction is 1/2 (in units of hbar), since spin is a vector[1] quantity with direction and magnitude. Quantum mechanics is weird and the projection of spin is only allowed to take on discrete values separated by units of hbar, so in the case of spin 1/2, the values of spin projection are either 1/2 or -1/2, with nothing in between.

[1] More accurately, it is a spinor, which is a kind of vector in the linear algebra sense, but somewhat different from a vector in the geometric sense.

7. Feb 18, 2015

### Orodruin

Staff Emeritus
No, it is the same mathematics, it is just that the half integer values do not appear for orbital angular momentum. This has to do with orbital angular momentum being based on the properties of a wave function in a rotationally symmetric potential.

8. Feb 18, 2015

Oh seems like i was confused since my book showed the half integer values through algebraic means. Now it makes sense. Thank you

9. Feb 18, 2015

### kith

If you are interested in more detail why half-integer values of orbital angular momentum are not possible, have a look at the spherical harmonics chapter in Sakurai. He gives several arguments.

10. Feb 18, 2015

The reason i had was because probability can not be multivariable, but i sure will check sakurai. Thank you.

11. Feb 18, 2015

### dextercioby

There's a simple explanation as to why the angular momentum eingenvalues need to be positive integers. It's a restatement of the last argument mentioned by Sakurai (but not expanded on, because his maths is a little sketchy): the Laplace-Beltrami operator on S2 is self-adjoint iff l is integer and non-negative. A proof of this appears in several books, see for example G. Teschl's "Mathematical Methods for Quantum Mechanics".

12. Feb 19, 2015