Is Spin Related to Angular Momentum in Elementary Particles?

[Ahmad Kishki]

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?

 

[Orodruin]

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.

 

[Ahmad Kishki]

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.

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.

 

[Orodruin]

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.

 

[Ahmad Kishki]

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.

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

 

[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.

 

[Orodruin]

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.

 

[Ahmad Kishki]

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.

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

 

[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.

 

[Ahmad Kishki]

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

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

 

[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".

 

[Ahmad Kishki]

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".

Oh well, i will go over this in a second reading of qm - i am currently self studying from griffiths and noticed the weaknesses of the book in angular momentum, i will remedy this asap from shankar and sakurai after i am done till the applications part.