Spin/ angular momentum, representations of SO(3), SU(2)

[plmokn2]

Homework Statement

I'm trying to understand why particles have both spin and angular momentum in terms of group theory. As I understand it orbital angular momentum comes from the normal generators SO(3) which are intuitively infintesimal rotations so d/d(theta) etc. Also spin comes from representations of SO(3) in terms of SU(2j+1) so generators of SU are equivilent to generators of SO(3) so are conserved.

The thing is I don't really get why a particle only has one spin degree of freedom (so particles are only ever spin 1/2 or spin 1 or whatever) which is equivilant to one of the SU representations but particles also have orbital angular momentum as well as spin.

The Attempt at a Solution

It seems reasonable, looking at the operators, that spin is a fundamentally different from orbital angular momentum, but I can't see why this should be the case.

Any help appreciated.
Thanks

 

[Jhero]

for your question! The concept of spin and angular momentum can be quite confusing, especially when trying to understand it in terms of group theory. Let me try to break it down for you.

First, let's define what spin and angular momentum are. Spin is a fundamental property of particles that describes their intrinsic angular momentum, while orbital angular momentum is the angular momentum associated with a particle's motion around a central point.

Now, as you mentioned, spin and orbital angular momentum are described by different mathematical groups. Spin is described by the special unitary group, SU(2), which has the property that its generators are equivalent to the generators of the three-dimensional rotation group, SO(3). This is why we can use SU(2) to represent spin in terms of rotations in three-dimensional space.

On the other hand, orbital angular momentum is described by the rotation group, SO(3). This group describes the rotations of objects in three-dimensional space, such as the motion of a particle around a central point.

So why do particles only have one spin degree of freedom? This is because spin is a fundamental property of particles, and it is not related to their motion in space. It is an intrinsic property that cannot change. On the other hand, orbital angular momentum is related to a particle's motion and can change depending on its position and velocity.

In terms of group theory, this means that spin is a symmetry of the particle itself, while orbital angular momentum is a symmetry of the system in which the particle is moving.

I hope this helps to clarify the difference between spin and orbital angular momentum. If you have any further questions, please don't hesitate to ask. Good luck with your studies!