I am reading about the total, orbital and spin angular momentum, and I am not clear as to what these generators actually do after exponentiating.
Could you give me a physical picture of what happens to the quantum set, after being acted upon by the operator obtained after exponentiating these?
For example does the total angular momentum rotate the ket in a circle, about the normal? What about the other two?
Best Answer
Did you try the wikipedia article about angular momentum operators?
The arrows schematically represent the internal state of the particle (its spin state). The blue and red dots represent two particles at different locations. Or, if you prefer, it could be two parts of the wavefunction of a single particle, with blue meaning negative phase and red meaning positive phase.
When you exponentiate Jz (the total angular momentum operator), you get real-world rotation about the z axis -- part A in the figure. The whole system and everything in it is rotated.
When you exponentiate Lz (the orbital angular momentum operator), you get "spatial-only" rotation about the z axis -- part B in the figure. The positions of particles get rotated but their spin states stay exactly the same.
When you exponentiate Sz (the spin operator), you get "internal-only" rotation about the z axis -- part C in the figure. The internal state of the particle is rotated, but the particle itself stays in the same place.
By the way, you should avoid phrases like "ket changes direction" and "rotate the ket". You're thinking about rotation in real three-dimensional space, but the ket is not an object in real three-dimensional space. The phrase "rotate the ket" sort of works for a spin-1/2 particle at a single point, but anyway using that phrase is a very bad habit.