quantum mechanics - Question on Total, Orbital and Spin Angular momentum

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?

Answer

Angular momenta are sets of three operators $\{\hat{L}_{i}\}$ such that

$$[\hat{L}_i,\hat{L}_j]=i\hbar\epsilon_{ijk}\hat{L}_{k} $$

where $\epsilon_{ijk}$ is the Levi-Civita symbol. Orbital angular momentum and spin are angular momenta because they satisfy that commutation relation. (That commutator can be derived if you define $\vec{L}=\hat{\vec{r}}\times\hat{\vec{p}}$, a formal vector product between vector operators)

As in classical physics, angular momentum is useful when you have spherical symmetry. Given the case, instead of using Cartesian coordinates $x,y,z$ you may want to use $r,\theta,\phi$ to represent the wave function $\Psi(\vec{r})$.

The eigenstates of these operators $|l,m\rangle$ can be used to write your wave function as

$$\Psi(\vec{r})=R(r)\varphi(\theta,\phi) $$

with

$$ \varphi(\theta,\phi)=\displaystyle\sum_{l,m}c_{lm}|l,m\rangle$$

You can write $|l,m\rangle$ as the spherical harmonics.

Angular momentum is related to rotations, they are the infinitesimal generators of rotations. What does that mean? It means that you can write a rotation of angle $\alpha$ over an axis $\vec{n}$ exponential of the angular momentum about that axis $\hat{L}_{n}$

$$\hat{R}(\vec{n},\alpha)=\exp(-i\frac{\alpha}{\hbar}\vec{n}\cdot\vec{L}) $$

where the dot product is understood as $n_{1}\hat{L}_{1}+n_{2}\hat{L}_{2}+n_{3}\hat{L}_{3}$. As you said the rotation operator $\hat{R}(\vec{n},\alpha)$ does rotate the quantum state represented by $\Psi(\vec{r})$.

Spin case is a little more tricky because the introduction of spin (e.g. $1/2$) in the formalism requires the Hilbert space of states of the system, say $\mathcal{H}$, as well as a two dimensional complex vector space $\mathbb{C}^2$ to the variables of spin. And this lead us to the introduction of quantities called spinors, usually represented by column vectors. The idea of the spin is the same as discussed above, the spin-space rotation operator can be written as exponentials of the spin operators and they do rotate the spinor of the state.

An arbitrary rotation operator acting upon an arbitrary state vector may be a little hard to write, but you can expand the state as a series of the angular momentum operator eigenstates and the spin operator eigenstates to simplify the calculations. The answer of course depends on the particular rotation and state, but as I said, it is always a rotation (in $\mathcal{H}$, in $\mathbb{C}^2$ or both).

Reference: http://students.washington.edu/tkarin/rotations.pdf it a crash course but it may be helpful