Quantum Mechanics measurement of spin

[Phymath]

Homework Statement

reading shankar he develops the measurement of angular momentum by discussing rotation state vectors in 3-D by the angle-and-axis parameterization so he creates these generators of rotation matrices and says these are what we use to measure angular momentum in analogy to classical mechanics.

He goes on to describe these $$D^{(j)} (\hat{n}\theta J^{(j)})$$ when we just use the $$J_i^{(j)}$$ i = 1,2,3 (designation of axis) to measure the angular momentum what are the $$D^{(j)} (\hat{n} \theta J^{(j)})$$ matrices used for? how are they related to the original rotation matrix we are all used to?

such as $$R(\textbf{n}\theta) |v> = |v'>$$ make |v> a typical 3-d vector...and make it rotated about the z axis
$$R(\textbf{z}\theta)|x,y,z> = |x cos(\theta) - y sin(\theta), x sin(\theta) + y cos(\theta),z>$$
how can i use the D matrices to get the same result?

 

[Final]

Hi...
The orbital angular momentum $$L=r \wedge p$$ is the generator of the rotation:
$$U \Psi(r) = e^{i L \cdot \omega}(r)$$. For an infinitesimal rotation $$\omega$$
$$U \Psi(r) \approx (1+ iL\cdot \omega)=(1+\omega_i \epsilon_{ijk}x_j \partial_k)\Psi \approx \Psi(r + \omega \wedge r )$$.
A generic operator transfor like $$O \rightarrow UOU^{\dagger}$$
If you take $$r \rightarrow r+\omega \wedge r$$ that is the action of a rotatation $$\omega$$.
For a wavefunction with spin s the generator is $$e^{i \omega \cdot (L+s)}$$.
For example a wavefuncion of spin 1/2 (it has 2 component) and L=0 transform with
$$U(\theta \bold{n})=cos(\theta/2)+i \bold{n}\cdot \bold{\sigma} sin (\theta /2)$$ where $$\bold{\sigma}$$ are the pauli matrices