## Wigner-Eckart theorem

The mathematical form of the theorem is, given a tensor operator of rank $k$, $T^{(k)}$, The expectation value on the eigen-state $\left|j,m\right>$ of total angular momentum $J$ is,

$\left = \left \left$

where, $\left$ is reduced matrix element. The power of the theorem is that, once the reduced matrix element is calculated for the system for a particular (may be the simplest) case, all other matrix element can be calculated.

The theorem works only in spherical symmetry. The state are eigen-state of total angular momentum. We can imagine, when the system rotated, there is something unchanged (which is the reduced matrix element). The quantum numbers $m, m'$ define some particular direction of the state, and these “direction” will cause an additional factor, which is the Clebsch-Gordan coefficient.

Another application is the Replacement theorem.

If any 2 spherical tensors $A^{(k)}, B^{(k)}$ of rank-k, using the theorem, we have,

$\displaystyle \left = \frac{\left}{\left} \left$

This can prove the Projection theorem, which is about rank-1 tensor.

$L , J$ are orbital and total angular momentum respectively. The projection of $L$ on  $J$ is

$L\cdot J = L_z J_z - L_+ J_- - L_-J_+$

The expectation value with same state $\left|j m\right>$,

$\left< L\cdot J\right> = \left< L_z J_z\right> - \left< L_+ J_-\right> - \left$

using Wigner-Eckart theorem, the right side becomes,

$\left< L \cdot J \right> = c_j \left$

where the coefficient $c_j$ only depends on $j$ as the dot-product is a scalar, which is isotropic. similarly,

$\left< J \cdot J \right> = c_j \left$,

Using the Replacement theorem,

$\displaystyle \left< L \right> = \frac{\left}{\left} \left$

Thus, we have,

$\displaystyle \left< L \right> = \frac{\left< L\cdot J \right>}{\left} \left$

as the state is arbitrary,

$\displaystyle L = \frac{L\cdot J}{J\cdot J} J$

this is same as the classical vector projection.

## Projection theorem

The simplest way to say is:

a operator can be projected on another one, for example, The orbital angular momentum cab be projected on the total angular momentum.

$L = L\cdot J \frac {J}{j(j+1)}$

a simple application is on the Zeeman effect on spin-orbital coupling. the Hamiltonian is:

$H_B = - \mu \cdot B = - ( \mu_l L + \mu_s S ) \cdot B$

by the Wigner-Eckart theorem:

$L = L\cdot J \frac {J}{j(j+1)}$

$S = S\cdot J \frac {J}{j(j+1)}$

then the Hamiltonian becomes:

$H_B = - \frac{1}{j(j+1)} ( \mu_l (L \cdot J) + \mu_s (S \cdot J) ) J\cdot B$

and introduce the Bohr Magneton and g-factor:

$H_B = - g \mu_B J \cdot B$

$g = - \frac{1}{j(j+1)} ( g_l (L \cdot J) + g_s (S \cdot J) )$