Wigner-Eckart theorem

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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<j m\right|T_q^{(k)} \left|j' m'\right> = \left<j' m' k q| j m \right> \left<j||T^{(k)}||j'\right>

where, \left<j||T^{(k)}||j\right> 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<j m|A^{(k)}|j' m' \right> = \frac{\left<j||A^{(k)}||j'\right>}{\left<j||B^{(k)}||j'\right>} \left<j m|B^{(k)}|j' m' \right>

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<L_- J_+\right>

using Wigner-Eckart theorem, the right side becomes,

\left< L \cdot J \right> = c_j \left<j||L||j\right>

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<j||J||j\right> ,

Using the Replacement theorem,

\displaystyle \left< L \right> = \frac{\left<j||L||j\right>}{\left<j||J||j\right>} \left<J \right>

Thus, we have,

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

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

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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) )