Wigner-Eckart theorem and reduced matrix element

[Tilde90]

Wigner-Eckart theorem and "reduced matrix element"

Hello,

I am studying the Wigner-Eckart theorem and I have found some difficulties understanding the reduced matrix element of a spherical tensor.
In fact, a spherical tensor is commonly defined through its transformation properties, and I imagine it as a "vector of angular operators": the Wigner-Eckart theorem evaluates one matrix element of a component of this vector. However, I cannot understand the meaning of the reduced matrix element involved in the expression of the theorem.
Please, could you explain to me the "idea" behind it, or where is the mistake in my idea of spherical tensors (if there is a mistake)?

Thank you very much for your help!

 

[Saraphysics123]

I have same problems

 

[DrDu]

The point is that you only have to calculate only one matrix element -- in most cases, the one with maximal $M_J$ is easiest to calculate -- divide it by the corresponding Clebsch-Gordan coefficient to get the reduced matrix element. From this you can calculate all the other matrix elements by multiplication with the corresponding CG coefficients.

 

[Saraphysics123]

Thank you

The point is that you only have to calculate only one matrix element -- in most cases, the one with maximal $M_J$ is easiest to calculate -- divide it by the corresponding Clebsch-Gordan coefficient to get the reduced matrix element. From this you can calculate all the other matrix elements by multiplication with the corresponding CG coefficients.

Thank you do you have problems for this theory