Calculating Irreducible Tensor Operators in a Given Basis

[msamp]

Perhaps very simple, but it eludes me:

How does one calculate an explicit form for the irreducible tensor operators in a given basis? In my particular case, I'm looking at expanding a 3X3 density matrix in the angular momentum basis. T_1n (n = -1, 0, 1) are simple enough : J+, J_z, J-. But what about T_2n (n = -2 ... 2)? I know the answer, but don't know how it was arrived at...

Clues?

(Note : '_', as usual, indicates that what follows is a subscript)


Oh - and if you can help out with a physical significance for the T_2n I would appreciate it. Again - T_1n are angular momenta, but T_2n?

 

[azntoon]

The explicit form of the irreducible tensor operators, T_2n, can be derived using the Wigner-Eckart theorem. This theorem states that for a given angular momentum state, the matrix elements of an irreducible tensor operator can be written as a product of a scalar coefficient and a reduced matrix element which is independent of the quantum numbers of the states. The scalar coefficient is dependent on the Clebsch-Gordan coefficients. The reduced matrix elements can be obtained from the Racah algebraic equations. The Racah algebraic equations are a set of equations that relate the reduced matrix elements to each other. They can be used to determine the explicit form of all irreducible tensor operators in the given basis. The physical significance of the T_2n operators is that they contain information about the quadrupole moments of a system. They describe the spatial arrangement of the charge distribution in space.