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.

 

[denian]

To calculate the explicit form of irreducible tensor operators in a given basis, one can use the Wigner-Eckart theorem. This theorem relates the matrix elements of a tensor operator in one basis to those in a different basis. The explicit form of the tensor operator can then be obtained by solving the resulting equations.

In the case of expanding a 3x3 density matrix in the angular momentum basis, the tensor operators T_1n are associated with the angular momentum operators J+, J_z, and J-. Similarly, the tensor operators T_2n correspond to the angular momentum operators J++, J_+, J_z, J_-, and J--. These operators represent the components of the quadrupole moment, which describes the shape of a system.

The physical significance of the T_2n operators can be understood by considering their relationship to the quadrupole moment. The quadrupole moment is a measure of the deviation from spherical symmetry in a system. In the case of atoms or molecules, this deviation can arise from the distribution of electrons or nuclei. The T_2n operators capture this deviation in a systematic way, allowing for a more detailed description of the system's shape and properties.

In summary, the explicit form of irreducible tensor operators in a given basis can be calculated using the Wigner-Eckart theorem. In the case of the angular momentum basis, the T_2n operators represent the components of the quadrupole moment, providing information about the shape and symmetry of a system.