Transformation Properties of the Hamiltonian

An irreducible tensor operator of rank k is an operator with the property that it transforms under rotations according to 

The simplest example is a scalar operator 2Tq0-1 with the transformation property 

This means that scalar operators are invariant with respect to rotations in coordinate or spin space. An example for a scalar operator is the Elamiltonian, i.e., the operator of the energy. 

A first-rank tensor operator 3 V) is also called a vector operator. It has three components, 2T and jH j. Operators of this type are the angular momentum operators, for instance. Relations between spherical and Cartesian components of first-rank tensor operators are given in Eqs. [36] and [37], Operating with the components of an arbitrary vector operator ( 11 on an eigenfunction u1fF) of the corresponding operators and 3 yields 

0 conserves the projection Mp of P on the z axis, increases Mp by one unit, and decreases Mp by one unit. The prefactors in Eq. [150] differ slightly from those obtained in Eq. [74] by operating with step operators on Uj1) (see the earlier section on step/shift/ladder and tensor operators). 

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