User:Xxxsemoi/Spherical tensor

Definition
A spherical tensor $$T^{(k)} $$ of rank k is defined by the two following properties :
 * $$[J_z, T^{(k)}_q] = \hbar q \,T^{(k)}_q$$

and
 * $$[J_\pm, T^{(k)}_q] = \hbar \sqrt{(k\mp q)(k\pm q+1)} \,T^{(k)}_{q\pm 1}$$.

In contrast a vector operator $$ V$$ is defined by
 * $$[V_m, J_n] = \hbar \epsilon_{n m k} \, V_k$$.

Examples
Examples for a spherical tensor of rank 1 are
 * $$T^{(1)}_{0}=z $$ and $$T^{(1)}_{\pm1}=\mp (x \pm i y)/{\sqrt{2}}$$

where $$x$$, $$y$$ and $$z$$ are the position operators in quantum mechanics, and
 * $$J^{(1)}_{0}=J_z $$ and $$J^{(1)}_{\pm1}=\mp (J_x \pm i J_y)/{\sqrt{2}}$$

where $$J_x$$, $$J_y$$ and $$J_z$$ are the operators for the total electronic angular momentum.