Editing Wigner–Eckart theorem (section)

===Motivating example: position operator matrix elements for 4d → 2p transition===

Let's say we want to calculate [[transition dipole moment]]s for an electron transition from a 4d to a 2p [[atomic orbital|orbital]] of a hydrogen atom, i.e. the matrix elements of the form <math>\langle 2p,m_1 | r_i | 4d,m_2 \rangle</math>, where ''r''<sub>''i''</sub> is either the ''x'', ''y'', or ''z'' component of the [[position operator]], and ''m''<sub>1</sub>, ''m''<sub>2</sub> are the [[magnetic quantum number]]s that distinguish different orbitals within the 2p or 4d [[Electron shell#Subshells|subshell]]. If we do this directly, it involves calculating 45 different integrals: there are 3 possibilities for ''m''<sub>1</sub> (−1, 0, 1), 5 possibilities for ''m''<sub>2</sub> (−2, −1, 0, 1, 2), and 3 possibilities for ''i'', so the total is 3 × 5 × 3 = 45.

The Wigner–Eckart theorem allows one to obtain the same information after evaluating just ''one'' of those 45 integrals (''any'' of them can be used, as long as it is nonzero). Then the other 44 integrals can be inferred from that first one—without the need to write down any wavefunctions or evaluate any integrals—with the help of [[Clebsch–Gordan coefficient]]s, which can be easily looked up in a table or computed by hand or computer.