Editing Wigner–Eckart theorem (section)

==Background and overview==

===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.

===Qualitative summary of proof===

The Wigner–Eckart theorem works because all 45 of these different calculations are related to each other by rotations. If an electron is in one of the 2p orbitals, rotating the system will generally move it into a ''different'' 2p orbital (usually it will wind up in a [[quantum superposition]] of all three basis states, ''m'' = +1, 0, −1). Similarly, if an electron is in one of the 4d orbitals, rotating the system will move it into a different 4d orbital. Finally, an analogous statement is true for the position operator: when the system is rotated, the three different components of the position operator are effectively interchanged or mixed.

If we start by knowing just one of the 45 values (say, we know that <math>\langle 2p,m_1 | r_i | 4d,m_2 \rangle = K</math>) and then we rotate the system, we can infer that ''K'' is also the matrix element between the rotated version of <math>\langle 2p,m_1 |</math>, the rotated version of <math>r_i</math>, and the rotated version of <math>| 4d,m_2 \rangle</math>. This gives an algebraic relation involving ''K'' and some or all of the 44 unknown matrix elements. Different rotations of the system lead to different algebraic relations, and it turns out that there is enough information to figure out all of the matrix elements in this way.

(In practice, when working through this math, we usually apply [[angular momentum operator]]s to the states, rather than rotating the states. But this is fundamentally the same thing, because of the close mathematical [[Angular momentum operator#Angular momentum as the generator of rotations|relation between rotations and angular momentum operators]].)

===In terms of representation theory===

To state these observations more precisely and to prove them, it helps to invoke the mathematics of [[representation theory]]. For example, the set of all possible 4d orbitals (i.e., the 5 states ''m'' = −2, −1, 0, 1, 2 and their [[quantum superposition]]s) form a 5-dimensional abstract [[vector space]]. Rotating the system transforms these states into each other, so this is an example of a "group representation", in this case, the 5-dimensional [[irreducible representation]] ("irrep") of the rotation group [[angular momentum operator#SU(2), SO(3), and 360° rotations|SU(2) or SO(3)]], also called the "spin-2 representation". Similarly, the 2p quantum states form a 3-dimensional irrep (called "spin-1"), and the components of the position operator also form the 3-dimensional "spin-1" irrep.

Now consider the matrix elements <math>\langle 2p,m_1 | r_i | 4d,m_2 \rangle</math>. It turns out that these are transformed by rotations according to the [[tensor product]] of those three representations, i.e. the spin-1 representation of the 2p orbitals, the spin-1 representation of the components of '''r''', and the spin-2 representation of the 4d orbitals. This direct product, a 45-dimensional representation of SU(2), is ''not'' an [[irreducible representation]], instead it is the [[direct sum]] of a spin-4 representation, two spin-3 representations, three spin-2 representations, two spin-1 representations, and a spin-0 (i.e. trivial) representation. The nonzero matrix elements can only come from the spin-0 subspace. The Wigner–Eckart theorem works because the direct product decomposition contains one and only one spin-0 subspace, which implies that all the matrix elements are determined by a single scale factor.

Apart from the overall scale factor, calculating the matrix element <math>\langle 2p,m_1 | r_i | 4d,m_2 \rangle</math> is equivalent to calculating the [[projection (linear algebra)|projection]] of the corresponding abstract vector (in 45-dimensional space) onto the spin-0 subspace. The results of this calculation are the [[Clebsch–Gordan coefficient]]s. The key qualitative aspect of the Clebsch–Gordan decomposition that makes the argument work is that in the decomposition of the tensor product of two irreducible representations, each irreducible representation occurs only once. This allows [[Schur's lemma]] to be used.<ref>{{harvnb|Hall|2015}} Appendix C.</ref>