Sum rule in quantum mechanics

In quantum mechanics, a sum rule is a formula for transitions between energy levels, in which the sum of the transition strengths is expressed in a simple form. Sum rules are used to describe the properties of many physical systems, including solids, atoms, atomic nuclei, and nuclear constituents such as protons and neutrons. The sum rules are derived from quite general principles, and are useful in situations where the behavior of individual energy levels is too complex to describe by a precise quantum-mechanical theory. In general, sum rules are derived by using Heisenberg's quantum-mechanical algebra to construct operator equalities, which are then applied to particles or the energy levels of a system.

Derivation of sum rules

Assume that the Hamiltonian $hat\{H\}$ has a completeset of eigenfunctions $|n\; angle$ with eigenvalues$epsilon\_n$::$hat\{H\}\; |n\; angle\; =\; epsilon\_n\; |n\; angle.$For the Hermitian operator $hat\{A\}$ we define therepeated commutator $hat\{C\}^\{(k)\}$ by::The operator $hat\{C\}^\{(0)\}$ is Hermitian since $hat\{A\}$is defined to be Hermitian. The operator $hat\{C\}^\{(1)\}$ isanti-Hermitian::$left(hat\{C\}^\{(1)\}\; ight)^dagger\; =\; (hat\{H\}hat\{A\})^dagger-(hat\{A\}hat\{H\})^dagger\; =\; hat\{A\}hat\{H\}\; -\; hat\{H\}hat\{A\}\; =\; -hat\{C\}^\{(1)\}.$By induction one finds::$left(hat\{C\}^\{(k)\}\; ight)^dagger\; =\; (-1)^k\; hat\{C\}^\{(k)\}$and also:$langle\; m\; |\; hat\{C\}^\{(k)\}\; |\; n\; angle\; =\; (E\_m-E\_n)^k\; langle\; m\; |\; hat\{A\}\; |\; n\; angle.$For a Hermitian operator we have:$$
langle m | hat{A} | n angle|^2 = langle m | hat{A} | n angle langle m | hat{A} | n angle^ast = langle m | hat{A} | n angle langle n | hat{A} | m angle.Using this relation we derive::The result can be written as:

For $k=1$ this gives::$langle\; m\; |\; [hat\{A\},\; [hat\{H\},hat\{A\}\; ]\; |\; m\; angle\; =\; 2\; sum\_n\; (E\_n-E\_m)\; |langle\; m\; |\; hat\{A\}\; |\; n\; angle|^2.$

[Sanwu Wang, {it Generalization of the Thomas-Reiche-Kuhn and the Bethe sum rules,} Physical Review A {f 60,} 262 (1999). http://prola.aps.org/abstract/PRA/v60/i1/p262_1]