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 general principles, and are useful in situations where the behavior of individual energy levels is too complex to be described 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 the particles or energy levels of a system.

Derivation of sum rules

Assume that the Hamiltonian has a complete
set of eigenfunctions with eigenvalues
For the Hermitian operator we define the
repeated commutator iteratively by:
The operator is Hermitian since
is defined to be Hermitian. The operator is
anti-Hermitian:
By induction one finds:
and also
For a Hermitian operator we have
Using this relation we derive:
The result can be written as
For this gives:

Example

See oscillator strength.