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[1]


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:


See also



  1. ^ Wang, Sanwu (1999-07-01). "Generalization of the Thomas-Reiche-Kuhn and the Bethe sum rules". Physical Review A. 60 (1). American Physical Society (APS): 262–266. Bibcode:1999PhRvA..60..262W. doi:10.1103/physreva.60.262. ISSN 1050-2947.