Sum rule in quantum mechanics
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quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
\hat has a complete set of eigenfunctions , n\rangle with eigenvalues E_n: : \hat , n\rangle = E_n , n\rangle. For the Hermitian operator \hat we define the repeated commutator \hat^ iteratively by: : \begin \hat^ & \equiv \hat\\ \hat^ & \equiv hat, \hat= \hat\hat-\hat\hat\\ \hat^ & \equiv hat, \hat^ \ \ \ k=1,2,\ldots \end The operator \hat^ is Hermitian since \hat is defined to be Hermitian. The operator \hat^ is anti-Hermitian: : \left(\hat^\right)^\dagger = (\hat\hat)^\dagger-(\hat\hat)^\dagger = \hat\hat - \hat\hat = -\hat^. By induction one finds: : \left(\hat^\right)^\dagger = (-1)^k \hat^ and also : \langle m , \hat^ , n \rangle = (E_m-E_n)^k \langle m , \hat , n \rangle. For a Hermitian operator we have : , \langle m , \hat , n \rangle, ^2 = \langle m , \hat , n \rangle \langle m , \hat , n \rangle^\ast = \langle m , \hat , n \rangle \langle n , \hat , m \rangle. Using this relation we derive: : \begin \langle m , hat, \hat^ , m \rangle &= \langle m , \hat \hat^ , m \rangle - \langle m , \hat^\hat , m \rangle\\ &= \sum_n \langle m , \hat , n\rangle\langle n, \hat^ , m \rangle - \langle m , \hat^ , n\rangle\langle n, \hat , m \rangle\\ &= \sum_n \langle m , \hat , n\rangle \langle n, \hat, m \rangle (E_n-E_m)^k - (E_m-E_n)^k \langle m , \hat , n\rangle\langle n, \hat , m \rangle \\ &= \sum_n (1-(-1)^k) (E_n-E_m)^k , \langle m , \hat , n \rangle, ^2. \end The result can be written as : \langle m , hat, \hat^ , m \rangle = \begin 0, & \mboxk\mbox \\ 2 \sum_n (E_n-E_m)^k , \langle m , \hat , n \rangle, ^2, & \mboxk\mbox. \end For k=1 this gives: : \langle m , hat, m \rangle = 2 \sum_n (E_n-E_m) ">\langle m , \hat , n \rangle, ^2.


See also

* Oscillator strength * Sum rules (quantum field theory) * QCD sum rules


References

Quantum mechanics {{quantum-stub