Schrödinger operator

Let V:n be a real-valued function. The Schroedinger operator H on the Hilbert spaceMathworldPlanetmath L2(n) is given by the action


This can be obviously re-written as:


where [-2+V(x)] is the Schrödinger operator, which is now called the Hamiltonian operatorPlanetmathPlanetmath (, H.

For stationary quantum systems such as electrons in ‘stable’ atoms the Schrödinger equation takes the very simple form :


, where E stands for energy eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the stationary quantum states. Thus, in quantum mechanics of systems with finite degrees of freedom that are ‘stationary’, the Schrödinger operator is used to calculate the (time-independent) energy states of a quantum system with potential energy V(x). Schrödinger called this operator the ‘Hamilton’ operator (, or the Hamiltonian (, and the latter name is currently used in almost all of quantum physics publications, etc. The eigenvalues give the energy levels, and the wavefunctions are given by the eigenfunctions. In the more general, non-stationary, or ‘dynamic’ case, the Schrödinger equation takes the general form:



Title Schrödinger operator
Canonical name SchrodingerOperator
Date of creation 2013-03-22 14:02:08
Last modified on 2013-03-22 14:02:08
Owner mhale (572)
Last modified by mhale (572)
Numerical id 30
Author mhale (572)
Entry type Definition
Classification msc 81Q10
Synonym Hamiltonian operator
Related topic HamiltonianOperatorOfAQuantumSystem
Related topic SchrodingersWaveEquation
Related topic CanonicalQuantization
Related topic QuantumOperatorAlgebrasInQuantumFieldTheories
Related topic QuantumSpaceTimes
Related topic SchrodingerOperator
Defines quantum system dynamics and eigenvalues