# Schrödinger operator

Let $V\colon\mathbb{R}^{n}\to\mathbb{R}$ be a real-valued function. The Schroedinger operator H on the Hilbert space  $L^{2}(\mathbb{R}^{n})$ is given by the action

 $\psi\mapsto-\nabla^{2}\psi+V(x)\psi,\quad\psi\in L^{2}(\mathbb{R}^{n}).$

This can be obviously re-written as:

 $\psi\mapsto[-\nabla^{2}+V(x)]\psi,\quad\psi\in L^{2}(\mathbb{R}^{n}),$

where $[-\nabla^{2}+V(x)]$ is the Schrödinger operator, which is now called the Hamiltonian operator  (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem), H.

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

 $\textbf{H}\psi=E\psi$

, where $E$ stands for energy eigenvalues     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 (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem), or the Hamiltonian (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem), 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:

 $\textbf{H}\psi=(-i)\partial\psi/\partial t$

.

 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