Schrödinger equation

Summary

The Schrödinger equation $i\hbar\, \partial_t \psi = -\frac{\hbar^2}{2m} \Delta \psi + V \psi$ describes how the wave function $\psi$ of a quantum particle of mass $m$ in a potential $V$ evolves in time. It is a linear dispersive partial differential equation with complex-valued solutions. Its right-hand side is a Schrödinger operator applied to $\psi$. The solutions are given by a unitary group of operators, so the total probability $\int \abs{\psi}^2 \, dx$ is conserved. It is the basic equation of non-relativistic quantum mechanics and a model problem for dispersive equations.