Schrödinger equation

Level SpecializationDifficulty ★★★★★Application⌖ Open in the map

What is it?

The PDE for the time evolution of a wave function: $i\hbar\,\partial_t\psi = \hat H\psi$ , with the Hamiltonian built from the Laplacian (kinetic energy) and a potential. Its operators — derivatives acting on functions — are where calculus and linear algebra meet; in quantum computing the same equation becomes $|\psi(t)\rangle = e^{-iHt/\hbar}|\psi(0)\rangle$ .

Formulas

$i\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\,\Delta\psi + V(x)\,\psi$; $\hat p = -i\hbar\,\frac{\partial}{\partial x}$
momentum is a derivative operator

The mathematics behind it

Laplacian★★★★★fundamental
Kinetic energy in quantum mechanics is $-\frac{\hbar^2}{2m}\Delta$ .

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.