Partial differential equation which describes the time evolution of the wavefunction of a quantum system. It is one of the first and most fundamental equations of quantum mechanics.
When to use this tag
Use the tag schrodinger-equation if you have a question specifically relating to the Schrödinger equation, such as its derivation or the particular form of the Schrödinger equation for a given problem. In general, you will also want to tag your question as quantum-mechanics.
Introduction
The general, time-dependent Schrödinger equation is
$$ i \hbar \frac{\partial}{\partial t} \Psi(x,t) = \hat H \Psi $$
with the Hamiltonian $\hat H$ and the wave function $\Psi(x,t)$. For a single, non-relativist particle, this is equal to
$$ i \hbar \frac{\partial}{\partial t} \Psi(x,t) = \left( \frac{-\hbar^2}{2m} \nabla^2 + V(x,t) \right) \Psi(x,t) \quad .$$
If the potential $V(x,t)$ is not time-dependent, this equation separates and gives the time-independent Schrödinger equation, which is just the eigenvalue equation for the Hamilton operator:
$$ \hat H \Psi(x,t) = E \Psi(x,t) $$
The Schrödinger equation describes the time evolution of states/wave functions in the Schrödinger picture. If one instead chooses to work in the Heisenberg picture, where states are time-independent and instead operators change in time, the governing equation is
$$ \frac{\mathrm{d}}{\mathrm{d}t} \hat A(t) = \frac{i}{\hbar} [ \hat H , \hat A(t) ] + \frac{\partial}{\partial t} A(t) \quad.$$
