# Definition The Schrödinger equation is the central equation of quantum mechanics. It is a linear partial differential equation. In its time-independent form it can be written as: $ H \ket{\psi} = E \ket{\psi} $ where $H$ is the [[Hamiltonian]] and $E$ is the [[energy]] and $\ket{\psi}$ is the [[Eigenstate]] of the Hamiltonian. For example, for a single particle in 1D, we have: $ \left(\frac{- \hbar^2}{2m} \frac{d^2}{dx^2} + V(x)\right) \psi(x) = E \psi(x) $ where $V(x)$ is the potential, and $\psi(x)$ is the [[wavefunction]]. The time-dependent Schrödinger equation, on the other hand, is given by $ H \ket{\psi} = i \frac{\partial}{\partial t} \ket{\psi} $ or, for a single particle in 1D $ \left(\frac{- \hbar^2}{2m} \frac{d^2}{dx^2} + V(x)\right) \psi(x,t) = i \frac{\partial \psi(x,t)}{\partial t} $