The Schrödinger equation determines how a nonrelativistic quantum state evolves in time.

For one particle in one dimension with potential energy $V(x,t)$,

$$i\hbar\frac{\partial\psi}{\partial t} =\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V\right)\psi.$$

The operator in parentheses is the Hamiltonian $\hat H$, representing total energy:

$$i\hbar\frac{\partial}{\partial t}|\psi\rangle=\hat H|\psi\rangle.$$

Why the equation is wave-like

The spatial second derivative makes neighboring values of the wavefunction influence its evolution, while the factor $i$ produces phase rotation rather than ordinary dissipative smoothing.

Stationary states

If the Hamiltonian does not depend explicitly on time, energy eigenstates satisfy

$$\hat H\phi=E\phi.$$

Their time evolution is

$$\psi(x,t)=\phi(x)e^{-iEt/\hbar}.$$

The probability density of a single energy eigenstate is therefore time-independent even though its complex phase evolves.

Superposition evolves linearly

Because the Schrödinger equation is linear, a superposition of solutions is also a solution. Different energy components acquire different phases over time, producing interference and changing probability distributions.