time-dependent Schrödinger equation

The time-dependent Schrödinger equation is the full, general version of Schrödinger's law. It states that the wavefunction's rate of change in time equals the Hamiltonian acting on the wavefunction, multiplied by a factor involving the imaginary unit and Planck's constant. In symbols it is written iℏ ∂ψ/∂t = Ĥψ, and it applies whether or not the system has a definite energy.

This is the equation you reach for whenever something is actually happening: a wave packet spreading out, an electron sloshing between two atoms, a spin flipping under a changing magnetic field. Because it is first order in time, knowing the wavefunction at one instant fixes it for all later instants — quantum evolution carries no hidden extra information beyond the present state.

Its solutions are smooth, continuous, and reversible: run the clock backward and the equation works equally well. The imaginary factor of i is what makes the wavefunction oscillate rather than simply grow or decay, giving quantum systems their wavelike, interfering behaviour rather than the exponential settling you see in ordinary diffusion.