Schrodinger equation

The Schrodinger equation is a linear differential equation used in various fields of physics to describe the time evolution of quantum states. It is a fundamental aspect of quantum mechanics. The equation is named for its discoverer, Erwin Schrodinger.

Mathematical forms

General time-dependent form

The Schrodinger equation may generally be written

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

where $i$ is the imaginary unit,
$\hbar$ is Planck's constant divided by $2\pi$,
$|\Psi \rangle$ is the quantum mechanical state or wavefunction (expressed here in Dirac notation), and
${\hat {H}}$ is the Hamiltonian operator.

The left side of the equation describes how the wavefunction changes with time; the right side is related to its energy.

Eigenvalue problems

In many instances, steady-state solutions to the equation are of great interest. Physically, these solutions correspond to situations in which the wavefunction has a well-defined energy. The energy is then said to be an eigenvalue for the equation, and the wavefunction corresponding to that energy is called an eigenfunction or eigenstate. In such cases, the Schrodinger equation is time-independent and is often written

$E\psi ={\hat {H}}\psi$

Here, E is energy, H is once again the Hamiltonian operator, and $\psi$ is the energy eigenstate for E.

One example of this type of eigenvalue problem is an electrons bound inside an atom.