Schrödinger Equation

quantum-mechanics quantum-computing

Definition

Schrödinger Equation

The Schrödinger equation describes the time evolution of a quantum state in a closed quantum system.

In its time-dependent form, it is

$$i\hslash \frac{\partial }{\partial t}|\psi (t)⟩=H|\psi (t)⟩,$$

where $\hslash$ is the reduced Planck constant and $H$ is the Hamiltonian operator.

Time Evolution

If $H$ is time-independent, then the solution can be written as

$$|\psi (t)⟩=e^{-\frac{i}{\hslash }tH}|\psi (0)⟩.$$

This evolution is generated by a unitary operator, so the norm of the state is preserved.