# Stationary States

### From FSUPhysicsWiki

Stationary states are the energy eigenstates of the Hamiltonian operator. These states are called "stationary" because their probability distributions are independent of time.

For a conservative system with a time independent potential, , the Schrödinger equation takes the form:

Since the potential and the Hamiltonian do not depend on time, solutions to this equation can be written as

- .

Obviously, for such state the probability density is

which is independent of time, hence the term, "stationary state".

The Schrödinger equation now becomes

which is an eigenvalue equation with eigenfunction and eigenvalue . This equation is known as the time-independent Schrödinger equation.

Something similar happens when calculating the expectation value of any dynamical variable.

For any time-independent operator

## Problem

The time-independent Schrodinger equation for a free particle is given by

Typically, one lets obtaining

Show that

**(a)** a plane wave and

**(b)** a spherical wave where

satisfy the equation. In either case, the wave length of the solution is given by and the momentum by de Broglie's relation