#### Stationary states

The solutions to Schrodinger's Time-Independent Equation,

are stationary states. Here we have plotted probability densities for the
harmonic oscillator at 8 different energies. Stationary states are defined
by a single energy.
For example, the quantum harmonic oscillator single state energy is depicted as the
darker horizontal red line at 2. Since
this is the time-independent solution, the red line never moves. The lighter
lines above and below it are upper and lower bounds of uncertainty. These
two never move either. The darker virtical red line
is the expectation value for the position of the "wave" of energy 2.
The lighter lines to the left and right are boundaries for the uncertainty of
the wave's position. The uncertainty never changes in time. Also, note
that as you increase in energy
level the position expectation value stays centered right at zero, while
the uncertainties broaden.

Below we have an anharmonic oscillator.
Say we are in a single state of energy 2, again with the solution
being time-independent, the horizontal red line never moves. This assymetric
potential shape causes different energy
levels to have different position expectation values seen in the green and blue
lines. Note that as the energy goes
up the position expectation values get further away from the energy minimum. The
uncertainties are not show here.

The point of this stationary state discussion was
that none of the above wave functions change in time.

back to Potential energy surfaces
go to Dynamic quantum states