In this Demonstration, the total wavefunction

is defined by a superposition of two eigenstates:

,

with

and

.

In this case, the wavefunction for the harmonic oscillator in spherical polar coordinates [4] reads

.

The velocity field

is calculated from the gradient of the phase from the total wavefunction in the eikonal form (often called polar form)

. The time-dependent phase function

from the total wavefunction

is

.

For

, both initial states are weighted equally. Since the wavefunction is in the single initial state

with

, the orbits of the particles are circles because of the magnetic quantum number

, and for

, the particles are at rest. For these stationary states such as for

and

, the distribution

with the complex conjugate

is a constant of time; that is, it does not change. For

and for

, the distributions

are

,

In Cartesian coordinates with

,

,

the total wavefunction combines to

.

With a little more effort, it could be shown that the wavefunctions in spherical coordinates are just linear combinations of the solutions in Cartesian coordinates.

The normalization condition for the total wavefunction of

yielding a result between

and

is defined by

because if the position of the particle is measured, there is 100% probability that it will be somewhere.

In the program, if

PlotPoints,

AccuracyGoal,

PrecisionGoal,

MaxSteps and

MaxIterations are enabled, increasing them will give more accurate results.