Associated Hermite polynomials arise as the solution of the Schrödinger equation:

,

with

,

, and so on. A degenerate, unnormalized, complex-valued wavefunction

for the three-dimensional case can be expressed by:

,

where

,

, and

are eigenfunctions, and

are permuted eigenvalues of the corresponding stationary one-dimensional Schrödinger equation with

. The eigenfunctions are defined by:

,

where

,

, and

are Hermite polynomials. The parameter

is a constant phase shift, and the energies depend on the quantum numbers, with

, with

.

For this Demonstration, the wavefunction

is defined by:

.

In this case, the square of the Schrödinger wavefunction

, where

is its complex conjugate, is time independent:

.

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:

.

The corresponding velocity field becomes time independent (autonomous) because of the gradient of the phase function.

In the program, if

PlotPoints,

AccuracyGoal,

PrecisionGoal,

MaxSteps, and

MaxIterations are enabled and increased, the results will be more accurate.