Schrödinger Equation: Numerical Experiments in One Dimensional Time Dependent Scattering

The red curve is the initial probability density, blue (yellow) curve the real (imaginary) part of the wave function.
The dark green curve is the potential energy V. Regions where V = 0 are said to be the external vacuum.
In order to modify the potential:

Left-click on the external vacuum to create a new potential well/barrier.
Left-click and drag an existing potential well/barrier to change its depth or height.
Control-Left-click and drag an existing potential well/barrier to change its width.
To delete an existing potential well/barrier, left-click and drag it toward the trashcan until the mouse is over the trashcan; then release the mouse.

Create record of scattering:


Evolution Method:

Nash-Chen	Cayley

Periodic Cayley

^{Initial wave function:}

# Frames =
WaveNumber =		\| Maximum Wave Number \| =
Center of Gaussian =		Fourier component of initial wave function at chosen wavenumber =
Initial Packet Spread =		Calculate Fourier component of initial wave function at wavenumber: FC=
Scale potential energy by:
^Scale ^by:

This program writes flic.avi to your DeskTop each time it is run. If you wish to save this file, then you should either move it to another folder or rename it.

Details Summary

0 <= x <= 1
N = #nodes in spatial lattice
a = delta x = spatial step size = 1 / (N-1)
x --> a x , where x = 0, 1, 2, ..., N-1
_Laplacian

Maximum wavenumber:
k_max a = 2 pi => -pi/a < k < pi/a => -pi/2a < k_{unique velocity} < pi/2a

Stability:
T = delta t = time step size
Evolution methods are stable if

is between 0 and 1; in these simulations we take m = 1 and
.

Non-dimensionalized potential energy:
V = potential energy
_UV
where U = non-dimensionalized potential energy = ±(rectangle.bottom - rectangle.top) · ( VALUE of 'Scale potential energy by' entry)

Unitary Evolution Methods

Nash-Chen

for x = 0, 1, 2, ..., N - 1. The J_n are Bessel functions of the first kind.
P. L. Nash and L.Y. Chen, Efficient finite difference solutions to the time-dependent Schrodinger equation, Journal of Computational Physics, 130, p266 (1997).

Cayley

This yields a tridiagonal system when
_Laplacian

Periodic Cayley

The tridiagonal Cayley system plus two additional matrix elements in the coefficient matrix at locations (first row, last column) and (last row, first column). This generates a sparse system that cannot be solved as efficiently as the tridiagonal system. The solution is periodic. When possible, it is more efficient to place potential barriers on the left and right boundaries and use the Cayley tridiagonal system.