The Time-Dependent Schrodinger Equation
This applet shows the evolution over time of a gaussian wave function, with either a series of potential wells, or a quadratic potential.
As usual, the depth, width and number of wells may be varied using the text fields provided. In addition, the initial width, position and energy of the the wave function can be chosen. The potential will be marked on the graph, either as a quadratic curve, or as pairs of vertical markers (edges of potential wells).
To start the animation, click the Start button; click this button again to pause/resume the animation once running. To start over from t=0 (after changing parameters, for instance), click the Reset button.
Quadratic potential
Selecting the "Quadratic potential" checkbox will replace the square well(s) with a quadratic curve.
Notes
While the animation is in progress, the integral of the wave function will be displayed. In a perfect simulation, this would be constant (and we would also normalize the function to make this equal to one). Due to inevitable rounding errors, the integral will slowly decay, although usually not at a high enough rate to have a noticable effect on the simulation. With extreme parameters this effect may be accelerated somewhat, so the integral is displayed as an indicator of this effect.
Since the interval considered is finite, the boundary conditions cause a small amount of interference to propagate from the edges of the simulation. By default, the width of the simulation is twice that of the displayed area, so this effect will not be immediately noticable. However, this does mean that any simulation, if left for long enough, will eventually decay into "bubbling soup". This is an artifact of the simulation, and should be ignored.
Documentation