Harmonic oscillator, 1 wave packet Harmonic oscillator, 2 wave packets Harmonic oscillator, 3 wave packets Harmonic oscillator, 4 wave packets Harmonic oscillator, 5 wave packets Harmonic oscillator, 6 wave packets Square well Perturbation 0 Perturbation 10^-6 Perturbation 10^-4 Perturbation 10^-2 5 × 5 modes 7 × 7 modes 9 × 9 modes 11 × 11 modes 13 × 13 modes 15 × 15 modes 17 × 17 modes

QuantumWell displays a wave function in either a square well or a harmonic oscillator potential (optionally evolving under a small perturbation that leaves the energy of the system unchanged, but is otherwise random). The contours are for the squared magnitude of the wave, with phase indicated by colour.

The harmonic oscillator begins with one or more roughly Gaussian wave packets oscillating back and forth in the potential. These wave packets are created by translating the oscillator’s Gaussian ground state away from the centre of the well, and if they were perfectly constructed they would bounce back and forth indefinitely without changing shape (they do overlap, though, so this is clearest when there’s only a single wave packet). However, the wave shown here is an approximation, built from a finite number of energy eigenstates, so the packets are not exactly Gaussian and their shape will change slightly as they move. In the longer term, they will also be degraded by any random perturbation that has been selected.

The square well begins with a localised wave, a finite-energy approximation to a randomly chosen rectangular step function. The original shape rapidly disperses, but will periodically recur (though again, if there is a random perturbation it will gradually disrupt that behaviour).

Click on the applet to pause or redraw. If the applet is not running smoothly, you can try to get a better frame rate by reducing the number of modes from which the wavefunction is built.

Applets Gallery / QuantumWell / created Sunday, 5 July 1998 / revised Monday, 26 June 2000