# Quantum Barrier Scattering

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Speed:   Real/imag  Density/phase   Grid
Wavepacket energy = 0.030 ± 0.005
Barrier energy = 0.030    Width = 20   Ramp = 0

This simulation shows a quantum mechanical wavepacket hitting a barrier. You can adjust the wavepacket’s nominal energy, the barrier energy, the barrier width, and the width of a “ramp” on either side of the barrier, to see how these affect the amount of the wavepacket that gets through (i.e., the tunneling probability). Drag the width slider all the way to the right to make a step instead of a barrier.

The wavefunction is always zero at the edges of the image, so the quantum particle is effectively trapped in an infinitely deep potential well. Thus, when the wavepacket hits the edges, it will reflect off of them.

You can plot either the real and imaginary parts of the wavefunction (shown in orange and blue, respectively), or the probability density and phase, with the phase represented by hues going from red (pure real and positive) to light green (pure imaginary and positive) to cyan (pure real and negative) to purple (pure imaginary and negative) and finally back to red.

Play with the simulation for a while, then try to predict what will happen when you change the various settings. How does the wavepacket behave when there is no barrier at all? How can you tell, when the simulation is paused, whether the wavepacket is moving to the left or right? How does the wavelength within the packet vary as you change its energy? Under what conditions will most of the wavepacket make it through the barrier? In what ways does the wavepacket behave like a classical particle?

Technical details: The simulation works by solving a discretized version of the time-dependent Schrödinger equation, as you can see by looking at the source code. Distances are measured in units of nominal screen pixels, and the grid spacing is 20 pixels. Other units are determined by setting h-bar and the particle mass to 1. This is a nonrelativistic particle, so its kinetic energy is p2/2m. From this formula and the de Broglie relation you can figure out how the energy of a wavefunction is related to its wavelength. A wavepacket is actually a mixture that includes a range of energies, so the uncertainty (standard deviation) in the energy is displayed next to the energy slider. Notice also that the phase velocity (of the individual waves within the wavepacket) differs from the group velocity (of the packet as a whole).

See PhET’s Quantum Tunneling and Wave Packets for a similar simulation with some more useful features.

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