Square-Barrier Scattering
A Gaussian wave packet is launched from the left at a rectangular potential of height V_0 and width a. The packet splits at the step: part of it is reflected, part is transmitted. Raise V_0 above the packet energy E and the transmitted part survives only by tunnelling. Make V_0 negative and the barrier becomes a well, where transmission oscillates with energy.
Units are \hbar = m = 1, so E = k_0^2/2 for a packet with central wavenumber k_0.
These demonstrations were built with generative AI.
- Time
- 0.0
- Reflected
- 0.000
- Transmitted
- 0.000
- Inside barrier
- 0.000
- Plane-wave T(E)
- 0.000
Reading the picture
The shaded block is the potential and the dashed line is the packet energy, both on the same vertical scale. The solid curve is |\psi(x)|^2, drawn on its own scale so the packet stays visible as it spreads.
The reflected and transmitted numbers are the probability found to the left and to the right of the barrier at the current time. They only mean what you expect once the packet has fully left the barrier region, so watch the inside barrier figure fall back to zero before reading them.
Why they disagree with the plane-wave value
The last readout is the textbook transmission coefficient for a plane wave at exactly the energy E:
T = \left[1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0 - E)}\right]^{-1}, \qquad \kappa = \sqrt{2(V_0 - E)} \quad (E < V_0),
with \sinh \to \sin and \kappa \to \sqrt{2(E - V_0)} when E > V_0.
A packet is not a plane wave. It carries a spread of wavenumbers around k_0, so its transmission is that curve averaged over the packet’s energy distribution. The two agree when the packet is long and narrow in energy, and differ most where T(E) varies quickly — near the top of a barrier, or at the resonances of a well.
Method
The evolution is the split-step Fourier method: half a step of the potential phase, a full step of the kinetic phase in Fourier space, then another half step of the potential. The grid is 2048 points over a box of length 420 with \Delta t = 0.01. Nothing absorbs at the walls, so the run pauses itself once the packet reaches the edge rather than letting it wrap around and interfere with itself.