near the barrier w(r) = exp(–0.04 r2)

Calculation of classical trajectories

Zeroes of the Jacobian

Left: red curves show the points where Jacobian of the transformation (y, t) –> (qx, qy) is zero.
Right: blue curves show the points where Jacobian of the transformation (y, t) –> (px, py) is zero.

Caustics

The same as above, but mapped into (qx, qy) plane.

Exact calculations

Animation

Contour plots

Absolute value of the wavefunction

Grey area - |psi| < 1, white - 1 < |psi| < 2, red - |psi| > 2.

Real part of the wavefunction

Pink - Re psi > 0, blue - Re psi < 0, darker - |Re psi| > 2. Animation

Semiclassical calculations

Number of contributing trajectories

Dark area corresponds to one or no trajectories, lighter areas correspond to multiple trajectories.

Animation

Contour plots

Absolute value of the wavefunction

Grey area - |psi| < 1, white - 1 < |psi| < 2, red - |psi| > 2.

Real part of the wavefunction

Pink - Re psi > 0, blue - Re psi < 0, darker - |Re psi| > 2. Animation

Comparison of exact and semiclassical wavefunctions

Real part. Left - exact, right - semiclassical

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More examples of potentials

Table of examples of two-dimensional potentials