below the barrier w(r) = exp(–0.04 r2)

The potential (red line is the energy)

Calculation of classical trajectories

Plot of trajectories on the potential surface

Mapping (y, t) –> (qx, qy)

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

Phase shift vs. angular momentum for partial waves

Density plots

Absolute value of the wavefunction

Real part of the wavefunction

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.

Density plots

Absolute value of the wavefunction

Real part of the wavefunction

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

Absolute value. Left - exact, right - semiclassical

Real part. Left - exact, right - semiclassical


Animate

Animate

R = 5

Dependence of angle chi on angle phi for the given R = 5

Primitive semiclassical (black) vs. exact wavefunction (red curves) for the given R = 5.
Plots show absolute value, real part, and number of contributing trajectories as a function of x

R = 10

Dependence of angle chi on angle phi for the given R = 10

Primitive semiclassical (black) vs. exact wavefunction (red curves) for the given R = 10.
Plots show absolute value, real part, and number of contributing trajectories as a function of x

R = 15

Dependence of angle chi on angle phi for the given R = 15

Primitive semiclassical (black) vs. exact wavefunction (red curves) for the given R = 15.
Plots show absolute value, real part, and number of contributing trajectories as a function of x

R = 20

Dependence of angle chi on angle phi for the given R = 20

Primitive semiclassical (black) vs. exact wavefunction (red curves) for the given R = 20.
Plots show absolute value, real part, and number of contributing trajectories as a function of x



More examples of potentials

Morse potential from a paper of Knudson et al., 1985 Gaussian repulsive potential Gaussian attractive potential near the barrier below the barrier
Gaussian multiplied by r^2 modified Gaussian, larger energy inverted modified Gaussian, larger energy Gaussian repulsive potential, large energy Gaussian attractive potential, large energy
Yukawa potential Oscillating Gaussian potential Inverted oscillating Gaussian potential Potential from M. Herman's notes Repulsive Yukawa potential

Table of examples of two-dimensional potentials