Gaussian repulsive potential, large energy 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

Absolute value of the wavefunction

Real part of the wavefunction

Animation

Semiclassical calculations

R = 0

Dependence of angle chi on angle phi for the given R = 0

Primitive semiclassical (black) vs. exact wavefunction (red curves) for the given R = 0.
Plots show absolute value, real part, and number of contributing trajectories as a function of x

R = 1

Dependence of angle chi on angle phi for the given R = 1

Primitive semiclassical (black) vs. exact wavefunction (red curves) for the given R = 1.
Plots show absolute value, real part, and number of contributing trajectories as a function of x

R = 2

Dependence of angle chi on angle phi for the given R = 2

Primitive semiclassical (black) vs. exact wavefunction (red curves) for the given R = 2.
Plots show absolute value, real part, and number of contributing trajectories as a function of x

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

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 Repulsive Yukawa potential

Table of examples of two-dimensional potentials