# Repulsive Yukawa potential w(r) = exp(–0.2 r)/r

### 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

### 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 ### 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 ### 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 ### 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 ### 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 ### 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 ### 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 ### Dependence of angle chi on angle phi for the given R = 25 ### Dependence of angle chi on angle phi for the given R = 30 ### More examples of potentials

Table of examples of two-dimensional potentials