# 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*) –> (*q*_{x}, *q*_{y})

### Zeroes of the Jacobian

Left: red curves show the points where Jacobian of the transformation (*y*, *t*) –> (*q*_{x}, *q*_{y}) is zero.

Right: blue curves show the points where Jacobian of the transformation (*y*, *t*) –> (*p*_{x}, *p*_{y}) is zero.
### Caustics

The same as above, but mapped into (*q*_{x}, *q*_{y}) 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

*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*

*R* = 25

### Dependence of angle chi on angle phi for the given *R* = 25

*R* = 30

### Dependence of angle chi on angle phi for the given *R* = 30

### More examples of potentials

Table of examples of two-dimensional potentials