potential

Potential module.

This module contains predefined potential functions that can be used to generate electrostatic potentials for the simulation. The potential functions take a coordinate array as input and return the corresponding potential energy values.

Examples:

>>> import wave_packet_dynamics as wpd
>>> x = wpd.Grid((-1, 1), 11).coordinates
>>> V = wpd.potential.harmonic(x, k=5, x0=0)
>>> V
array([2.5, 1.6, 0.9, 0.4, 0.1, 0. , 0.1, 0.4, 0.9, 1.6, 2.5])

With the help of functools.partial, the potential function can be converted into a Callable that only takes the coordinate array as an argument.

>>> from functools import partial
>>> V = partial(wpd.potential.barrier, w=0.5, h=1)
>>> V(x)
array([0., 0., 0., 0., 1., 1., 1., 0., 0., 0., 0.])

`zero(x: NDArray[np.float64]) -> NDArray[np.float64]`

Zero potential function.

Parameters:

Name	Type	Description	Default
`x`	`NDArray[float64]`	Coordinate array.	required

Returns:

Type	Description
`NDArray[float64]`	Array of zeros.

Notes

\[V\left( x \right) = 0\]

`harmonic(x: NDArray[np.float64], *, k: float, x0: float = 0.0) -> NDArray[np.float64]`

Harmonic potential function.

Parameters:

Name	Type	Description	Default
`x`	`NDArray[float64]`	Coordinate array.	required
`k`	`float`	Force constant of the harmonic potential.	required
`x0`	`float`	Coordinate of the potential minimum.	`0.0`

Returns:

Type	Description
`NDArray[float64]`	Discretized harmonic potential.

Notes

\[V(x) = 0.5k \left( x-x_0 \right) ^2\]

`barrier(x: NDArray[np.float64], *, h: float, w: float, x0: float = 0.0) -> NDArray[np.float64]`

Rectangular barrier potential function.

Parameters:

Name	Type	Description	Default
`x`	`NDArray[float64]`	Coordinate array.	required
`h`	`float`	Height of the potential.	required
`w`	`float`	Width of the potential.	required
`x0`	`float`	Center of the potential.	`0.0`

Returns:

Type	Description
`NDArray[float64]`	Discretized rectangular barrier potential.

Notes

\[V(x)= \begin{cases} 0 &\text{for} & &x &\le x_0 - w / 2 \\ h &\text{for} &x_0 - w / 2 \le &x &\le x_0 + w / 2 \\ 0 &\text{for} &x_0 + w / 2 \le &x & \end{cases}\]

`step(x: NDArray[np.float64], *, h: float, x0: float = 0.0) -> NDArray[np.float64]`

Step potential function.

Parameters:

Name	Type	Description	Default
`x`	`NDArray[float64]`	Coordinate array.	required
`h`	`float`	Height of the potential.	required
`x0`	`float`	Coordinate of the step.	`0.0`

Returns:

Type	Description
`NDArray[float64]`	Discretized step potential.

Notes

\[V(x)= \begin{cases} 0 &\text{for} & &x &\le x_0\\ h &\text{for} &x_0 \le &x & \end{cases}\]