simulation

Simulation module.

Provides a class for the simulation of one-dimensional single particle quantum dynamics in a time-independent external potential. The wave packet is propagated in time by applying the time evolution operator at each time step. While the simulation is running for a specified number of time steps, wave function, probability density and expectation values are saved at specified intervals.

Examples:

To set up a simulation, a grid, a particle, its initial wave function and a potential function are required.

>>> from functools import partial
>>> import wave_packet_dynamics as wpd
>>> simulation = wpd.Simulation(
...     grid=wpd.Grid((0, 10), 1000),
...     particle=wpd.particles.Electron(),
...     wave_function=partial(
...         wpd.wave_function.gaussian, sigma=1, x0=5, k0=10
...     ),
...     potential=wpd.potential.zero,
... )

Creating a simulation object also normalizes the wave function.

>>> simulation.expectation_value("total_density")
np.float64(1.0000005734276094)

At any point of the simulation, expectation values can be calculated.

>>> simulation.expectation_value("position")
np.float64(5.000002867138047)
>>> simulation.expectation_value("momentum")
np.float64(9.983189000651)
>>> simulation.expectation_value("kinetic_energy")
np.float64(50.08273993919016)
>>> simulation.expectation_value("potential_energy")
np.float64(0.0)

Calling the simulation object will run the simulation for a specified number of time steps with a given time interval. Additionally, simulation data is saved at specified intervals.

>>> simulation(
...     time_interval=0.01,
...     total_time_steps=1000,
...     save_step=1001,
...     save_directory="output",
... )
>>> simulation.time
9.999999999999831

Simulation

Simulation of a single particle in an electrostatic potential.

Parameters:

Name	Type	Description	Default
grid	Grid	Grid on which the simulation is performed.	required
particle	Particle	The particle to simulate.	required
wave_function	Callable[[NDArray[float64]], NDArray[complex128]]	The initial wave function of the particle.	required
potential	Callable[[NDArray[float64]], NDArray[float64]]	The electrostatic potential acting on the particle.	required

Attributes:

Name	Type	Description
time	float	The current time of the simulation.

__call__(time_interval: float, total_time_steps: int, save_step: int = 10, save_directory: str | Path = 'simulation') -> None

Run the simulation.

Parameters:

Name	Type	Description	Default
time_interval	float	Time interval between two consecutive time steps.	required
total_time_steps	int	Total number of time steps.	required
save_step	int	Save interval in time steps.	10
save_directory	str or Path	Output directory.	'simulation'

Raises:

Type	Description
FileExistsError	If the output directory already exists.

density: NDArray[np.float64] property

Calculate the probability density.

Notes

The probability density is defined as the absolute square of the wave function.

\[|\psi(x)|^{2} = \psi^{\ast}(x) \psi(x) \quad \text{with} \quad |\psi(x)|^{2} \in \mathbb{R}_{\geq 0}\]

normalize() -> None

Normalize the wave function.

expectation_value(observable: str) -> float

Calculate the expectation value of an observable.

Notes

The expectation value of an observable (\(A\)) is calculated using the corresponding self-adjoint linear operator (\(\hat{A}\)).

\[\langle A \rangle_\psi = \langle \psi | \hat{A} | \psi \rangle = \int_{-\infty}^{\infty} \psi^{\ast}(x) \hat{A} \psi(x) \, dx\]