wave_function

Wave function module.

This module contains continuous functions representing wave packets suitable for simulation of quantum dynamics. The wave functions take a coordinate array as input and return the corresponding discretized complex-valued wave functions.

Examples:

>>> import wave_packet_dynamics as wpd
>>> x = wpd.Grid((-5, 5), 101).coordinates
>>> psi = wpd.wave_function.gaussian(x, sigma=1, x0=0, k0=10)
>>> x[50], psi[50]
(np.float64(0.0), np.complex128(0.6316187777460647+0j))

All wave functions are normalized.

>>> from scipy import integrate
>>> norm = integrate.trapezoid(np.real(psi.conj() * psi), x)
>>> norm
np.float64(0.9999994143527634)

gaussian(x: NDArray[np.float64], *, sigma: float, x0: float = 0.0, k0: float = 0.0) -> NDArray[np.complex128]

Gaussian wave function.

Parameters:

Name	Type	Description	Default
x	NDArray[float64]	Coordinate array.	required
sigma	float	Standard deviation of the Gaussian distribution.	required
x0	float	Initial most probable coordinate.	0.0
k0	float	Initial wave number of the particle.	0.0

Returns:

Type	Description
NDArray[complex128]	Discretized Gaussian wave function.

Notes

\[\Psi(x) = \left( 2 \pi \sigma ^2 \right) ^{-1/4} \cdot e^{ -\left( x-x_0 \right) ^2 / 4 \sigma ^2 } \cdot e^{ \text{i} k_0 \left( x-x_0 \right) }\]