Wavepackets

Initial wavepackets for time propagation.

Wavepackets are evaluated through psi(x), where x may be a scalar or a NumPy array of positions.

WavePacket

Base class for initial wavepackets.

Derived classes implement psi(r), which must accept array-valued positions.

WavepacketSuperposition(psis)

Bases: WavePacket

Create a superposition of several wave packets.

Create a superposition of several wavepackets.

Parameters:

Name	Type	Description	Default
psis	list	List of WavePacket objects to be added.	required

Gaussian1d(x0, sigma, p0)

Bases: WavePacket

A Gaussian wavepacket in one dimension created from momentum and position space width.

Create a Gaussian wavepacket \(\psi(x) = N \exp(-(x-x_0)^2/(2\sigma^2) + i p_0 (x-x_0))\).

Parameters:

Name	Type	Description	Default
x0	float	Center position.	required
sigma	float	Position space width.	required
p0	float	Momentum of corresponding classical particle.	required

sigma property writable

Standard deviation.

Gaussian1dEnergy(x0, E0, delta_E, direction=+1)

Bases: Gaussian1d

A Gaussian wavepacket in one dimension created from energy parameters.

Create a Gaussian wavepacket from energy parameters.

Parameters:

Name	Type	Description	Default
x0	float	Center position.	required
E0	float	Center energy.	required
delta_E	float	Full width half maximum of the Gaussian in energy space. Using this and E0, sigma_x will be derived.	required
direction	float	Positive for propagation to the right, negative for propagation to the left.	+1

Notes

The narrower the Gaussian in energy space (~ momentum space), the broader the Gaussian will be in position space. However, choosing a broad energy distribution will lead to high dispersion.

Journal reference for formulas: Hussain and Roberts, Phys. Rev. A 63, 012703 (2000)

CoherentState(x0, omega, p0)

Bases: Gaussian1d

A Gaussian wavepacket that does not spread in harmonic potentials.

Create a coherent state for an oscillator of frequency \(\omega\). The momentum can also be specified.

Parameters:

Name	Type	Description	Default
x0	float	Center position.	required
omega	float	Harmonic oscillator frequency (as in \(V=\omega^2 x^2 / 2\)).	required
p0	float	Momentum of corresponding classical particle.	required

Gaussian2d(x0, y0, sigma_x, sigma_y=None, px=0.0, py=0.0)

Bases: WavePacket

A Gaussian wavepacket in two dimensions, evaluated as psi((X, Y)).

Create a 2d Gaussian wavepacket, the product of two 1d Gaussians \(\psi = \psi_x(x)\,\psi_y(y)\).

Parameters:

Name	Type	Description	Default
x0	float	Center position.	required
y0	float	Center position.	required
sigma_x	float	Position space widths; sigma_y defaults to sigma_x.	required
sigma_y	float	Position space widths; sigma_y defaults to sigma_x.	required
px	float	Center momentum.	0.0
py	float	Center momentum.	0.0