wavepacket.special

Special functions for use in Wavepacket.

These mostly encompass typical pulse shapes or initial states.

Classes

`Gaussian`	Callable that defines a one-dimensional Gaussian function.
`PlaneWave`	Callable that defines a plane wave.
`SphericalHarmonic`	Callable that returns a spherical harmonic Y_l^m(theta, phi=0).
`SinSquare`	Shape function for a squared sinusoidal laser pulse.
`SoftRectangularFunction`	Shape function for a rectangular pulse with soft (cosine) turn-on.

Package Contents

class wavepacket.special.Gaussian(x: float = 0.0, p: float = 0.0, *, rms: float | None = None, fwhm: float | None = None)

Bases: wavepacket.typing.Generator

Callable that defines a one-dimensional Gaussian function.

This callable can be supplied wherever a callable is required. An example would be an initial wave function for wavepacket.builder.product_wave_function(), or a potential wrapped in a wavepacket.operator.Potential1D.

Parameters:

xfloat, default=0: The center of the Gaussian.
pfloat, default=0: The momentum of the Gaussian.
rms, fwhmfloat: You must specify the width of the Gaussian using exactly one of these keyword parameters, either the root-mean-square width, or the full-width-at-half-maximum.

Raises:

wp.InvalidValueError: If the width of the Gaussian is not positive.
wp.BadFunctionCall: If both rms and fwhm have either been set or not supplied.

Notes

Up to scaling, the functional form of the Gaussian is $f(x) = e^{-(x-x_0)^2 / 2 \sigma^2 + \imath p (x-x_0)}$ . Here, sigma is the rms width, which is connected to the FWHM by $\sigma = \mathrm{FWHM} / \sqrt{8 \ln 2}$ .

class wavepacket.special.PlaneWave(k: float)

Bases: wavepacket.typing.Generator

Callable that defines a plane wave.

You will typically use this callable for initial states. There are often better options, especially if your FBR already defines a plane wave basis, but sometimes you may just want to represent a reasonable plane wave and not count indices to get the correct wave vector.

Parameters:

kfloat: The wave vector of the plane wave.

class wavepacket.special.SphericalHarmonic(l: int, m: int)

Bases: wavepacket.typing.RealGenerator

Callable that returns a spherical harmonic Y_l^m(theta, phi=0).

Usually, this callable will be used for initial states. Note that the phi-dependence of a spherical harmonic is trivial exp(i m phi), and usually not needed (we fix m and the phi-integration yields a constant). For this reason, the functor takes only the theta-values as single parameters, and returns the spherical harmonic at phi = 0.

Parameters:

lint: The rotational quantum number / angular momentum
mint: The minor rotational quantum number -l <= m <= l

Raises:

wp.InvalidValueError: If l is negative or if (-l <= m <= l) does not hold.

class wavepacket.special.SinSquare(t0: float, half_width: float)

Shape function for a squared sinusoidal laser pulse.

Usually, you use this function to describe a smooth laser pulse with a definite start and end point (which a Gaussian does not have). The exact shape is $cos^2(\pi \frac{t - t_0}{2 \Delta})$ , and zero outside the interval $[t_0-\Delta, t_0+\Delta]$ .

Parameters:

t0: float: The center of the pulse.
half_width: The half-width of the laser pulse, $\Delta$ .

Raises:

wp.InvalidValueException: if the half-width is not positive.

class wavepacket.special.SoftRectangularFunction(t0: float, half_width: float, border: float | None = None)

Shape function for a rectangular pulse with soft (cosine) turn-on.

This shape function is a rectangular function with a given half-width, with an added soft turn-on and turn-off of the form $cos(\frac{t}{B})$ with the border width B.

Parameters:

t0: float: Center of the rectangular pulse.
half_width: float: Half-width of the rectangular part of the pulse.
border: float = half_width/10: Optional width of the turn-on / turn-off region.

Raises:

wp.InvalidValueError: If the half-width or the border region is not positive.