Grids

Grids hold the discretized wavefunction and potential.

Grid(dim, potential, psi0, absorber=None, dtype=np.complex128)

A grid object holds discrete representations of the wavepacket and the potential at a given time t. It also knows about the initial wavepacket.

Create a grid.

Parameters:

Name	Type	Description	Default
dim	int	Spatial dimension.	required
potential	Potential object	The potential used in potential discretization.	required
psi0	WavePacket object	Initial wavepacket object.	required
absorber	Potential object	Absorbing boundary potential.	None
dtype	(complex128, complex64)	Precision of the wavefunction and the potential.	numpy.complex128

psi property writable

Current wavefunction \(\psi(t)\).

V property

Current potential \(V(t)\), including the absorber.

psi_norm property

Norm \(\Vert\psi\Vert\).

density property

Probability density \(\vert\psi\vert^2\).

expectation_kinetic()

Kinetic energy \(\langle T\rangle = \langle k^2/2\rangle\), computed in momentum space.

expectation_potential()

Potential energy \(\langle V\rangle\); absorbers (the imaginary part of \(V\)) do not contribute.

energy()

Energy \(\langle H\rangle = \langle T\rangle + \langle V\rangle\).

update_potential(t)

Update potential for time \(t\).

renormalize_psi()

Renormalize \(\psi\) such that \(\Vert\psi\Vert = 1\).

suggest_time_step(safety=3.0)

Suggest a time step from the potential landscape elevation. The highest difference in total energy that is possible is \(\Delta E = \Delta V\), which corresponds to \(\delta\omega = \Delta V\) (atomic units!) or \(\delta t = 2\pi/\Delta V\).

Do not use this for tiny or vanishing potentials as e.g. in free-particle propagation.

Parameters:

Name	Type	Description	Default
safety	float	Divide computed dt by this number for safer propagation.	3.0

Returns:

Name	Type	Description
dt	float	Suggested time step.

Grid1d(size, grid_samples, potential, psi0, absorber=None, dtype=np.complex128)

Bases: Grid

One dimensional periodic grid.

The grid points are \(x_j = x_\mathrm{min} + j\,\Delta x\) with \(\Delta x = (x_\mathrm{max} - x_\mathrm{min})/N\), i.e. the right boundary is not included, as appropriate for the periodicity implied by the FFT.

Create a 1d grid.

Parameters:

Name	Type	Description	Default
size	tuple of float	Grid boundaries (x_min, x_max).	required
grid_samples	int	Number of grid points; powers of two are fastest.	required
potential	Potential object	The potential used in potential discretization.	required
psi0	WavePacket object	Initial wavepacket object.	required
absorber	Potential object	Absorbing boundary potential.	None
dtype	(complex128, complex64)	Precision of the wavefunction and the potential; complex64 halves memory and is faster for large grids, at a relative accuracy of about 1e-6.	numpy.complex128

make_psi_periodic()

Try to avoid Gibbs' phenomenon by manually making the wavefunction periodic.

expectation_x()

Position expectation value \(\langle x\rangle\).

expectation_p()

Momentum expectation value \(\langle p\rangle\), computed in momentum space.

width_x()

Position uncertainty \(\Delta x\).

Grid2d(size, grid_samples, potential, psi0, absorber=None, dtype=np.complex128)

Bases: Grid

Two dimensional periodic grid.

Arrays are indexed as psi[i, j] with \(x_i\) along the first and \(y_j\) along the second axis (indexing="ij"). As in 1d, the upper boundaries are not included. 2d potentials and wavepackets are evaluated with r = (X, Y), a tuple of coordinate arrays.

Create a 2d grid.

Parameters:

Name	Type	Description	Default
size	tuple	Grid boundaries ((x_min, x_max), (y_min, y_max)).	required
grid_samples	int or tuple of int	Number of grid points (N_x, N_y); powers of two are fastest.	required
potential	Potential object	The potential used in potential discretization.	required
psi0	WavePacket object	Initial wavepacket object.	required
absorber	Potential object	Absorbing boundary potential.	None
dtype	(complex128, complex64)	Precision of the wavefunction and the potential; complex64 halves memory and is faster for large grids, at a relative accuracy of about 1e-6.	numpy.complex128

extent property

(x_min, x_max, y_min, y_max), e.g. for imshow.

expectation_r()

Position expectation value \((\langle x\rangle, \langle y\rangle)\).

width_r()

Position uncertainties \((\Delta x, \Delta y)\).

expectation_p()

Momentum expectation value \((\langle p_x\rangle, \langle p_y\rangle)\).