Propagation

Simple quantum time propagation in 1d and 2d.

All physical quantities are in (Hartree) atomic units (numerical values: hbar = e = m = k_B = 1; c=1/alpha), cmp. also http://en.wikipedia.org/wiki/Atomic_units

Propagator

Base class for wavepacket time-propagation.

step(grid, dt)

To be implemented in subclasses.

SplitOperator(renormalize_every=0, workers=-1, order=2)

Bases: Propagator

Split operator time propagation.

Uses the symmetrized (Strang) splitting \(e^{-iH\Delta t} \approx e^{-iV\Delta t/2} e^{-iT\Delta t} e^{-iV\Delta t/2}\) with a local error of \(O(\Delta t^3)\). The kinetic part is applied in momentum space via FFT.

For a time-dependent potential, the opening half step uses \(V(t)\) and the closing one \(V(t+\Delta t)\), \(e^{-iV(t+\Delta t)\Delta t/2} e^{-iT\Delta t} e^{-iV(t)\Delta t/2}\), which keeps the scheme symmetric in time and second order accurate.

In :meth:propagate, the closing potential half step of one step and the opening half step of the next one use the same potential and are merged into a single full step as long as no callback or renormalization comes in between. This gives the same result with one array multiplication less per step, and the potential is evaluated only once per step.

With order=4 or order=6, one step is a symmetric composition of 3 or 9 Strang steps \(S(c_j\Delta t)\) (Yoshida's triple jump, applied once or twice), which makes the global error \(O(\Delta t^4)\) or \(O(\Delta t^6)\) for the cost of 3 or 9 FFT pairs per step. Some of the \(c_j\) are negative, so the substeps briefly go back in time; this is harmless in real time but not in imaginary time, where a backward kinetic step amplifies high momenta, so higher orders need a real \(\Delta t\).

A time step with negative imaginary part, \(\Delta t = -i\tau\), performs imaginary time evolution, which relaxes to the ground state.

Create a split operator propagator.

Parameters:

Name	Type	Description	Default
renormalize_every	int	Renormalize the wavefunction after this many steps have been made. Use 0 if you do not want renormalization.	0
order	(2, 4, 6)	Order of the splitting in the time step.	2
workers	int	Number of threads for the FFTs, passed on to scipy.fft; -1 uses all CPU cores. Other FFT libraries can be plugged in with scipy.fft.set_backend(), e.g. pyfftw.interfaces.scipy_fft.	-1

step(grid, dt)

Advance time by one step dt.

Parameters:

Name	Type	Description	Default
grid	Grid object	The grid that holds all information needed to advance time.	required
dt	complex	Time step.	required

propagate(grid, dt, n_steps, callback=None, every=1)

Do several time steps.

Parameters:

Name	Type	Description	Default
grid	Grid object	The grid to be propagated.	required
dt	complex	Time step.	required
n_steps	int	Number of steps.	required
callback	callable or sequence of callables	Called as callback(grid) after every every steps, e.g. an analyzer.	None
every	int	Callback interval.	1