SemiclassicalPropagator

About the SemiclassicalPropagator class

The WaveBlocks Project

@author: R. Bourquin @copyright: Copyright (C) 2010, 2011, 2012, 2013, 2014, 2015, 2016 R. Bourquin @license: Modified BSD License

Inheritance diagram

Inheritance diagram of SemiclassicalPropagator

Class documentation

class WaveBlocksND.SemiclassicalPropagator(parameters, potential, packets=[])[source]

This class can numerically propagate given initial values \Psi in a potential V(x). The propagation is done for a given set of homogeneous Hagedorn wavepackets neglecting interaction.

__init__(parameters, potential, packets=[])[source]

Initialize a new SemiclassicalPropagator instance.

Parameters:
  • parameters (A ParameterProvider instance) – A ParameterProvider instance containing at least the key dt for providing the timestep \tau.
  • potential – The potential V(x) the wavepacket \Psi feels during the time propagation.
  • packet – The initial homogeneous Hagedorn wavepacket \Psi we propagate in time.
Raises:

ValueError – If the number of components of \Psi does not match the number of energy levels \lambda_i of the potential.
add_wavepacket(packet)[source]

Add a new wavepacket \Psi to the list of propagated wavepackets.

Parameters:packet (A tuple (\Psi, \chi) with \Psi a HagedornWavepacket instance and \chi an integer.) – The new wavepacket \Psi and its leading component \chi \in [0,N-1].
build(method)
Parameters:method – A string specifying the method for time integration.
Returns:Two arrays a and b.
Method Order Authors Reference
LT 1 Lie/Trotter [1], [3] page 42, equation 5.2
S2 2 Strang [2], [3] page 42, equation 5.3
SS 2 Strang [2], [3] page 42, equation 5.3
PRKS6 4 Blanes/Moan [4] page 318, table 2, ‘S6’
BM42 4 Blanes/Moan [4] page 318, table 3, ‘SRKNb6’
Y4 4 Yoshida [5], [3] page 40, equation 4.4
Y61 6 Yoshida [5], [3] page 144, equation 3.11
BM63 6 Blanes/Moan [4] page 318, table 3, ‘SRKNa14’
KL6 6 Kahan/Li [6], [3] page 144, equation 3.12
KL8 8 Kahan/Li [6], [3] page 145, equation 3.14
KL10 10 Kahan/Li [6], [3] page 146, equation 3.15
[1]H.F. Trotter, “On the product of semi-groups of operators”, Proc. Am. Math. Soc.1O (1959) 545-551.
[2](1, 2) G. Strang, “On the construction and comparison of difference schemes”, SIAM J. Numer. Anal. 5 (1968) 506-517.
[3](1, 2, 3, 4, 5, 6, 7, 8) E. Hairer, C. Lubich, and G. Wanner, “Geometric Numerical Integration - Structure-Preserving Algorithms for Ordinary Differential Equations”, Springer-Verlag, New York, 2002 (Corrected second printing 2004).
[4](1, 2, 3) S. Blanes and P.C. Moan, “Practical Symplectic Partitioned Runge-Kutta and Runge-Kutta-Nystrom Methods”, J. Computational and Applied Mathematics, Volume 142, Issue 2, (2002) 313-330.
[5](1, 2) H. Yoshida, “Construction of higher order symplectic integrators”, Phys. Lett. A 150 (1990) 262-268.
[6](1, 2, 3) W. Kahan and R.-c. Li, “Composition constants for raising the orders of unconventional schemes for ordinary differential equations”, Math. Comput. 66 (1997) 1089-1099.
get_number_components()[source]
Returns:The number N of components \Phi_i of \Psi.
get_potential()

Returns the potential V(x) used for time propagation.

Returns:A MatrixPotential subclass instance.
get_wavepackets(packet=None)[source]

Return the wavepackets \{\Psi_i\}_i that take part in the time propagation by the current SemiclassicalPropagator instance.

Parameters:packet (Integer or None) – The index i (in this list) of a single packet \Psi_i that is to be returned. If set to None (default) return the full list with all packets.
Returns:A list of HagedornWavepacket instances or a single instance.
intsplit(psi1, psi2, a, b, tspan, N, args1=[], args2=[])

Compute a single, full propagation step by operator splitting.

Parameters:
  • psi1 – First evolution operator \Psi_a
  • psi2 – Second evolution operator \Psi_b
  • a – Parameters for evolution with \Psi_a
  • b – Parameters for evolution with \Psi_b
  • tspan – Timespan t of a single, full splitting step
  • N – Number of substeps to perform
  • args1 – Additional optional arguments of \Psi_a
  • args2 – Additional optional arguments of \Psi_b

Note

The values for args1 and args2 have to be of type list even in case of single items.

order(method)
Parameters:method – A string specifying the method for time integration.
Returns:The order of this method.
post_propagate()

Given the wavefunction \psi at final time T, perform some computations exactly once after running the ordinary time propagation and before each time simulation data will be saved.

This method does not raise an exception but instead just does nothing and returns.

pre_propagate()

Given the wavefunction \psi at initial time t_0, perform some computations exactly once before running the ordinary time propagation and after each time simulation data was saved.

This method does not raise an exception but instead just does nothing and returns.

propagate()[source]

Given a wavepacket \Psi at time t compute the propagated wavepacket at time t + \tau. We perform exactly one timestep of size \tau here. This propagation is done for all packets in the list \{\Psi_i\}_i and neglects any interaction between two packets. The semiclassical propagation scheme is used.

More details can be found in [7].

[7]V. Gradinaru and G.A. Hagedorn, “Convergence of a semiclassical wavepacket based time-splitting for the Schroedinger equation”, Numerische Mathematik, volume 126 number 1 (2013) 53-73.
set_wavepackets(packetlist)[source]

Set the list \{\Psi_i\}_i of wavepackets that the propagator will propagate.

Parameters:packetlist (A list of (\Psi_i, \chi_i) tuples.) – A list of new wavepackets \Psi_i and their leading components \chi_i to propagate.