def propagate(self):
r"""Given a wavepacket :math:`\Psi` at time :math:`t` compute the propagated
wavepacket at time :math:`t + \tau`. We perform exactly one timestep of size
:math:`\tau` here. This propagation is done for all packets in the list
:math:`\{\Psi_i\}_i` and neglects any interaction between two packets.
More details can be found in [#]_.
.. [#] E. Faou, V. Gradinaru and C. Lubich, "Computing semiclassical quantum dynamics with Hagedorn wavepackets",
SIAM Journal on Scientific Computing, volume 31 number 4 (2009) 3027-3041.
"""
# Cache some parameter values
dt = self._dt
Mi = self._Minv
key = ("q", "p", "Q", "P", "S", "adQ")
# Propagate all packets
for packet, leading_chi in self._packets:
eps = packet.get_eps()
# Do a kinetic step of dt/2
q, p, Q, P, S, adQ = packet.get_parameters(key=key)
q = q + 0.5 * dt * dot(Mi, p)
Q = Q + 0.5 * dt * dot(Mi, P)
S = S + 0.25 * dt * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), key=key)
# Do a potential step with the local quadratic part
q, p, Q, P, S = packet.get_parameters()
V = self._potential.evaluate_local_quadratic_at(
q, diagonal_component=leading_chi)
p = p - dt * V[1]
P = P - dt * dot(V[2], Q)
S = S - dt * V[0]
packet.set_parameters((q, p, Q, P, S))
# Do a potential step with the local non-quadratic Taylor remainder
innerproduct = packet.get_innerproduct()
F = innerproduct.build_matrix(
packet,
operator=partial(self._potential.evaluate_local_remainder_at,
diagonal_component=leading_chi))
coefficients = packet.get_coefficient_vector()
coefficients = self._matrix_exponential(F, coefficients,
-1.0j * dt / eps**2)
packet.set_coefficient_vector(coefficients)
# Do a kinetic step of dt/2
q, p, Q, P, S, adQ = packet.get_parameters(key=key)
q = q + 0.5 * dt * dot(Mi, p)
Q = Q + 0.5 * dt * dot(Mi, P)
S = S + 0.25 * dt * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), key=key)
def propagate(self):
r"""Given a wavepacket :math:`\Psi` at time :math:`t` compute the propagated
wavepacket at time :math:`t + \tau`. We perform exactly one timestep of size
:math:`\tau` here. This propagation is done for all packets in the list
:math:`\{\Psi_i\}_i` and neglects any interaction between two packets.
More details can be found in [#]_.
.. [#] E. Faou, V. Gradinaru and C. Lubich, "Computing semiclassical quantum dynamics with Hagedorn wavepackets",
SIAM Journal on Scientific Computing, volume 31 number 4 (2009) 3027-3041.
"""
# Cache some parameter values
dt = self._dt
Mi = self._Minv
# Propagate all packets
for packet in self._packets:
# Unpack, no codata:
packet = packet[0]
eps = packet.get_eps()
key = ("q", "p", "Q", "P", "S", "adQ")
# Do a kinetic step of dt/2
for component in range(self._number_components):
q, p, Q, P, S, adQ = packet.get_parameters(component=component, key=key)
q = q + 0.5 * dt * dot(Mi, p)
Q = Q + 0.5 * dt * dot(Mi, P)
S = S + 0.25 * dt * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), component=component, key=key)
# Do a potential step with the local quadratic part
for component in range(self._number_components):
q, p, Q, P, S = packet.get_parameters(component=component)
V = self._potential.evaluate_local_quadratic_at(q, diagonal_component=component)
p = p - dt * V[1]
P = P - dt * dot(V[2], Q)
S = S - dt * V[0]
packet.set_parameters((q, p, Q, P, S), component=component)
# Do a potential step with the local non-quadratic Taylor remainder
innerproduct = packet.get_innerproduct()
F = innerproduct.build_matrix(packet, packet, self._potential.evaluate_local_remainder_at)
coefficients = packet.get_coefficient_vector()
coefficients = self._matrix_exponential(F, coefficients, -1.0j * dt / eps**2)
packet.set_coefficient_vector(coefficients)
# Do a kinetic step of dt/2
for component in range(self._number_components):
q, p, Q, P, S, adQ = packet.get_parameters(component=component, key=key)
q = q + 0.5 * dt * dot(Mi, p)
Q = Q + 0.5 * dt * dot(Mi, P)
S = S + 0.25 * dt * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), component=component, key=key)
def _propkin(self, h, packet):
"""Do a kinetic step of size h.
"""
Mi = self._Minv
key = ("q", "p", "Q", "P", "S", "adQ")
q, p, Q, P, S, adQ = packet.get_parameters(key=key)
q = q + h * dot(Mi, p)
Q = Q + h * dot(Mi, P)
S = S + 0.5 * h * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), key=key)
def _propkin(self, h, packet):
"""Do a kinetic step of size h.
"""
Mi = self._Minv
key = ("q", "p", "Q", "P", "S", "adQ")
q, p, Q, P, S, adQ = packet.get_parameters(key=key)
q = q + h * dot(Mi, p)
Q = Q + h * dot(Mi, P)
S = S + 0.5 * h * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), key=key)
def propagate(self):
r"""Given a wavepacket :math:`\Psi` at time :math:`t` compute the propagated
wavepacket at time :math:`t + \tau`. We perform exactly one timestep of size
:math:`\tau` here. This propagation is done for all packets in the list
:math:`\{\Psi_i\}_i` and neglects any interaction between two packets.
More details can be found in [#]_.
.. [#] E. Faou, V. Gradinaru and C. Lubich, "Computing semiclassical quantum dynamics with Hagedorn wavepackets",
SIAM Journal on Scientific Computing, volume 31 number 4 (2009) 3027-3041.
"""
# Cache some parameter values
dt = self._dt
Mi = self._Minv
key = ("q", "p", "Q", "P", "S", "adQ")
# Propagate all packets
for packet, leading_chi in self._packets:
eps = packet.get_eps()
# Do a kinetic step of dt/2
q, p, Q, P, S, adQ = packet.get_parameters(key=key)
q = q + 0.5 * dt * dot(Mi, p)
Q = Q + 0.5 * dt * dot(Mi, P)
S = S + 0.25 * dt * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), key=key)
# Do a potential step with the local quadratic part
q, p, Q, P, S = packet.get_parameters()
V = self._potential.evaluate_local_quadratic_at(q, diagonal_component=leading_chi)
p = p - dt * V[1]
P = P - dt * dot(V[2], Q)
S = S - dt * V[0]
packet.set_parameters((q, p, Q, P, S))
# Do a potential step with the local non-quadratic Taylor remainder
innerproduct = packet.get_innerproduct()
# Transform to psi
packet2psi = self._TR.transform_phi_to_psi(packet)
# G is F but in the new basis <psi|W|psi> at actual time t_{1/2}
G = innerproduct.build_matrix(packet2psi, operator=partial(self._potential.evaluate_local_remainder_at, diagonal_component=leading_chi))
coefficients = packet2psi.get_coefficient_vector()
coefficients = self._matrix_exponential(G, coefficients, -1.0j * dt / eps**2)
packet2psi.set_coefficient_vector(coefficients)
# Transform back to phi
packet2phi = self._TR.transform_psi_to_phi(packet2psi)
packet.set_coefficient_vector(packet2phi.get_coefficient_vector())
# Do a kinetic step of dt/2
q, p, Q, P, S, adQ = packet.get_parameters(key=key)
q = q + 0.5 * dt * dot(Mi, p)
Q = Q + 0.5 * dt * dot(Mi, P)
S = S + 0.25 * dt * dot(p.T, dot(Mi, p))
adQn = cont_angle(det(Q), reference=adQ)[0]
packet.set_parameters((q, p, Q, P, S, adQn), key=key)