In [TM98] the potential term
has been interpreted as a quantum force
giving rise to dynamic particle trajectories. They nicely
explain the
tunneling process but yet cannot solve Equation 6.52
because the
quantum force itself
depends on the solution .
In [NDRJ96] a particle approach to the coherent Wigner equation has been proposed. The Wigner potential has been treated exactly by the iteration series of the equation. The convergence of the obtained backward Monte Carlo method has been theoretically investigated.
Recently the coherent Equation 6.52
has been solved numerically by using particles
[SF01] which cross the device by collisionless drift
over classical trajectories.
The information about is retained as particle affinity.
The phase space propagator
is
defined to give the time evolution of the Wigner function
![]() |
(6.82) |
Quantum propagators can be quite different from classical
propagators. The classical equations of motion define classical
trajectories in phase space for the coordinates , and
momenta
, as a function of the time
. The classical
propagators for Liouville distributions in phase space are
therefore
functions over the classical trajectories
defining a one-to-one mapping between single points of the initial
and final distributions. In quantum mechanics, on the other
hand, a single point of the phase space distribution at one
time can in principle be causally connected to many points of
this phase space distribution at any other time. Furthermore,
quantum propagators may include quantum jumps, sudden changes
in the momentum or quick change in the coordinate of the phase space
quasi distributions. Two points of the phase space distribution
at two different times can be causally connected even when no classical
trajectory connects them. These are essential quantum features.
For any Hamiltonian
with potential energy
and constant mass
that
is, at most, quadratic in the coordinate
, the phase space
propagator is a
function on the classical paths.
For the free propagator (
) we get
![]() |
(6.83) |
In this case the classical and quantum dynamics are basically
the same as Wigner's integro-differential equation for quadratic
potentials is equivalent to Liouville's classical equation with
a constant force term. The main difference is in the initial
conditions: In the quantum case the initial distribution
must be chosen in such a way that it corresponds to a
positive definite density matrix . In the classical case
any positive distribution function
can be chosen.
For more complicated potentials the phase space propagator
is no longer a function. To deal with this
in a particle picture
we develop a stochastic approach suitable for
Monte Carlo methods in Chapter 9.
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