For the *NANOTCAD* project we modeled electron-phonon scattering
from first principles using a Fröhlich Hamiltonian
[NKKS01].
The state space is built from a single electron wave function
and the phonon Fock space. This is a Schrödinger equation
in a larger space describing electron
phonon interaction. However,
the subsequently derived equation
for the electron density function is no longer
separable and describes a non-Markovian process.
In the derivation of the reduced equation
the separability of the equation is lost. As discussed
in Section 7.2.6 no Schrödinger
equation can be derived.

Here we only consider electron-phonon scattering as a classical stochastic process which has to be coupled with a quantum mechanical system describing charge carrier motion in a potential. Such systems are often denoted as hybrid. Reference [DGS00] suggests a general procedure for coupling classical and quantum systems.

It is not clear a priori, that quantization of the classical scattering model really leaves it invariant. In a fully quantum mechanical treatment the classical scattering probabilities should be replaced by the correct quantum mechanical quantities [Pru84]. However, we derived Fermi's golden rule in a special case [KNRS03] as a scaling limit from a fully quantum mechanical description of electron-phonon interaction. This is a hint, that the model adopted for the Monte Carlo simulations is physically admissible.

For Monte Carlo simulations, the classical Boltzmann scattering term was included into the Wigner equation. This contradicts the interpretation of scattering as a classical stochastic process, as the Wigner function may take on negative values and the interpretation of scattering probability breaks down.

One way out of this dilemma is to use the Husimi function which is positive everywhere as the probability distribution for calculating scattering probabilities. The scattering of particles is then interpreted as a transition between coherent state wave packets, which present the closest quantum mechanical analogue to classical particles.

Then the numerical problem remains to calculate the time evolution of a coherent wave packet in an arbitrary potential. Classically this time evolution is a simple trajectory, but quantum mechanically the problem is not so simple as even the free propagator (zero potential) is not given by a trajectory in the Husimi representation.

For reuse of classical Monte Carlo codes this numerical problem has to be solved very efficiently, otherwise performance is too poor. The development and implementation of such fast methods is an open task. In this thesis we only consider scattering in the Wigner formulation.

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R. Kosik: Numerical Challenges on the Road to NanoTCAD