Chemistry Reference
In-Depth Information
3. Direct Processes
4. Raman Processes
E. The Realistic Phonon Spectrum
F. Ground-State Tunneling and Relaxation
1. The Two-Level Model
2. Ground-Ground-State Resonance
3. Dynamics of the Ground-Ground-State Resonance via Effective Classical Spin
4. Coherence in the Ground-Ground-State Resonance
5. Relaxation Rate Between two Tunnel-Split States
6. Ground-Excited State Resonance
G. Numerical Implementation and Illustrations
H. Discussion
Acknowledgments
References
I.
GENERAL THEORY
Density matrix is used to describe properties of a system that is a part of a larger
system with which it interacts. Whereas isolated systems (e.g., the above mentioned
larger system) can be described by a Schr odinger equation, systems that interact
with their environments cannot. Starting from the Schr odinger equation for the
isolated whole system, small system + environment (or
bath
), and eliminating
the environmental variables, one can, in principle, construct an object that can be
used to calculate observables of the small system, in a short way. This object is the
density matrix of the small system. Of course, integrating or taking matrix elements
in two steps, at first over the environment and then over the small system, is not
a big simplification. This approach becomes really useful if one obtains a closed
equation of motion for the density matrix of the small system, the density matrix
equation (DME). This is possible if the interaction between the small system and
its environment is small and can be considered as a perturbation, and the small
system does not strongly perturb the state of the environment. The derivation of the
DME for a bathed small system will be presented in Section II. Here the necessary
components of the formalism will be introduced.
A.
From the Wave Function to the Density Matrix
The general wave function or state
of an isolated quantum system can be
expanded over a set of complete basis states
|
|
m
as:
|
=
C
m
|
m
(1)
m
