Abstract
In this chapter, basic quantum tools such as time-evolution operators, transition rates and amplitudes, statistical and projector operators, and interaction and density matrix representations are employed to characterize the open and interacting quantum systems with the aid of Schrödinger, quantum master, Fokker-Planck, and Feynman path integral equations and formulations.
Top1.2. Schrödinger Equation
We start with the celebrated Schrödinger equation
(1.1) which has the coordinate
that spans a set of
degrees of freedom (DOFs); its wave function solution provides the rule for the expectation value of the operator
(Accardi et al. 1995):
(1.2)For an open system (S) that is coupled with a reservoir (R) bath (at temperature T), one can list the following processes to be encountered (May & Kühn 2000):
- •
Relaxation: The energy transfer from S to its surroundings;
- •
Recurrence: For a small environment, all DOFs may become noticeably excited, and it may be possible that the energy moves back into S;
- •
Dissipation: If there is no chance for the energy to move back into S, the energy flow is unidirectional into the environment;
- •
Short-Time-Limit: The regime where the interaction of S with its surroundings is negligible;
- •
Incoherent Motion: If the coupling to the environment become predominant, the time–dependent occupation probabilities 
fulfill the so-called Master Pauli equation:
(1.3)