Open Systems

This chapter gives an introduction to some basic concepts. There is a fundamental theoretical problem in modeling and understanding the origin of dissipation and loss. In mesoscopic physics the primary focus is on quantum transport and thermodynamic properties that originate from flow and fluctuation of charge or current that is intricately connected to dissipation and loss. This chapter will provide a novel approach to this old problem and how it is done for mesoscopic systems. Next we will orient the reader to the von-Neumann approach to quantum mechanics, which is believed to be the correct theory of quantum mechanics, in dealing with unbounded operators that model the continuum. That will show the importance of studying the problem using first quantization and a basic introduction will be given as to how to treat boundary conditions leading to continuity of wave-functions and conservation of currents. The boundary condition approach can naturally deal with the situation when there are evanescent modes in the system that most matrix diagonalization approaches overlook. Such evanescent modes will become very important in later chapters. Also, a typical thermodynamic property, i.e., magnetization of a mesoscopic system due to AB effect will be explained in detail using the continuum approach. Special emphasis will be given to explain the topological nature of the problem. An elementary introduction to density of states and its connection to scattering phase shift will be provided.