This chapter addresses generalizations of the Schrödinger equation. It tries to convey that the Schrödinger equation is not the whole story when it comes to quantum physics. This is illustrated by expanding the framework in two rather orthogonal directions: relativistic quantum physics and open quantum systems. The former is introduced by taking the Klein–Gordon equation as the starting point, before shifting attention to the Dirac equation. Its time-independent version is solved numerically for a one-dimensional example, and its relation to the Schrödinger equation is derived. Also here, the Pauli matrices play crucial roles. The notion of open quantum systems is motivated by the fact that it is hard to keep a quantum system completely isolated from its surroundings – and that this necessitates a different approach than the one provided by wave functions. To this end, reduced density matrices and the notion of master equations are introduced. It is explained why master equations of the form of the generic Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation are desirable. Two particular phenomena following this equation are studied quantitatively: amplitude damping for a single quantum bit system and particle capture in a confining potential. Again, these examples draw directly on previous ones.

This appendix provide a self-contained presentation of the open systems and quantum optics methods used in other parts of the book (e.g., studying the relaxation of a microwave cavity or a qubit, the driving of a quantum amplifier, etc.). Half of the appendix is devoted to the derivation of master equations for small systems that are in contact with Markovian environments. The other half of the appendix is devoted to the development of an alternative input–output description of how those systems absorb information from the environment and reflect it back.

Effusion is one of the most elementary kinetic processes, to which the concepts of nonequilibrium statistical mechanics can be applied. Remarkably, the stationary probability distribution can be exactly constructed as a so-called Poisson suspension, explicitly showing that time reversal is broken at the statistical level of description under nonequilibrium conditions. The multivariate fluctuation relation for the energy and particle currents can be directly deduced from the underlying microscopic dynamics. Moreover, temporal disorder and its nonequilibrium time asymmetry can be fully characterized and shown to be related to the thermodynamic entropy production. The multivariate fluctuation relation can also be applied to mass separation by effusion.

This chapter discusses the spin degrees of freedom in an atom. We first review how atomic energy levels can be classified in terms of the electron spin and orbital angular momenta and how this couples to the nuclear spin. We then describe how atoms interact with each other, and how the spins affect this interaction. The effect of electromagnetic radiation on the energy levels of an atom is described, and the Hamiltonian for energy levels transitions . After briefly describing how the important phenomena of the ac Stark shift and Feshbach resonances occur, we then turn to describing how dissipative dynamics affect atomic systems. Specifically, we examine master equations for spontaneous emission and atom loss, and look at these can be solved. Finally, we consider an alternative framework for solving such open systems using the quantum jump method, which allows for a stochastic approach to solving the dynamics