Schuch D. Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental Physics

Springer International Publishing AG, 2018. — 261 p. — (Fundamental Theories of Physics 191) — ISBN: 3319655922.This book provides a unique survey displaying the power of Riccati equations to describe reversible and irreversible processes in physics and, in particular, quantum physics. Quantum mechanics is supposedly linear, invariant under time-reversal, conserving energy and, in contrast to classical theories, essentially based on the use of complex quantities. However, on a macroscopic level, processes apparently obey nonlinear irreversible evolution equations and dissipate energy.
The Riccati equation, a nonlinear equation that can be linearized, has the potential to link these two worlds when applied to complex quantities. The nonlinearity can provide information about the phase-amplitude correlations of the complex quantities that cannot be obtained from the linearized form. As revealed in this wide ranging treatment, Riccati equations can also be found in many diverse fields of physics from Bose-Einstein-condensates to cosmology. The book will appeal to graduate students and theoretical physicists interested in a consistent mathematical description of physical laws.Time-Dependent Schrödinger Equation and Gaussian Wave Packets
Time-Independent Schrödinger and Riccati Equations
Dissipative Systems with Irreversible Dynamics
Irreversible Dynamics and Dissipative Energetics of Gaussian Wave Packet Solutions
Dissipative Version of Time-Independent Nonlinear Quantum Mechanics
Nonlinear Riccati Equations in Other Fields of Physics
Summary, Conclusions and Perspectives
Appendixes
Method of Linear and Quadratic Invariants
Position and Momentum Uncertainties in the Dissipative Case
Classical Lagrange–Hamilton Formalism in Expanding Coordinates
On the Connection Between the Bateman Hamiltonian and the Hamiltonian in Expanding Coordinates
Logarithmic Nonlinear Schrödinger Equation via Complex Hydrodynamic Equation of Motion