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. Contents Introduction 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
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