Quantum Theory from a Nonlinear Perspective: Riccati Equations in Fundamental PhysicsThis 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
1 | |
9 | |
3 TimeIndependent Schrödinger and Riccati Equations | 69 |
4 Dissipative Systems with Irreversible Dynamics | 85 |
5 Irreversible Dynamics and Dissipative Energetics of Gaussian Wave Packet Solutions | 133 |
6 Dissipative Version of TimeIndependent Nonlinear Quantum Mechanics | 178 |
7 Nonlinear Riccati Equations in Other Fields of Physics | 187 |
8 Summary Conclusions and Perspectives | 210 |
Appendix A Method of Linear and Quadratic Invariants | 229 |
Appendix B Position and Momentum Uncertainties in the Dissipative Case | 233 |
Appendix C Classical LagrangeHamilton Formalism in Expanding Coordinates | 240 |
Appendix D On the Connection Between the Bateman Hamiltonian and the Hamiltonian in Expanding Coordinates | 245 |
D2 The Case a 0 | 246 |
Appendix E Logarithmic Nonlinear Schrödinger Equation via Complex Hydrodynamic Equation of Motion | 248 |
251 | |
Other editions - View all
Quantum Theory from a Nonlinear Perspective: Riccati Equations in ... Dieter Schuch No preview available - 2018 |
Quantum Theory from a Nonlinear Perspective: Riccati Equations in ... Dieter Schuch No preview available - 2019 |
Common terms and phrases
amplitude annihilation operators approach Bateman Bernoulli equation Bose–Einstein condensates canonical level canonical transformations classical coefficient complex Riccati equation connection conservation constant of motion constant width continuity equation corresponding diffusion term discussed dissipative systems eigenvalues energy equation of motion Ermakov equation Ermakov invariant expressed in terms formalism frequency friction force fulfils Gaussian WP Green function Hamiltonian irreversible irreversible dynamics Langevin equation leads Lett linear logarithmic derivative logarithmic NLSE Math mean value momentum space momentum uncertainties Newtonian equation non-dissipative obtained parameter phase space Phys physical level position and momentum position space position uncertainty potential problem quadratic quantization quantum mechanics quantum systems quantum theory relation replaced Riccati equation 2.4 Schrödinger equation Schuch Sect superpotential TD Green function TDSE time-dependent time-evolution time-reversal TISE variable velocity wave function Wigner function WP solution WP width written