Phase and amplitude of the complex wave function are not independent of each other, but coupled, which becomes obvious looking at Madelung's hydrodynamic formulation of quantum mechanics. In the time-independent case, this leads to a kind of conservation law that allows for the reformulation of the linear Schrodinger equation in terms of a nonlinear Ermakov equation which is equivalent to a complex Riccati equation where the quadratic term in this equation explains the origin of the phase-amplitude coupling. A similar conservation law and corresponding nonlinear equations can also be found in the time-dependent case. The gain from the nonlinear formulations emerges when open systems with dissipation and irreversibility are considered. Describing this kind of systems by an effective nonlinear Schrodinger equation leads to a modification of the above-mentioned equations with new qualitative effects like Hopf bifurcations.
Phase and amplitude of the complex wave function are not independent of each other, but coupled, which becomes obvious looking at Madelung's hydrodynamic formulation of quantum mechanics. In the time-independent case, this leads to a kind of conservation law that allows for the reformulation of the linear Schrodinger equation in terms of a nonlinear Ermakov equation which is equivalent to a complex Riccati equation where the quadratic term in this equation explains the origin of the phase-amplitude coupling. A similar conservation law and corresponding nonlinear equations can also be found in the time-dependent case. The gain from the nonlinear formulations emerges when open systems with dissipation and irreversibility are considered. Describing this kind of systems by an effective nonlinear Schrodinger equation leads to a modification of the above-mentioned equations with new qualitative effects like Hopf bifurcations.
Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.
Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex quantities are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics as well as in in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. It turns out that this dissipative effect can either affect the amplitude or the phase of the complex quantity describing the open quantum system. This suggests a relation between non-unitary transformations and gauge-transformations for these systems. A change of the amplitude also seems to be connected with a second quantum of action for "radial" changes, compared to ħ for "angular" changes, leading to an interpretation of Sommerfeld's constant.
The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system.The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem.The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation.The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation.It can be expressed in terms of a complex variable that linearizes the Riccati equation.This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown.Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations.This provides the link to a nonlinear formulation of timeindependent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law.Comparison with SUSY and the time-dependent problems concludes our analysis.
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
In contrast with classical physics, complex quantities have a fundamental physical meaning in quantum physics and action, being essentially the quantized entity, should be given more attention instead of focusing mainly on Hamiltonians or Lagrangians that have the dimension of energy. Phase and amplitude of the complex quantities in (time-dependent and time-independent) quantum mechanics are not independent of each other but coupled via some conservation law. This coupling can be understood if the systems are described in terms of complex nonlinear Riccati equations. These equations not only enable a connection to the Pythagorean triples, probably the oldest and most abstract 'quantization' problem, but also lead to dynamical invariants with the dimension of action. Factorization of the corresponding operator provides generalized creation and annihilation operators, which is also possible for dissipative systems where no conventional Hamiltonian formalism exists. Formal similarities with other fields, particularly with nonlinear dynamics, are shown.
Taking into account wave-particle duality and the uncertainty principle, a nonlinear Schrödinger wave equation (NLSE) for the description of dissipative systems can be constructed by applying only Newton's form of classical mechanics. A comparison between our NLSE for dissipative systems and the linear time-dependent SE for conservative systems shows that, for free motion and motion in a harmonic potential, both equations have Gaussian wave-packet solutions. The time dependence of the maximum of these functions is governed by the classical (real) Newtonian equation of motion. While the time dependence of the width of the wave packet can be determined from a nonlinear complex Bernoulli/Ricatti type differential equation. This can be done by transformation into a Newton-type equation for a real quantity α (which is proportional to the square root of the width), including a kind of “centrifugal force” term. This equation can be solved analytically and when taken with Newton's equation of motion for the mean value of the position, it forms a so-called Ermakov system. For the linear SE equation, this implies the existence of a dynamical invariant. For our NLSE we also obtain such an exact invariant, but now for a dissipative system. Furthermore, this complex nonlinear equation can be linearised using the logarithmic derivative of a complex quantity λ, thus yielding a complex Newtonian equation of motion. The absolute value of λ is equivalent to α, the phase angle can be determined using a relation that connects real and imaginary parts of λ. This relation is linked to a kind of “conservation of angular momentum” for the quantity λ in a complex plane. The relations between the above-mentioned quantities will be discussed in detail.