This paper proposes a quantum-fluid density-functional theory (an interlinking of quantum-fluid dynamics and density-functional theory) for dealing with molecular collisions, in order to incorporate both time dependence and excited states. Using a new kinetic energy density functional for ion-atom collisions, we have derived a single-particle time-dependent single density equation for many-electron systems. The equation is a new generalized nonlinear Schr\"odinger equation whose solution directly yields the time-dependent charge density, current density, and a pulsating effective potential surface on which the process occurs. The new equation is also derived through Nelson's stochastic interpretation of the single-particle Schr\"odinger equation. A ``thermodynamics'' of the entire time-evolving system is suggested in terms of space- and time-dependent quantities. New algorithms have been devised for solving the above equation in one and two spatial dimensions, in a succession of time steps. Results have been obtained and analyzed in the approach regime for proton-neon high-energy (25-keV) collisions, which permit excitation but not ionization. These results show distinct nonlinear features.

Density-functional and Hydrodynamical Approach to Ion-Atom Collisions Through a New Generalized Nonlinear Schrödinger Equation

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.

Lie systems: theory, generalisations, and applications

A simple treatment to the problem of finding exact invariants and related auxiliary equations for time‐dependent oscillators with friction is presented. The treatment is based on the use of a time‐dependent canonical transformation and an auxiliary transformation.

Canonical transformations and exact invariants for dissipative systems

Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex variable that fulfills a nonlinear Riccati equation which determines the time-evolution of the wave packet width. The solutions of this Riccati equation depend on the physical system under consideration and on the (complex) initial conditions and have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics. The definition of the generalized creation and annihilation operator is also in agreement with a factorization of the operator corresponding to the Ermakov invariant that exists in all cases considered.

Generalized coherent states for time-dependent and nonlinear Hamiltonians via complex Riccati equations

Quantum tunneling: A general study in multi-dimensional potential barriers with and without dissipative coupling

Via the hydrodynamical formulation of quantum mechanics, a unified protocol to treat the quantum time‐dependent harmonic oscillator with friction is presented, described by two different models: an explicitly time‐dependent, linear Schrodinger equation (Caldirola–Kanai model) and a logarithmic nonlinear Schrodinger equation (Kostin model) For the former model, an Ermakov system that makes it possible to obtain an invariant of Ermakov–Lewis‐type is derived For the latter model, a non‐Ermakov system is derived instead and it is shown that neither an exact nor an approximate invariant of Ermakov–Lewis‐type exists

Ermakov and non-Ermakov systems in quantum dissipative models

Via the quantum‐hydrodynamical method, a time‐dependent invariant associated to the quantum dissipative time‐dependent harmonic oscillator (TDHO), described by two classes of nonlinear Schrodinger–Langevin equations with the following frictional nonlinear terms is constructed: (i) W1=−iℏν(ln ψ−〈ln ψ〉), which is the Schuch–Chung–Hartman frictional nonlinear term, and (ii) W2=ν{[x−〈x〉][cp+(1−c)〈 p〉]− 1/2 iℏc}, which includes the Sussmann (c=1), the Hasse (c= 1/2 ), and the Albrecht–Kostin (c=0) frictional nonlinear operators. The associated invariant found is exact for the Schuch–Chung–Hartman and Hasse models, and only approximate for the Sussman and Albrecht–Kostin models.

Antônio B. Nassar

Papers

Ermakov and non-Ermakov systems in quantum dissipative models

Time‐dependent invariant associated to nonlinear Schrödinger–Langevin equations