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Dynamics

Dynamical Invariant for Dissipative Systems via Complex Quantum Hydrodynamics

Dieter Schuch, Moise Bonilla–Licea

Published Jan 16, 2023• 0 citations
Publisher

Abstract

For Hamiltonian systems with time-dependent potential, the Hamiltonian, and thus the energy, is no longer a constant of motion. However, for such systems as the parametric oscillator, i.e., an oscillator with time-dependent frequency ω(t), still, a dynamical invariant can be found that now has the dimension of action. The question, if such an invariant still exists after the addition of a dissipative friction force is analyzed for the classical as well as for the quantum mechanical case from different perspectives, particularly from that of a complex hydrodynamic formulation of quantum mechanics.

DOI
10.3390/dynamics3010002

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For Hamiltonian systems with time-dependent potential, the Hamiltonian, and thus the energy, is no longer a constant of motion. However, for such systems as the parametric oscillator, i.e., an oscillator with time-dependent frequency ω(t), still, a dynamical invariant can be found that now has the dimension of action. The question, if such an invariant still exists after the addition of a dissipative friction force is analyzed for the classical as well as for the quantum mechanical case from different perspectives, particularly from that of a complex hydrodynamic formulation of quantum mechanics.
For time dependent Hamiltonians like the parametric oscillator with time-dependent frequency, the energy is no longer a constant of motion. Nevertheless, in 1880, Ermakov found a dynamical invariant for this system using the corresponding Newtonian equation of motion and an auxiliary equation. In this paper it is shown that the same invariant can be obtained from Bohmian mechanics using complex Hamiltonian equations of motion in position and momentum space and corresponding complex Riccati equations. It is pointed out that this invariant is equivalent to the conservation of angular momentum for the motion in the complex plane. Furthermore, the effect of a linear potential on the Ermakov invariant is analysed.
In a former publication [D. Schuch and M. Moshinsky, Phys. Rev. A 73, 062111 (2006)] we have shown that the time evolution of a quantum system with at most quadratic Hamiltonian that can be described by different methods, namely, the time-dependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function is connected via a dynamical invariant, the so-called Ermakov invariant. Since exact invariants of this type also exist for certain effective descriptions of dissipative quantum systems, we will show the relations between these descriptions and the corresponding invariants and will discuss how the results obtained for the conservative systems must be modified in the presence of a linear velocity dependent frictional force.
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 Schrödinger equation (Caldirola–Kanai model) and a logarithmic nonlinear Schrödinger 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.
Abstract The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined by solving a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied by linearizing the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. The effect of a dissipative, linear velocity dependent friction force on these systems is discussed.
An aim of this paper is to show that the major difference between classic and quantum physics is not only that in the latter case complex quantities are a conceptional element, but also that real and imaginary parts, or phase and amplitude, respectively, are not independent of each other but coupled via some conservation laws. This also holds for systems where the Hamiltonian is no longer a constant of motion but a so-called Ermakov invariant with the dimension of an action still exists. In particular, this is valid for certain dissipative system where a transition between formal Hamiltonian level and physical nonlinear level can be achieved via a non-unitary transformation.
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.