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Journal of Mathematical Physics

Madelung representation of damped parametric quantum oscillator and exactly solvable Schrödinger–Burgers equations

Şirin A. Büyükaşık, Oktay K. Pashaev

Published Dec 1, 2010• 8 citations
Publisher

Abstract

We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger–Burgers equation, for which we obtain exact solutions using complex Cole–Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola–Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.

DOI
10.1063/1.3524505

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We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger–Burgers equation, for which we obtain exact solutions using complex Cole–Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola–Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.
We construct a Madelung fluid model with specific time variable parameters as dissipative quantum fluid and linearize it in terms of Schrodinger equation with time dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrodinger equation and the corresponding classical linear ODE with variable frequency and damping. For the complex velocity field the Madelung system takes the form of a nonlinear complex Schrodinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we discuss and give exact results for nonlinear Madelung systems related with Caldirola-Kanai type dissipative harmonic oscillator.
The Schrödinger equation is a linear partial differential equation that describes the temporal evolution of the state of a quantum mechanical system. One of the pillars of quantum mechanics, the Schrödinger equation has been used to describe a wide variety of phenomena such as wave-particle duality, atomic orbitals, quantum oscillations, and even some fluid mechanics problems. Despite its established importance in the development of quantum mechanics, a solution of the Schrödinger equation for an arbitrary distribution of quantum mechanical potential or for an arbitrary number of space dimensions remains unknown. This work seeks to rectify this theoretical roadblock by providing an analytical solution through a recasting of the well-known Cole-Hopf transformation which is used to linearize a broad class of PDEs of which the Schrödinger equation becomes a member after an earlier change of variables. These two successive transformations render the general wave function proportional to the solution of a forced heat equation, allowing an explicit analytical solution applicable to any system in any number of space dimensions. Two practical applications to deriving the explicit wave functions of multielectronic atoms and to the forced Burgers equation of fluid mechanics are discussed, enabling greater mathematical insight into quantum entanglement and nonlinear fluid systems.
We present a mapping between a Schrödinger equation with a shifted nonlinear potential and the Navier–Stokes equation. Following a generalization of the Madelung transformations, we show that the inclusion of the Bohm quantum potential plus the laplacian of the phase field in the nonlinear term leads to continuity and momentum equations for a dissipative incompressible Navier–Stokes fluid. An alternative solution, built using a complex quantum diffusion, is also discussed. The present models may capture dissipative effects in quantum fluids, such as Bose–Einstein condensates, as well as facilitate the formulation of quantum algorithms for classical dissipative fluids.
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.
Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrödinger equation. A simple generalization of this result leads to the reduction of the Schrödinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive real solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.
The model of a quantum damped oscillator based on the Caldirola-Kanai equation is investigated by exploiting its coherent state basis. It is shown how these coherent states facilitate the derivation of fluid-dynamical equations corresponding to this model in the classical limit. From the conservative form of these equations it is concluded that the system represented by the Caldirola-Kanai equation cannot be considered as the quantum analogue of the classical damped oscillator. This is contrasted with the case of a harmonic oscillator interacting with a heat bath.