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Journal of Physics A Mathematical and General

Diffraction in time with dissipation

M. Moshińsky, Dieter Schuch

Published May 2, 2001• 11 citations
Publisher

Abstract

The more usual problems discussed in classical mechanics in elementary textbooks are the non-dissipative ones in which there is a Hamiltonian representing the energy as a constant of motion. The translation of this type of problem to quantum mechanics is very well known. Conversely, there are very simple classical mechanical problems that involve dissipation, but whose translation to quantum mechanics on the basis of a corresponding Hamiltonian is sometimes misinterpreted, since the underlying classical formalism involves non-canonical transformations that lead to non-unitary transformations of the quantum mechanical wavefunctions. In this paper we shall discuss the problem of a beam of particles of given momentum incident from the left on a shutter that is opened at time t = 0. The solution has the well known properties of diffraction in time, but we will analyse it quantum mechanically both when dissipation is and is not present. The objective is to get a better understanding of the effect of dissipation in the quantum mechanical picture and to estimate the influence of the above mentioned non-unitary transformations on this particular problem.

DOI
10.1088/0305-4470/34/19/317

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The more usual problems discussed in classical mechanics in elementary textbooks are the non-dissipative ones in which there is a Hamiltonian representing the energy as a constant of motion. The translation of this type of problem to quantum mechanics is very well known. Conversely, there are very simple classical mechanical problems that involve dissipation, but whose translation to quantum mechanics on the basis of a corresponding Hamiltonian is sometimes misinterpreted, since the underlying classical formalism involves non-canonical transformations that lead to non-unitary transformations of the quantum mechanical wavefunctions. In this paper we shall discuss the problem of a beam of particles of given momentum incident from the left on a shutter that is opened at time t = 0. The solution has the well known properties of diffraction in time, but we will analyse it quantum mechanically both when dissipation is and is not present. The objective is to get a better understanding of the effect of dissipation in the quantum mechanical picture and to estimate the influence of the above mentioned non-unitary transformations on this particular problem.
The subject of transient effects in quantum mechanics has been of interest to one of the authors (MM) since long before it became possible to study them experimentally. In particular, the problem of opening a shutter instantaneously led him to the concept of diffraction in time [1]. Physically, it is only possible to open a shutter as a function of time and this complicates the problem greatly, as it is then not invariant under time translations and thus, among other difficulties, the energy of the system is not a constant of the motion. Kleber and Scheitler [2] analyzed the problem describing the shutter as a ±¡function potential at the origin x = 0, but whose strength was an inverse function of time. In this paper, we follow another procedure by adding to our initial two particle channel another one, and make them interact through appropriate time independent boundary conditions at the point of coincidence. The full problem conserves the total energy, but this does not happen if we restrict ourselves to a description in the first channel only. Thus, we have a rough but analytically solvable problem using a Laplace transform, which mimics some aspects of a time-dependent shutter, and compare our results with some of those derived by the different procedure of reference [2].
The translation from classical to quantum mechanics using the hamiltonian formalism for a non-dissipative systems is very well known. If in the classical picture there is dissipation, the equation of motion is given in terms of newtonian mechanics whose quantum equivalent is not obvious. Using results of previous works we show a way of obtaining a corresponding hamiltonian and discuss the wave function in physical space. We then proceed to analyze the quantum mechanical motion of gaussian wave packets with dissipation.
The purpose of this paper is to discuss two forms in which the time–energy uncertainty relation can appear. One of them concerns the time of preparation of a given state and the other the lifetime of compound systems. Both will be illustrated here through exactly solvable examples. The first one will be the opening and subsequent closing of a shutter on which we have an incident beam of particles. This problem shows a diffraction-in-time effect closely connected with the first type of time–energy uncertainty relation. The second will be the decay of a compound state described through a schematic R matrix formalism. There one can clearly see the transition between the time–energy uncertainty relation as associated with the interval in which the decay process has been in operation, and as associated with the lifetime of the compound state.
The problem of the dissipative systems from the quantum point of view has been of increasing interest, but it is far from having a satisfactory solution (1). There are many problems where the dissipation has an important role such as in quantum optics, in quantum analysis of fields, in quantum gravity (2). Dissipation can be observed in interactions between two systems, the observed system and another one often called reservoir or the bath, into which the energy flows via an irreversiblemanner (3). The system is embedded in some environmentwhich is in principle unknown. For an effective description of such systems, we can use time dependent Hamiltonians which in classical physics yield the proper equation of motion. If the friction is a linear funcion of the velocity with friction constant γ the Hamiltonian is the well known Caldirola Kanai Hamiltonian
The usual treatment of damping forces in quantum mechanics starts from the introduction of the explicitly time dependent Kanai Hamiltonian, which actually represents a variable mass particle, and the misinterpretation of this Hamiltonian as representing a particle of constant mass leads to certain physical difficulties. However the Hamiltonian can be modified so that it can be reinterpreted as describing a constant mass particle. Here we explicitly introduce the mass as a new dynamical variable, which allows us to write a linear, time independent Hamiltonian for the system, which can be solved by conventional methods. The damped harmonic oscillator and damped free particle are treated in detail, both for the Kanai Hamiltonian and for our case, and the solutions are compared. Our solution can be reduced to the Kanai one in appropriate circumstances, but in general it has a much greater versatility, as a result of which it can be more easily reinterpreted as describing a constant mass particle subject to a damping force, which reinterpretation is of course necessary if the method is to have practical applicability. We also show how such a reinterpretation can be carried out in detail by introducing a ’’dissipation variable’’, in terms of which one may avoid the concept of a variable mass altogether.
In a previous note a dynamical description of resonance scattering was given, and transient terms appeared in the wave function describing the process. To understand the physical significance of these terms, the transient effects that appear when a shutter is opened are discussed in this paper. For a nonrelativistic beam of particles, the transient current has a close mathematical resemblance with the intensity of light in the Fresnel diffraction by a straight edge. This is the reason for calling the transient phenomena by the name of diffraction in time.The shutter problem is discussed for particles whose wave functions satisfy the Schr\"odinger equation, the ordinary wave equation, and the Klein-Gordon equation. Only for the Schr\"odinger time-dependent equation do the transient wave functions resemble the solutions that appear in Sommerfeld's theory of diffraction. The connection of transient phenomena with the time-energy uncertainty relation, and the interpretation of the transient current in a scattering process, are briefly discussed. The relativistic wave functions for the shutter problem may play an important role in the dynamical description of a relativistic scattering process.