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Transient effects in two channel interactions and an application to the behavior of a time dependent shutter

Tobias Kramer, M. Moshińsky

Published Jan 1, 2005• 1 citation
OpenAlex

Abstract

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].

Related papers

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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.

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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.

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The analytical solution to the time-dependent Schr\"odinger equation for tunneling using cutoff plane-wave initial conditions is in general given by the sum of two types of terms that exhibit a transient behavior. The time evolution of the probability density for the $\ensuremath{\delta}$ potential is compared with the free case to investigate in this case the role of these transient terms for the delay time. We find, by a dynamical calculation, that the delay time arises from the interference between these transient terms and we show that at very long times it goes into the phase delay time, given by the energy derivative of the phase of the transmission amplitude.

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The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.

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The transient solution and delay time for a $\ensuremath{\delta}$ potential scatterer with a movable quantum shutter is calculated by solving analytically the time-dependent Schr\"odinger equation. The delay time is analyzed as a function of the distance between the shutter and the potential barrier and also as a function of the distance between the potential barrier and the detector. In both cases, it is found that the delay time exhibits a dynamical behavior and that it tends to a saturation value $\ensuremath{\Delta}{t}_{\mathrm{sat}}$ in the limit of very short distances, which represents the maximum delay produced by the potential barrier near the interaction region. The phase time ${\ensuremath{\tau}}_{\ensuremath{\theta}}$, on the other hand, is not an appropriate time scale for measuring the time delay near the interaction region, except if the shutter is moved far away from the potential. The role played by the antibound state of the system on the behavior of the delay time is also discussed.

Physical Review A•May 28, 2010•4 citations
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].
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.
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 analytical solution to the time-dependent Schr\"odinger equation for tunneling using cutoff plane-wave initial conditions is in general given by the sum of two types of terms that exhibit a transient behavior. The time evolution of the probability density for the $\ensuremath{\delta}$ potential is compared with the free case to investigate in this case the role of these transient terms for the delay time. We find, by a dynamical calculation, that the delay time arises from the interference between these transient terms and we show that at very long times it goes into the phase delay time, given by the energy derivative of the phase of the transmission amplitude.
The transient behavior of wave packets of electrons in crystals may possibly be accessible to experiments with present techniques. Thus it is interesting to study this problem theoretically. The simplest example is a Kronig-Penney lattice, with δ function potentials, and with a wave packet in one of its segments. We study the behavior of this packet as function of time t if the position x is fixed or vice versa, and derive the general characteristics of the phenomena. In particular we compare our results with the diffraction in time behavior studied long ago by one of the authors (M.M.) when the potential vanish, i.e. for the free packet. It is shown that the characteristic oscillations of the diffraction in time can also appear in the Kronig-Penney lattice. In addition, we have observed that the situation when the energy of the initial state of the particles lies within the bands is essentially indistinguishable from that where it lies within the gaps. The connection with the nonrelativistic diffusion problem in quantum mechanics is also briefly discussed.
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.
The transient solution and delay time for a $\ensuremath{\delta}$ potential scatterer with a movable quantum shutter is calculated by solving analytically the time-dependent Schr\"odinger equation. The delay time is analyzed as a function of the distance between the shutter and the potential barrier and also as a function of the distance between the potential barrier and the detector. In both cases, it is found that the delay time exhibits a dynamical behavior and that it tends to a saturation value $\ensuremath{\Delta}{t}_{\mathrm{sat}}$ in the limit of very short distances, which represents the maximum delay produced by the potential barrier near the interaction region. The phase time ${\ensuremath{\tau}}_{\ensuremath{\theta}}$, on the other hand, is not an appropriate time scale for measuring the time delay near the interaction region, except if the shutter is moved far away from the potential. The role played by the antibound state of the system on the behavior of the delay time is also discussed.