We present a contribution to the discussion of how best to minimize time-evolving wave functions near the edge of a space-time grid. Our suggestion builds on the work of several groups and adopts an optical potential with real and imaginary parts separated in space (though not necessarily in time) to bring about an artificial acceleration and subsequent suppression of the wave-function amplitude as it approaches the boundary of discretized space-time. The emphasis of the proposal is on the absorption of incident wave functions characterized by low kinetic energy (long de Broglie wavelength) and/or a broad spread of momenta. The action of the optical potential is quantified by calculations of transmission and reflection as a function of incident wave-function kinetic energy, relative bandwidth, and de Broglie wavelength. Numerical constraints on the maximum by which a wave packet may be accelerated and the time and space over which the accelerating potential may act are discussed within the terms of the split-operator method for wave-function propagation. The perceived advantages of spatially overlapping real and imaginary potentials with respect to the minimization of broad-bandwidth wave functions are investigated. It is found that the addition of a simple, though not totally arbitrary, energy shift applied in space prior to absorption by a negative imaginary potential greatly enhances the absorption of wave packets constructed from multiple momentum components.
We present a contribution to the discussion of how best to minimize time-evolving wave functions near the edge of a space-time grid. Our suggestion builds on the work of several groups and adopts an optical potential with real and imaginary parts separated in space (though not necessarily in time) to bring about an artificial acceleration and subsequent suppression of the wave-function amplitude as it approaches the boundary of discretized space-time. The emphasis of the proposal is on the absorption of incident wave functions characterized by low kinetic energy (long de Broglie wavelength) and/or a broad spread of momenta. The action of the optical potential is quantified by calculations of transmission and reflection as a function of incident wave-function kinetic energy, relative bandwidth, and de Broglie wavelength. Numerical constraints on the maximum by which a wave packet may be accelerated and the time and space over which the accelerating potential may act are discussed within the terms of the split-operator method for wave-function propagation. The perceived advantages of spatially overlapping real and imaginary potentials with respect to the minimization of broad-bandwidth wave functions are investigated. It is found that the addition of a simple, though not totally arbitrary, energy shift applied in space prior to absorption by a negative imaginary potential greatly enhances the absorption of wave packets constructed from multiple momentum components.
In this paper, the reflection and transmission of plane waves are examined from a complex potential. Such potentials have the property of absorbing wave packets incident on them and are used widely in time-dependent quantum scattering theory. The purpose of the study is to determine the optimal form of potential to be used for absorbing wave packets near the edges of finite grids in coordinate space. The best potentials for such purposes lead to the minimum possible transmission and reflection of the incident wave packet. The Jeffreys–Wentzel–Kramers–Brillouin (JWKB) theory is used to address this problem and a new form for the optimal complex potential is proposed. A scaled dimensionless form of the Schrödinger equation is also derived, so that the parameters of any optimized potential obtained for a particular collision energy and mass combination may be readily converted to apply to a new set of masses and energies.
The time-dependent scattering of a wave packet from a Gaussian barrier is investigated computationally in the complex z-plane. The initial wave packet and the potential energy are obtained through analytic continuation from functions specified on the real-axis. The wave packet is then propagated on the two-dimensional grid. For a low initial wave packet energy, the time evolution is followed by plotting the following functions: |ψ(z,t)|, real(ψ(z,t)), and the quantum momentum function (QMF), p(z,t). In the reflected packet, an important role is played by ripples (quasi-nodes) forming above the real axis. As these quasi-nodes move down across the real axis, they are 'detected' as 'interference oscillations' in the density. In contrast, the component of the packet below the real axis makes a significant contribution to the transmitted packet. Vector maps of the QMF show hyperbolic flow around quasi-nodes and counterclockwise circular flow around transient stagnation points, where the QMF vanishes. However, when the Pólya vector field (defined by P(z,t) = p*(z,t)) is plotted, circular counterclockwise flow is obtained near the quasi-nodes. The real and imaginary parts of the quantum action function S(z,t) are plotted and the vorticity, defined by the curl of the Pólya field, is used to pinpoint regions of nonanalyticity in the QMF.
The criterion for the validity of the JWKB approximation is used to derive a first order differential equation for a negative imaginary absorbing potential, −iε(r). The resulting absorbing potential switches on as rapidly as possible with increasing r without causing too much reflection, and it has a second order pole at the end of the absorbing region which eliminates all transmission. It is completely specified by a single physical parameter: the minimum energy Emin at which absorption is required; the width of the absorbing region can be chosen to correspond to the de Broglie wavelength at this minimum energy. With this choice, the exact quantum mechanical reflection probability of the potential is <1% at E=Emin, <0.1% at E=2Emin, and <0.01% for all E⩾3Emin. Because of its lack of parameters and its convenient reflection properties, we anticipate that the new potential (and other potentials of a similar form) will find useful application in both time-dependent and time-independent quantum scattering calculations.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x) exceeds the value of the total energy E, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=a at about the time at which the incident packet reaches the point x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
We propose and experimentally demonstrate a method to prepare a nonspreading atomic wave packet. Our technique relies on a spatially modulated absorption constantly chiseling away from an initially broad de Broglie wave. The resulting contraction is balanced by dispersion due to Heisenberg's uncertainty principle. This quantum evolution results in the formation of a nonspreading wave packet of Gaussian form with a spatially quadratic phase. Experimentally, we confirm these predictions by observing the evolution of the momentum distribution. Moreover, by employing interferometric techniques, we measure the predicted quadratic phase across the wave packet. Nonspreading wave packets of this kind also exist in two space dimensions and we can control their amplitude and phase using optical elements.
The quantum trajectory dynamics is extended to the wave function evolution in imaginary time. For a nodeless wave function a simple exponential form leads to the classical-like equations of motion of trajectories, representing the wave function, in the presence of the momentum-dependent quantum potential in addition to the external potential. For a Gaussian wave function this quantum potential is a time-dependent constant, generating zero quantum force yet contributing to the total energy. For anharmonic potentials the momentum-dependent quantum potential is cheaply estimated from the global Least-squares Fit to the trajectory momenta in the Taylor basis. Wave functions with nodes are described in the mixed coordinate space/trajectory representation at little additional computational cost. The nodeless wave function, represented by the trajectory ensemble, decays to the ground state. The mixed representation wave functions, with lower energy contributions projected out at each time step, decay to the excited energy states. The approach, illustrated by computing energy levels for anharmonic oscillators and energy level splitting for the double-well potential, can be used for the Boltzmann operator evolution.