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2 Asymptotic Behavior of the Wave Packet Propagation through a Barrier: the Green’s Function Approach Revisited

Bogdan Mihaila, Shmuel A. Gurvitzc, David Deana, Witold Nazarewicza E F

Published Aug 16, 2016• 0 citations
OpenAlex

Abstract

To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann [1,2]. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state [5]. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schrödinger equation is available today.

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Asymptotic behavior of the wave packet propagation through a barrier: The Green’s function approach revisited

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To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann [1,2]. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state [5]. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schrödinger equation is available today.

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To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann [1,2]. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state [5]. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schrödinger equation is available today.
To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann [1,2]. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state [5]. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schrödinger equation is available today.
We demonstrate the application of transition state theory to wave packet dynamics in metastable Schrödinger systems which are approached by means of a variational ansatz for the wavefunction and whose dynamics is described within the framework of a time-dependent variational principle. The application of classical transition state theory, which requires knowledge of a classical Hamilton function, is made possible by mapping the variational parameters to classical phase space coordinates and constructing an appropriate Hamiltonian in action variables. This mapping, which is performed by a normal form expansion of the equations of motion and an additional adaptation to the energy functional, as well as the requirements to the variational ansatz are discussed in detail. The applicability of the procedure is demonstrated for a cubic model potential for which we calculate thermal decay rates of a frozen Gaussian wavefunction. The decay rate obtained with a narrow trial wavefunction agrees perfectly with the results using the classical normal form of the corresponding point particle. The results with a broader trial wavefunction go even beyond the classical approach, i.e. they agree with those using the quantum normal form. The method presented here will be applied to Bose–Einstein condensates in the following paper (Junginger et al 2012 J. Phys. A: Math. Theor. 45 155202).
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.
We consider the expansion of wave packets governed by the free Schrödinger equation. This seemingly simple task plays an important role in simulations of various quantum experiments, especially in the field of matter-wave interferometry. The initial tight confinement of quantum particles results in a very fast expansion of the wave function at later times which significantly complicates an efficient and precise numerical evaluation. In many practical cases the expansion time is too short for the validity of the stationary phase approximation and too long for an efficient application of Fourier collocation-based methods. We develop an alternative method based on a discretization of the free-particle propagator. This simple approach yields highly accurate results which readily follows from the exceptionally fast convergence of the trapezoidal rule approximation of integrals involving smooth, rapidly decaying functions. We discuss and analyze our approach in detail and demonstrate how to estimate the numerical error in the one-dimensional setting. Furthermore, we show that by exploiting the separability of the Green's function, the numerical effort of the multi-dimensional approximation is considerably reduced. Our method is very fast, highly accurate, and easy to implement on modern hardware.
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.