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AIP conference proceedings

Diffraction in time in tunneling phenomena

Gastón García‐Calderón, Kurt Bernardo Wolf, Luis Benet, Juan Mauricio Torres, Peter Oto Hess

Published Jan 1, 2010• 0 citations
Publisher

Abstract

This paper reviews exact analytical results on diffraction in time for tunneling of a particle through an arbitrary potential of finite range. It emphasizes the essential role played by the formalism of resonant states to derive the analytical solution to the time‐dependent Schrödinger equation as an initial value problem and in the analysis of the corresponding transient behavior.

DOI
10.1063/1.3537870

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This paper reviews exact analytical results on diffraction in time for tunneling of a particle through an arbitrary potential of finite range. It emphasizes the essential role played by the formalism of resonant states to derive the analytical solution to the time‐dependent Schrödinger equation as an initial value problem and in the analysis of the corresponding transient behavior.
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.
A formal analytic solution of the time-dependent Schr\"odinger equation is obtained for cutoff Gaussian wavepackets incident on finite range potentials of arbitrary shape. We derive closed analytical expressions for transmitted wavepackets with a small cutoff and the solutions are applied to investigate the dynamics of wavepacket tunneling. We showed that the analytic solutions satisfy a useful rescaling property that was applied to study and characterize the delay time in delta potential barriers.
Abstract We discuss the procedure of solving the time-dependent Schrödinger equation via a separation ansatz that leads to the time-independent Schrödinger equation. The resulting equation is in the form of an eigenvalue problem for the Hamiltonian, whose solutions are the stationary states. We derive and discuss the bound states and corresponding energy levels for several one-dimensional potentials of interest, including the finite and infinite potential well, the Dirac-delta potential, and the square double-well modelling the spatial configuration of the ammonia molecule. We then turn to the scattered solutions and discuss reflection from and transmission through a finite potential barrier, thus encountering the tunnel effect, which we in turn formulate more generally using the Gamow factor. This leads us to the interesting phenomenon of tunnelling resonances
Based on the similarity between time-independent Schrodinger equation and Helmholtz equation, the analytical transfer matrix method was utilized to analyze macroscopic quantum tunneling effect, and the potential filed and eigenvalue were corresponded to the refractive index and propagation constant, respectively.Without solving the Schrodinger equation and any approximation,the exact analytic expressions of reflection coefficient and reflection probability of the particle were obtained.Initial results show the equations obtained here are exact and have clear physical meanings.It is expected to be applied in the researches of many basic quantum phenomena,such as quantum tunneling effect,quantum reflection, quantum particles and tunneling resonance.