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American Journal of Physics

Tunneling through a truncated harmonic oscillator potential barrier

Jeff D. Chalk

Published Feb 1, 1990• 3 citations
Publisher

Abstract

The sum of a one-dimensional, truncated harmonic oscillator potential and square well of the same range, defined in the positive half-space, serves as a convenient and instructive example for which the Schrödinger equation may have both bound-state and continuum solutions. A superposition of these solutions is used in a study of barrier penetration by a wave packet representing a particle with an initial position in the region of the potential well. The presence or absence of a bound state in the superposition is shown to be the key factor determining the evolution of the wave packet. If no bound state exists, the probability of the particle having a position within the potential well is a monotonically decreasing function of time. If the superposition includes a bound state, however, this probability oscillates slightly because of an interference between the bound-state and continuum components of the wavefunction.

DOI
10.1119/1.16223

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The sum of a one-dimensional, truncated harmonic oscillator potential and square well of the same range, defined in the positive half-space, serves as a convenient and instructive example for which the Schrödinger equation may have both bound-state and continuum solutions. A superposition of these solutions is used in a study of barrier penetration by a wave packet representing a particle with an initial position in the region of the potential well. The presence or absence of a bound state in the superposition is shown to be the key factor determining the evolution of the wave packet. If no bound state exists, the probability of the particle having a position within the potential well is a monotonically decreasing function of time. If the superposition includes a bound state, however, this probability oscillates slightly because of an interference between the bound-state and continuum components of the wavefunction.
We consider a time independent one dimensional finite range and repulsive constant potential barrier between two impenetrable walls. For a nonrelativistic massive particle projected towards the potential with energies less than the barrier and irrespective of the spatial positioning of the potential allowing for quantum tunneling, analytically we solve the corresponding Schrodinger equation. For a set of suitable parameters utilizing Mathematica we display the wave functions along with their associated probabilities for the entire region. We investigate the sensitivity of the probability distributions as a function of the potential range and display a gallery of our analysis. We extend our analysis for bound state particles confined within constant attractive potentials.
An interesting analytical expression is obtained for the coordinate probability distribution of a wave packet transmitted through a potential barrier. In the large-time asymptotics the distribution is given in terms of the transmission amplitude and the phase-space quasidistribution of the initial (not necessarily pure) state. The distortion of a (broad) wave packet due to the barrier penetration is determined by two timelike parameters, given by the energy derivative of the complex transmission amplitude. The resulting change of the packet shape reveals a dependence on the detector position and incorporates a possible speed-up effect and a negative dispersion.
A simple exact solution for the spacetime particle propagator in the presence of a one-dimensional rectangular potential is obtained. This propagator solves the Cauchy problem for the corresponding time-dependent Schrodinger equation in terms of the forward-moving (positive momentum) and backward-moving (negative momentum) components of the wavefunction. The exact formulae for the time-dependent wavefunction in different spatial regions (before, inside and after the barrier) are found. These results demonstrate that generally the forward- and backward-moving components of the wavefunction as well as their interference contribute to the probability density of finding a particle in different spacetime regions. The time of arrival to and the dwell time in the above-mentioned spatial regions are also obtained. The relative contribution of the positive- and negative-momentum components of the wavefunction as well as those of the below- and above-barrier energy regions are discussed for the case of an initial Gaussian wave packet. Numerical visualization of the probability density to find a particle in the different spacetime regions is performed and discussed in the tunneling regime.
The time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time-dependent Schrödinger equation as an initial-value problem by the method of Laplace transform. The wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion. A concise expression is derived for the probability current into the outer space. At long times the current oscillates about zero before decaying with a t −4 long-time tail. A comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states.
Abstract This paper examines tunnelling through potential well protected by double barriers as well as the motion of a wave packet encountering this type of potential. Various options for the left and right barriers’ penetration factors are taken into consideration. When one of the penetration factors has limiting value of 0.5, the transmission coefficients turns out to be independent of incoming energy of the particle and the width of potential well. When both penetration factors have a limiting value of 0.5, a scenario similar to resonance tunneling is achieved. Furthermore, by doing the wave packet analysis, we found that the width and height of potential barriers have a negligible impact on the probability of finding the particle inside the potential well in the limit of large penetration factors.
Scanning tunneling microscopy is introduced as an analytic tool for characterizing nanostructures and dynamics on the nanoscale. To analyze the underlying tunneling phenomenon, the Schrödinger equation is solved for a particle confined between finite potential energy steps. For a single potential energy well, the proper solutions to the Schrödinger equation are similar to those of a particle in an infinite box, except for the emerging “tails” of the stationary wave functions in the “classically forbidden” regions of the external confining potentials. For a symmetric double well potential, wave function penetration of the stationary solutions manifests in their even distribution among the wells. An attempt to localize the particle in one of them would lead to periodic oscillations between the wells, as if the particle can “tunnel” under the separating energy barrier. The wave function penetration and tunneling phenomena are discussed also in relation to energy band formation in periodic lattices.