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American Journal of Physics

A study of barrier penetration in quantum mechanics

Jeff D. Chalk

Published Jan 1, 1988• 16 citations
Publisher

Abstract

The Schrödinger equation is solved for the truncated Gaussian potential barrier using a power series solution and a general expression is obtained for the transmission coefficient. For this potential barrier, computer calculations show only a trace of the oscillatory behavior that characterizes the penetration of a rectangular barrier. The oscillations diminish rapidly as the discontinuity in the potential function approaches zero. The results indicate that it is the unrealistic discontinuities in potential models that cause such oscillatory behavior.

DOI
10.1119/1.15425

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The Schrödinger equation is solved for the truncated Gaussian potential barrier using a power series solution and a general expression is obtained for the transmission coefficient. For this potential barrier, computer calculations show only a trace of the oscillatory behavior that characterizes the penetration of a rectangular barrier. The oscillations diminish rapidly as the discontinuity in the potential function approaches zero. The results indicate that it is the unrealistic discontinuities in potential models that cause such oscillatory behavior.
Elementary numerical methods are used to evaluate solutions to Schrödinger’s equation for a variety of one-dimensional potential barriers and to determine the transmission coefficients of free particles. These techniques are used to study the oscillatory variations in transmission coefficients for both continuous and discontinuous potentials. The amplitude of the variations is found to be strongly increased by discontinuities in the potential function but small variations exist even for a smooth, continuous potential function. The techniques described are suitable for intermediate-level undergraduate physics students.
Transmission coefficients for arbitrary potential barriers are obtained using a spreadsheet approach to solve the pertinent Schrödinger equation for the corresponding dispersion eigenfunctions in one dimension. Potentials should be of finite range, otherwise arbitrary. The method of implementation in personal computers is presented here, and it is also shown that the graphical capabilities of the software make it more useful in teaching. This approach reproduces exact solutions for square step barriers and it is illustrated with Gaussian potential energy functions. As a practical example, we simulate a chemical reaction whose rate shows an Arrhenius-like behavior.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
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 the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.