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American Journal of Physics

A numerical study of quantum barrier penetration in one dimension

Charles W. Leming, Andrew Van Smith

Published May 1, 1991• 11 citations
Publisher

Abstract

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.

DOI
10.1119/1.16525

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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.
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.
A method, based on the use of Milne's function www, for the numerical determination of quantum-mechanical reflection coefficients in one-dimensional problems to any preassigned degree of accuracy is outlined. The case in which a "barrier" is both preceded and followed by a field-free space is considered in detail, sections 2-4, and a numerical example is worked out, section 5. The procedure in the case in which a "barrier" is preceded by a field-free space and is followed by a potential for which Schroedinger's equation can be solved analytically is outlined, sections 6-7, and the special case in which a "barrier" is followed by a uniform field is considered in some detail, section 8. The case in which a "barrier" is both preceded and followed by potentials for which Schroedinger's equation can be solved analytically is mentioned, section 8. Formulas are given for the evaluation in terms of $\ensuremath{\psi}$ and its first derivative, of the respective densities of the dextral and of the sinistral current flowing past any point xxx in regions where the total energy is greater than the potential energy, section 6. A procedure is given for finding solutions representing unidirectional beams at infinity, section 6.
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.
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.