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Journal of Chemical Education

Quantum Mechanical Tunneling through Barriers: A Spreadsheet Approach

Andrés Cedillo

Published Apr 1, 2000• 26 citations
Publisher

Abstract

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.

DOI
10.1021/ed077p528

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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.
Quantum tunneling is a phenomenon where particles have a probability of penetrating a potential barrier despite having total energy llower than the barrier height. This study analyzes the tunneling effect by solving the one-dimensional (1D) Schrodinger equation using the finite difference method to obtain the wave function evolution for various potential barrier configurations. The solution is implemented in a Graphical User Interface (GUI) MATLAB to facilitate analysis and visualization, allowing users to interactively adjust potential parameters, energy, and other conditions. Simualation results demonstratehow transmission probability depends on energy, height and width of potential barrier. This GUI provides an intiuitive tool for exploring quantun tunneling, making it valuable for both education and research in quantum physics.
I use the Laplace transform to obtain the transmission coefficient for each number of potential barriers.There are a little number of close-form solution of Schröginger equation, and this is another exact solution.It is possible to obtain the tunnelling for electronic components (resonant-tunnelling diode), or neutronand proton diffusion in astrophysics (stellar nucleosyntheis) and nuclear physics (nuclear fission and fusion),or chemical tunnelling for two molecules reaction (that I solve in a generic case), or to obtain a quantumfilter for the energy spectrum (that I solve for some coating layers).
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
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.