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Journal of Nantong University

Quantum Reflection in Step-Exponential Barrier and Parabolic Barrier

Fang Jing-huai

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

We calculate transition and reflection coefficients of arbitrary barrier by analytical transfer matrix method(ATMM),and then apply the method to the quantum reflection above the barrier.The results in step-exponential barrier and parabolic barrier by ATMM are more accurate than those by WKB approximation.Furthermore,when the energy of the particle is greater than the barrier maximum,only ATMM rather than WKB can get precise results.The results show that reflection coefficients gradually change to zero,where quantum reflection happens.These problems can be solved by ATMM exactly.

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We calculate transition and reflection coefficients of arbitrary barrier by analytical transfer matrix method(ATMM),and then apply the method to the quantum reflection above the barrier.The results in step-exponential barrier and parabolic barrier by ATMM are more accurate than those by WKB approximation.Furthermore,when the energy of the particle is greater than the barrier maximum,only ATMM rather than WKB can get precise results.The results show that reflection coefficients gradually change to zero,where quantum reflection happens.These problems can be solved by ATMM exactly.
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.
People are familiar with quantum mechanical reflection and transmission coefficient. In all those cases corresponding potentials are usually assumed as of constant height and depth. For the cases of varying potential, corresponding reflection and transmission coefficients can be found out using WKB approximation method. But due to change of barrier height, reflection and transmission coefficients should be changed from point to point. Here we show the analytical expressions of the instantaneous reflection and transmission coefficients. Here as if we apply the WKB approximation at each point, so we call it as Instantaneous WKB method or IWKB method. Once we know the forward and backward wave amplitudes we can find out corresponding wave function by calculating Eiconal. For the case of analytically complicated potential corresponding differential equation seems to be unsolvable analytically. If the potentials are well behaved then obviously these could be replaced by functions of simple expression of same behaviour which can be integrated analytically and the equation is now possible to solve analytically.
Quantum tunneling is a phenomenon in quantum mechanics. In this phenomenon, Quantum particle can go through potential barrier that potential energy is higher than total energy of quantum particle. Physicists are interested in transmitted and reflected value of Quantum particle that is transmission and reflection probabilities. Then, Schrödinger equation is used to find transmission and reflection probabilities by directly solving. However, some potential barriers are too complex to solve directly. Then, we use another method to solve more complex problem. That method is called "quasilinearization''. Quasilinearization method (QLM) is a numerical method to solve nonlinear differential equations. In this thesis, we compare the quasilinearization method with the well-known WKB approximation method. The results show that the transmission and reflection probabilities calculated by the QLM and WKB approximation have the same trend as the transmission and reflection probabilities calculated from the exact solution.
Application of the variational RRR-matrix method to one-dimensional quantum tunneling
We have applied the variational RRR-matrix method to calculate the reflection and tunneling probabilities of particles tunneling through one-dimensional potential barriers for five different types of potential profiles -- truncated linear step, truncated exponential step, truncated parabolic, bell-shaped, and Eckart. Our variational results for the transmission and reflection coefficients are compared with exact analytical results and results obtained from other numerical methods. We find that our results are in good agreement with them. We conclude that the variational RRR-matrix method is a simple, non-iterative, and effective method to solve one-dimensional quantum tunneling problems.