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Journal of Jiangxi Normal University

The Transmission Probability of Position-Dependent-Mass Particle Through the Free Potential Field

Sang Ming-huang

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Based on the transmission formula derived from the analytical transfer matrix method,two cases of the transmission probabilities of the position-dependent-mass particle through the free potential field are calculated.Combined with the point canonical transformation,we conclude that the transmission probabilities are closely connected to the structure of their effective potential.If the effective potential is a well,the transmission resonance peaks lies exactly at the eigen-energies.If the effective potential is a barrier,the transmission probabilities are very low at the below-barrier energies and oscillating around 1 at the beyond-barrier energies.

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Based on the transmission formula derived from the analytical transfer matrix method,two cases of the transmission probabilities of the position-dependent-mass particle through the free potential field are calculated.Combined with the point canonical transformation,we conclude that the transmission probabilities are closely connected to the structure of their effective potential.If the effective potential is a well,the transmission resonance peaks lies exactly at the eigen-energies.If the effective potential is a barrier,the transmission probabilities are very low at the below-barrier energies and oscillating around 1 at the beyond-barrier energies.
A transfer-matrix technique is formulated to treat the scattering of a particle incident on a piecewise constant potential and interacting with an oscillatory field. A study of tunneling through a single barrier shows in the transmission probabilities, for absorption or emission of quanta from the field, replicas below the barrier of the overbarrier field-free transmissivity, with its resonant structure. In the case of resonant tunneling through a double barrier, an appropriate choice of parameters gives evidence for a change in the regime, with inhibition of transmission at a resonance energy of the intermediate well when the period of the field approaches the lifetime of the resonant state. The technique can be extended to treat scattering of a particle by more general potentials in the presence of an oscillatory perturbation. As an illustrative example, the transmissivity as a function of a bias potential in a double-barrier structure is calculated.
Based on the analytical transfer matrix method (ATMM), the universal dispersion equation determining the energy spectrum for an arbitrary potential function experienced by a particle with arbitrary position-dependent effective mass distribution is obtained. As an illustrative example, for a given mass distribution m ( x ) and a potential function V ( x ), our predicted results are in complete agreement with the exact ones. The energy spectrum of a harmonic oscillator with nonconstant mass is obtained. Since the exact solutions to Schrödinger equation with nonconstant effective mass are a valuable means for checking and improving models and numerical methods in the study of electronic properties, the ATMM will find wide applications in material science and condensed matter.
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.
Some aspects of quantum tunnelling of a composite particle are clarified by a simple model. We calculate the probability that a couple of point particles bound to each other by a square well potential with a hard core tunnel through a rectangular potential barrier. It is shown that resonance structures appear in the transmission spectrum in some cases, which reflects the existence of quasi-bound-states of the centre of mass motion around the potential barrier. It is also shown that an inelastic tunnelling occurs in which the transitions in the relative motion are induced by the tunnelling of the centre of mass.