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Acta Physica Sinica

The bound energy spectrum of position-dependent mass particle in the null potential

Dai Man-Yuan, Yi-you Nie, Sang Ming-huang, Xianping Wang, Yin Cheng, Zhuangqi Cao

Published Jan 1, 2010• 0 citations
Publisher

Abstract

The bound energy spectrum of a position-dependent mass particle which is trapped in a null potential is obtained by combining the point canonical transformation (PCT) method and the analytical transfer matrix (ATM) method. With the calculation for the transmission of a position-dependent mass particle passing through the null potential, the one-to-one correspondence between transmission resonance peak and eigenenergy is found and explained.

DOI
10.7498/aps.59.7586

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The bound energy spectrum of a position-dependent mass particle which is trapped in a null potential is obtained by combining the point canonical transformation (PCT) method and the analytical transfer matrix (ATM) method. With the calculation for the transmission of a position-dependent mass particle passing through the null potential, the one-to-one correspondence between transmission resonance peak and eigenenergy is found and explained.
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.
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.
On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrödinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schrödinger equation is exactly solvable.This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions.The general wavefunctions have however remained unknown in explicit form because for their determination one would need the solutions of a rather tricky differential-difference equation.Here we show that solving this equation may be avoided by combining the deformed shape invariance technique with the point canonical transformation method in a novel way.It consists in employing our previous knowledge of the PDM problem energy spectrum to construct a constant-mass Schrödinger equation with similar characteristics and in deducing the PDM wavefunctions from the known constantmass ones.Finally, the equivalence of the wavefunctions coming from both approaches is checked.
We study the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the framework of the position-dependent effective mass Dirac equation. The Dirac equation is mapped into the exactly solvable Schrödinger-like equation endowed with position-dependent effective mass that we present a new procedure to solve it. The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function, the imaginary potential can be obtained. The spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials. We also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials.