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arXiv:1007.2094 · Journal of Physics A Mathematical and Theoretical

Position-dependent-mass: cylindrical coordinates, separability, exact solvability and \mathcal {PT} -symmetry

Omar Mustafa

Published Aug 27, 2010• 13 citations
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Abstract

The kinetic energy operator with position-dependent-mass in cylindrical coordinates is obtained. The separability of the corresponding Schrödinger equation is discussed within radial cylindrical mass settings. Azimuthal symmetry is assumed and spectral signatures of various z -dependent interaction potentials (Hermitian and non-Hermitian -symmetric) are reported.

DOI
10.1088/1751-8113/43/38/385310

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arXiv:quant-ph/0601004•Dec 31, 2005•1 citation
The kinetic energy operator with position-dependent-mass in cylindrical coordinates is obtained. The separability of the corresponding Schrödinger equation is discussed within radial cylindrical mass settings. Azimuthal symmetry is assumed and spectral signatures of various z -dependent interaction potentials (Hermitian and non-Hermitian -symmetric) are reported.
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Jia and Dutra ( 2006 J. Phys. A: Math. Gen. 39 11877 ) have considered the one- dimensional non-Hermitian complexified potentials with real spectra in the context of position-dependent mass in the Dirac equation. In their second example, a smooth step shape mass distribution is considered and a non-Hermitian non- PT -symmetric Lorentz vector potential is obtained. They have mapped this problem into an exactly solvable Rosen–Morse Schrödinger model and claimed that the energy spectrum is real. The energy spectrum they have reported is pure imaginary or at best forms an empty set. Their claim on the reality of the energy spectrum is fragile, therefore.
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Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass(PDM). The explicit expressions for the potentials, energy eigenvalues and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term VmV_mVm​ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of VmV_mVm​ for several mass functions are discussed.