2020   03   en   p.31-35 E.I. Jafarov and A.M. Mammadova,
On the exact solution of the confined position-dependent mass harmonic oscillator model under the kinetic energy operator compatible with galilean invariance


We propose exactly-solvable model of the confined harmonic oscillator in the framework of the effective mass formalism varying with position. Analytical expression of the position-dependent effective mass is chosen by such a way that it provides confinement effect for the via the infinitely high borders at value of position x = ±a. Wave functions of the stationary states of the oscillator model under study have been obtained by solving exactly corresponding Schrödinger equation, which free Hamiltonian is compatible with Galilean invariance. Analytical expression of the wave function is described by the Gegenbauer polynomials, whereas obtained energy spectrum is discrete, but non-equidistant. It is shown that both energy spectrum and wave function completely recover known expressions of the so-called Hermite oscillator equidistant energy spectrum and wave function of the stationary states under the limit a → ∞.

Keywords: Position-dependent effective mass, quantum harmonic oscillator, Gegenbauer polynomials, non-equidistant energy spectrum.
PACS: 03.65.-w, 02.30.Hq, 03.65.Ge

Received: 19.10.2020


Institute of Physics, Azerbaijan National Academy of Sciences Javid av. 131, AZ 1143, Baku, Azerbaijan
E-mail: ejafarov@physics.science.az

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