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Exact solution of the position-dependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter

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Abstract
We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel-Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to infinity, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.
Keywords
ALGEBRAIC APPROACH, QUANTUM-WELLS, ENERGY-GAP, POINT, LAYER, TIME, position-dependent effective mass and angular frequency, confined, harmonic oscillator, associated Legendre and Gegenbauer polynomials, Schr&#246, dinger equation, BenDaniel&#8211, Duke kinetic energy, operator, quantized confinement parameter

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MLA
Jafarov, E., I., et al. “Exact Solution of the Position-Dependent Effective Mass and Angular Frequency Schrodinger Equation : Harmonic Oscillator Model with Quantized Confinement Parameter.” JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, vol. 53, no. 48, 2020, doi:10.1088/1751-8121/abbd1a.
APA
Jafarov, E., I., Nagiyev, S. M., Oste, R., & Van der Jeugt, J. (2020). Exact solution of the position-dependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 53(48). https://doi.org/10.1088/1751-8121/abbd1a
Chicago author-date
Jafarov, E., I, S. M. Nagiyev, Roy Oste, and Joris Van der Jeugt. 2020. “Exact Solution of the Position-Dependent Effective Mass and Angular Frequency Schrodinger Equation : Harmonic Oscillator Model with Quantized Confinement Parameter.” JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 53 (48). https://doi.org/10.1088/1751-8121/abbd1a.
Chicago author-date (all authors)
Jafarov, E., I, S. M. Nagiyev, Roy Oste, and Joris Van der Jeugt. 2020. “Exact Solution of the Position-Dependent Effective Mass and Angular Frequency Schrodinger Equation : Harmonic Oscillator Model with Quantized Confinement Parameter.” JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 53 (48). doi:10.1088/1751-8121/abbd1a.
Vancouver
1.
Jafarov, E. I, Nagiyev SM, Oste R, Van der Jeugt J. Exact solution of the position-dependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 2020;53(48).
IEEE
[1]
I. Jafarov, E., S. M. Nagiyev, R. Oste, and J. Van der Jeugt, “Exact solution of the position-dependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter,” JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, vol. 53, no. 48, 2020.
@article{8682691,
  abstract     = {{We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel-Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to infinity, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.}},
  articleno    = {{485301}},
  author       = {{Jafarov, E., I and Nagiyev, S. M. and Oste, Roy and Van der Jeugt, Joris}},
  issn         = {{1751-8113}},
  journal      = {{JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL}},
  keywords     = {{ALGEBRAIC APPROACH,QUANTUM-WELLS,ENERGY-GAP,POINT,LAYER,TIME,position-dependent effective mass and angular frequency,confined,harmonic oscillator,associated Legendre and Gegenbauer polynomials,Schr&#246,dinger equation,BenDaniel&#8211,Duke kinetic energy,operator,quantized confinement parameter}},
  language     = {{eng}},
  number       = {{48}},
  pages        = {{14}},
  title        = {{Exact solution of the position-dependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter}},
  url          = {{http://dx.doi.org/10.1088/1751-8121/abbd1a}},
  volume       = {{53}},
  year         = {{2020}},
}

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