Exact solution of the positiondependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter
 Author
 I Jafarov, E., S. M. Nagiyev, Roy Oste (UGent) and Joris Van der Jeugt (UGent)
 Organization
 Abstract
 We present an exact solution of a confined model of the nonrelativistic 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 BenDanielDuke kinetic energy operator. The positiondependency 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 positiondependent mass and angular frequency is finite, has a nonequidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with positiondependent 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 wellknown equidistant energy spectrum and the wave functions of the stationary nonrelativistic harmonic oscillator expressed in terms of Hermite polynomials. The positiondependent effective mass and angular frequency also become constant under this limit.
 Keywords
 ALGEBRAIC APPROACH, QUANTUMWELLS, ENERGYGAP, POINT, LAYER, TIME, positiondependent effective mass and angular frequency, confined, harmonic oscillator, associated Legendre and Gegenbauer polynomials, Schrö, dinger equation, BenDaniel–, Duke kinetic energy, operator, quantized confinement parameter
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU8682691
 MLA
 Jafarov, E., I., et al. “Exact Solution of the PositionDependent Effective Mass and Angular Frequency Schrodinger Equation : Harmonic Oscillator Model with Quantized Confinement Parameter.” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, vol. 53, no. 48, 2020, doi:10.1088/17518121/abbd1a.
 APA
 Jafarov, E., I., Nagiyev, S. M., Oste, R., & Van der Jeugt, J. (2020). Exact solution of the positiondependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter. JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, 53(48). https://doi.org/10.1088/17518121/abbd1a
 Chicago authordate
 Jafarov, E., I, S. M. Nagiyev, Roy Oste, and Joris Van der Jeugt. 2020. “Exact Solution of the PositionDependent Effective Mass and Angular Frequency Schrodinger Equation : Harmonic Oscillator Model with Quantized Confinement Parameter.” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL 53 (48). https://doi.org/10.1088/17518121/abbd1a.
 Chicago authordate (all authors)
 Jafarov, E., I, S. M. Nagiyev, Roy Oste, and Joris Van der Jeugt. 2020. “Exact Solution of the PositionDependent Effective Mass and Angular Frequency Schrodinger Equation : Harmonic Oscillator Model with Quantized Confinement Parameter.” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL 53 (48). doi:10.1088/17518121/abbd1a.
 Vancouver
 1.Jafarov, E. I, Nagiyev SM, Oste R, Van der Jeugt J. Exact solution of the positiondependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter. JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL. 2020;53(48).
 IEEE
 [1]I. Jafarov, E., S. M. Nagiyev, R. Oste, and J. Van der Jeugt, “Exact solution of the positiondependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter,” JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, vol. 53, no. 48, 2020.
@article{8682691, abstract = {{We present an exact solution of a confined model of the nonrelativistic 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 BenDanielDuke kinetic energy operator. The positiondependency 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 positiondependent mass and angular frequency is finite, has a nonequidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with positiondependent 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 wellknown equidistant energy spectrum and the wave functions of the stationary nonrelativistic harmonic oscillator expressed in terms of Hermite polynomials. The positiondependent 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 = {{17518113}}, journal = {{JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL}}, keywords = {{ALGEBRAIC APPROACH,QUANTUMWELLS,ENERGYGAP,POINT,LAYER,TIME,positiondependent effective mass and angular frequency,confined,harmonic oscillator,associated Legendre and Gegenbauer polynomials,Schrö,dinger equation,BenDaniel–,Duke kinetic energy,operator,quantized confinement parameter}}, language = {{eng}}, number = {{48}}, pages = {{14}}, title = {{Exact solution of the positiondependent effective mass and angular frequency Schrodinger equation : harmonic oscillator model with quantized confinement parameter}}, url = {{http://dx.doi.org/10.1088/17518121/abbd1a}}, volume = {{53}}, year = {{2020}}, }
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