2023   01   en   p.56-64 Shakir M. Nagiyev, Shovqiyya A. Amirova,
The Wigner distribution function of a semiconfined harmonic oscillator model with a position-dependent mass and frequency in an external homogeneous field. The case of parabolic well


The phase space representation for a semiconfined harmonic oscillator model with the position-dependent mass and frequency in an external homogeneous field is constructed in terms of the Wigner distribution function. It is expressed through the Bessel function and Laguerre polynomials. Some of the special cases and limits are also discussed.

Keywords: Harmonic oscillator; external homogeneous field; position-dependent mass; Wigner function; limit relations.
PACS: 03.65.-w; 02.30. Hq; 03.65.Ge


Received: 27.02.2023


Institute of Physics Ministry of Science and Education of Azerbaijan, AZ 1143, Baku, H. Javid Ave., 131

[1]   E.P. Wigner. On the quantum correction for thermodynamic equilibrium, Phys.Rev., 40 749 1932.
[2]   M. Hillery, R.F. O’Connell, M.O. Scully and E.P. Wigner. Distribution functions in physics: Fundamentals, Phys. Rep., 106 121, 1984.
[3]   H.-W. Lee. Theory and application of the quantum phase-space distribution functions, Phys. Rep. 259, 1995, 147-211.
[4]   V.I. Tatarski˘ı. The Wigner representation of quantum mechanics, Sov. Phys. Usp., 26 311, 1983.
[5]   K. Husimi, Some formal properties of the density matrix, Proc. Phys.-Math. Soc. Japan, 22 264, 1940.
[6]   H. Weyl. “Quantenmechanik und Gruppentheorie” Z. Phys 46 (1927) 1-33.
[7]   R.W. Davies and K.T.R. Davies. On the Wigner distribution function for an oscillator, Ann.Phys. (N.Y.), 89 261-273, 1975.
[8]   N.M. Atakishiyev, Sh.M. Nagiyev and K.B. Wolf. Wigner distribution functions for a relativistic linear oscillator, Theor.Math,Phys., 114 322 (1998).
[9]   E.I. Jafarov, S. Lievens, S.M. Nagiyev, J. Van der Jeught. The Wigner functions of q-deformed harmonic oscillator model, J.Phys. A: Math. Theor., 40 5427-5441, 2007.
[10]  E. I. Jafarov, S. Lievens and J. Van der Jeught. The Wigner distribution functions for the one-dimensional parabose oscillator, J.Phys. Theor., 41 235301, 2008.
[11]  S.M. Nagiyev, G.H. Guliyeva and E.I. Jafarov. The Wigner functions of the relativistic finite-difference oscillator in an external field, J.Phys. A: Math. Theor., 42 454015, 2009.
[12]  K. Li, J. Wang, S. Dulat and K. Ma. Wigner functions for Klein-Gordon oscillators in non-commutative space, Int.J.Theor.,Phys., 49 134-143, 2010.
[13]  M. Kai, W. Jian-Hua and Y. Yi. Wigner function for the Dirac oscillator in spinor space, Chinese Phys.C, 35 11-15, 2011.
[14]  J. Van der Jeugt. A Wigner distribution function for finite oscillator systems, J.Phys. A: Math. Theor., 46 475302, 2013.
[15]  S. Hassanabadi and M. Ghominejad. Wigner function for Klein-Gordon oscillator in commutative and non-commutative space, Eur. Phys. J. Plus, 131 212, 2016.
[16]  Z.-d. Chen and G. Chen. Wigner function of the position-dependent effective Schrodinger equation, Phys. Scr., 73 354-358, 2006.
[17]  A. de Souza Dutra and J.A. de Oliveira. Wigner distribution for a class of isospectral position-dependent mass systems, Phys. Scr., 78 035009, 2008.
[18]  E.I. Jafarov, S.M. Nagiyev, R. Oste, J. Van der Jeugt. “Exact solution of the position-dependent effective mass and angular frequency Schr¨odinger equation: harmonic oscillator model with quantized confinement parameter”, J. Phys. A: Math. Theor., 53:48, 2020, 485301, 14 pp.
[19]  D.J. BenDaniel, C.B. Duke. “Space-charge effects on electron tunneling”, Phys. Rev., 152:2, 1966, 683–692.
[20]  O. Von Roos. “Position-dependent effective masses in semiconductor theory”, Phys. Rev. B, 27:12, 1983, 7547–7552.
[21]  J.-M. L´evy-Leblond. “Position-dependent effective mass and Galilean invariance”, Phys. Rev. A, 52:3, 1995, 1845–1849.
[22]  G. Bastard. Wave Mechanics Applied to Semiconductor Heterostructure, Les Edition de Physique, Paris, 1988.
[23]  P. Harrison. Quantum Wells, Wires and Dots: Theoretical and Computational Physics, John Wiley and Sons, New York, 2000.
[24]  M. Barranco, M. Pi, S.M. Gatica, E.S. Hern´andez, J. Navarro. “Structure and energetics of mixed 4He–3He drops”, Phys. Rev. B, 56:14, 1997, 8997–9003.
[25]  F. Arias de Saavedra, J. Boronat, A. Polls, A. Fabrocini. “Effective mass of one 4He atom in liquid 3He”, Phys. Rev. B, 50:6, 1994, 4248–4251.
[26]  T. Gora, F. Williams. “Theory of electronic states and transport in graded mixed semiconductors”, Phys. Rev., 177:3, 1969, 1179–1182.
[27]  Q.-G. Zhu, H. Kroemer. “Interface connection rules for effective-mass wave functions at an abrupt heterojunction between two different semiconductors”, Phys. Rev. B, 27:6, 1983, 3519–3527.
[28]  H. Raibongshi, N.N. Singh. “Construction of exactly solvable potentials in the D-dimensional Schrödinger equation with coordinate-dependent mass using the transformation method”, TMF, 183:2, 2015, 312–328.
[29]  N. Amir, S. Iqbal. “Algebraic solutions of shape-invariant position-dependent effective mass systems”, J. Math. Phys., 57:6, 2016, 062105.
[30]  B. Roy. “Lie algebraic approach to singular oscillator with a position-dependent mass”, Europhys. Lett., 72:1, 2005, 1–6.
[31]  J. Yu, S.-H. Dong. “Exactly solvable potentials for the Schr¨odinger equation with spatially dependent mass”, Phys. Lett. A, 325, 2004, 194–198.
[32]  J.R.F. Lima, M.Vieira, C. Furtado, F. Moraes, C. Filgueiras. “Yet another position-dependent mass quantum model”, J. Math. Phys., 53:7, 2012, 072101.
[33]  C. Quesne, V.M. Tkachuk. “Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem”, J. Phys. A: Math. Gen., 37:14, 2004, 4267–4281, arXiv: math-ph/0403047.
[34]  J.F. Cari˜nena, M.F. Ra˜nada, M. Santander. “Quantization of Hamiltonian systems with a position dependent mass: Killing vector fields and Noether momenta approach”, J. Phys. A: Math. Theor., 50:46, 2017, 465202, 20 pp.
[35]  E.I. Jafarov, S.M. Nagiyev, A.M. Jafarova. “Quantum-mechanical explicit solution for the confined harmonic oscillator model with the von Roos kinetic energy operator”, Rep. Math. Phys., 86:1, 2020, 25–37.
[36]  E.I. Jafarov, Sh.M. Nagiyev. “Angular part of the Schrödinger equation for the potential Oto as a harmonic oscillator with coordinate-dependent mass in a homogeneous gravitational field”, TMF, 207:1, 2021, 58–71.
[37]  A. de Souza Dutra, A. de Oliveira. “Two-dimensional position-dependent massive particles in the presence of magnetic fields”, J. Phys. A: Math. Theor., 42:2, 2009, 025304, 13 pp.
[38]  Shakir M. Nagiyev, Shovqiyya A. Amirova, Model of a linear harmonic oscillator with a position-dependent mass in the external homogeneous field. The case of a parabolic well, AJP Physics 28:4, 2022 section En, 36-41.
[39]  S.M. Nagiyev. On two direct limits relating pseudo-Jacobi polynomials to Hermite polynomials and the pseudo-Jacobi oscillator in a homogeneous gravitational field. Theor. Math. Phys. 210, 2022, 121.
[40]  S.M. Nagiyev, C. Aydin, A.I. Ahmadov, S.A. Amirova. “Exactly solvable model of the linear harmonic oscillator with a positiondependent mass under external homogeneous gravitational field”, Eur. Phys. J. Plus., 540: 137, 2022,13.
[41]  A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, “Integrals and series”, vol. 2: Special functions, Nauka, M., 1983.
[42]  H. Bateman and A. Erd´elyi. “Higher Transcendental Functions”: 2 (McGraw Hill Publications, New York, 1953).
[43]  H. Bateman and A. Erd´elyi. “Higher Transcendental Functions”: 1 (McGraw Hill Publications, New York, 1953).
[44]  R. Koekoek, P.A. Lesky and R.F. Swarttouw. “Hypergeometric Orthogonal Polynomials and their q-Analogues”, (Springer, Berlin 2010).