2021   03   en   p.20-27 E.I. Jafarov, A.M. Jafarova, S.M. Nagiyev, S.K. Novruzova,
The confined harmonic oscillator as an explicit solution of the position-dependent effective mass Schrodinger equation with Morrow-Brownstein Hamiltonian		  pdf 

ABSTRACT

Exactly-solvable confined model of the non-relativistic quantum harmonic oscillator with Morrow-Brownstein kinetic energy operator
H0=Mα(x)p̂x Mβ (x)p̂xMα(x)/2 (with 2α+β=-1) is proposed. Corresponding position-dependent effective mass Schrödinger equation in the canonical approach is solved in position representation. Explicit expressions of both wavefunctions of the stationary states and discrete e nergy spectrum have been found. It is shown that the energy spectrum has non-equidistant form and depends on both confinement parameter a and Morrow-Brownstein parameter α. Wavefunctions of the stationary states in position representation are expressed in terms of the Gegenbauer polynomials. At limit α→∞, both energy spectrum and wavefunctions recover well-known equidistant energy spectrum and wavefunctions of the stationary non-relativistic harmonic oscillator expressed by Hermite polynomials. Position dependence of the effective mass also disappears under the same limit.

Keywords: Morrow-Brownstein kinetic energy operator, confined harmonic oscillator model, exact solution, Gegenbauer polynomials, non-equidistant energy spectrum, position-dependent effective mass
PACS: 03.65.-w; 02.30.Hq; 03.65.Ge

DOI:-

Received: 14.07.2021

AUTHORS & AFFILIATIONS

Institute of Physics, Azerbaijan National Academy of Sciences Javid ave. 131, AZ1143, Baku, Azerbaijan
E-mail: 		REFERENCIES

[1]   S.C. Bloch. Introduction to Classical and Quantum Harmonic Oscillators. Wiley, New-York, 384 p.,1997.
[2]   S. Flügge. Practical Quantum Mechanics: Vol I Springer, Berlin, 620 p., 1971.
[3]   Y. Ohnuki and S. Kamefuchi. Quantum Field Theory and Parastatistics. Springer Verslag, New-York, 490 p., 1982).
[4]   N.M. Atakishiev, R.M. Mir-Kasimov and Sh.M. Nagiev. Theor. Math. Phys.44, 592,1980.
[5]   M. Moshinsky and S. Szczepaniak. J. Phys. A: Math. Gen.22, L817,1989.
[6]   S. Bruce and M. Minning. Nuovu Cimento A 106,711,1993.
[7]   N.M. Atakishiev and S.K. Suslov. Theor. Math. Phys.85, 1055,1990.
[8]   N.M. Atakishiyev, E.I. Jafarov, S.M. Nagiyev and K.B.Wolf. Rev. Mex. Fis.44, 235,1998.
[9]   E.I. Jafarov, N.I. Stoilova and J. Van der Jeugt. J. Phys. A: Math. Theor.44, 265203, 2011.
[10]  E.I. Jafarov, N.I. Stoilova and J. Van der Jeugt. J. Phys. A: Math. Theor. 44, 355205, 2011.
[11]  E.I. Jafarov and J.Van der Jeugt. J. Phys. A: Math. Theor. 45, 275301, 2012.
[12]  E.I. Jafarov, A.M. Jafarova and J.Van der Jeugt. J. Phys. Conf. Ser. 597, 012047, 2015.
[13]  R. Oste and J. Van der Jeugt. J. Phys. A: Math. Theor.49, 175204, 2016.
[14]  R. Oste and J. Van der Jeugt. Phys. Atom. Nucl.80, 786, 2017.
[15]  E.I. Jafarov and J. Van der Jeugt. J. Phys. A: Math. Theor.45, 485201, 2012.
[16]  F.C. Auluck. Proc. Indian Nat. Sci. Acad.7, 133,1941.
[17]  S. Chandrasekhar. Astrophys. J 97, 263,1943.
[18]  F.C. Auluck and D.S. Kothari. Math. Proc. Camb. Phil. Soc.41, 175,1945.
[19]  A. Consortini and B.R. Frieden. Nuovo Cim. B 35, 153,1976.
[20]  F.C. Rotbart. J. Phys. A: Math. Gen. 11, 2363,1978.
[21]  A.H. MacDonald and P. Středa. Phys. Rev. B 29, 1616, 1984.
[221  ]M. Grinberg, W. Jaskólski, Cz. Coepke, J.Planelles and M.Janowicz. Phys. Rev. B 50, 6504, 1994.
[23]  Lj. Stevanović and K.D. Sen. J. Phys. B: At. Mol. Opt. Phys. 41, 225002, 2008.
[24]  W.A. Harrison. Phys. Rev. 123, 85, 1961.
[25]  I. Giaever. Phys. Rev. Lett.5, 147, 1960.
[26]  I. Giaever. Phys. Rev. Lett.5, 464,1960.
[27]  D. J. BenDaniel and C.B.Duke. Phys. Rev.152, 683,1966.
[28]  Q-G. Zhu and H. Kroemer. Phys. Rev. B27 3519, 1983.
[29]  R.A. Morrow and K.R. Brownstein. Phys. Rev. B 30, 678, 1984.
[30]  R.A. Morrow. Phys. Rev. B 35, 8074,1987.
[31]  G.X. Ju, Ch.Y. Cai, Y. Xiang and Zh.Zh. Ren. Commun. Theor. Phys. 47, 1001, 2007.
[32]  G.X. Ju, Ch.Y. Cai and Zh.Zh. Ren. Commun. Theor. Phys. 51, 797, 2009.
[33]  N. Amir and Sh. Iqbal. Commun. Theor. Phys. 62, 790, 2014.
[34]  M.K. Bahar and S. Yasuk. Canadian J. Phys. 91, 191, 2013.
[35]  M. K. Bahar and S.Yasuk. Canadian J. Phys. 92, 1565, 2014.
[36]  R. Koekoek, P.A. Lesky and R.F. Swarttouw. Hypergeometric orthogonal polynomials and their q-analogues. Springer Verslag, Berlin, 578p., 2010.
[37]  F. Nikiforov and V.B. Uvarov. Special Functions of Mathematical Physics: A Unified Introduction with Applications. Birkhauser, Basel, 427p., 1988.