Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions

Christiane Quesne

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schrödinger equations.

Détail

Keywords: Schrödinger equation; position-dependent mass; quadratic algebra.

Bibliographie

[1] Bastard G., Wave mechanics applied to semiconductor heterostructures, Editions de Physique, Les Ulis, 1988

[2] Serra L., Lipparini E., “Spin response of unpolarized quantum dots”, Europhys. Lett., 40 (1997), 667–672 | DOI

[3] Ring P., Schuck P., The nuclear many body problem, Springer, New York, 1980 | MR

[4] Arias de Saavedra F., Boronat J., Polls A., Fabrocini A., “Effective mass of one ${}^4$He atom in liquid ${}^3$He”, Phys. Rev. B, 50 (1994), 4248–4251 ; cond-mat/9403075 | DOI

[5] Barranco M., Pi M., Gatica S. M., Hernández E. S., Navarro J., “Structure and energetics of mixed ${}^4$He-${}^3$He drops”, Phys. Rev. B, 56 (1997), 8997–9003 | DOI

[6] Puente A., Serra Ll., Casas M., “Dipole excitation of Na clusters with a non-local energy density functional”, Z. Phys. D, 31 (1994), 283–286 | DOI

[7] Quesne C., “First-order intertwining operators and position-dependent mass Schrödinger equations in $d$ dimensions”, Ann. Physics, 321 (2006), 1221–1239 ; quant-ph/0508216 | DOI | MR | Zbl

[8] Bhattacharjie A., Sudarshan E. C. G., “A class of solvable potentials”, Nuovo Cimento, 25 (1962), 864–879 | DOI | MR | Zbl

[9] Natanzon G. A., “General properties of potentials for which the Schrödinger equation can be solved by means of hypergeometric functions”, Theoret. and Math. Phys., 38:2 (1979), 146–153 | DOI | MR | Zbl

[10] Lévai G., “A search for shape-invariant solvable potentials”, J. Phys. A: Math. Gen., 22 (1989), 689–702 | DOI | MR | Zbl

[11] Alhassid Y., Gürsey F., Iachello F., “Group theory approach to scattering. II. The Euclidean connection”, Ann. Physics, 167 (1986), 181–200 | DOI | MR | Zbl

[12] Wu J., Alhassid Y., “The potential group approach and hypergeometric differential equations”, J. Math. Phys., 31 (1990), 557–562 | DOI | MR | Zbl

[13] Englefield M. J., Quesne C., “Dynamical potential algebras for Gendenshtein and Morse potentials”, J. Phys. A: Math. Gen., 24 (1991), 3557–3574 | DOI | MR | Zbl

[14] Lévai G., “Solvable potentials associated with su(1,1) algebras: a systematic study”, J. Phys. A: Math. Gen., 27 (1994), 3809–3828 | DOI | MR | Zbl

[15] Cooper F., Khare A., Sukhatme U., “Supersymmetry and quantum mechanics”, Phys. Rep., 251 (1995), 267–385 ; hep-th/9405029 | DOI | MR

[16] Bagchi B., Supersymmetry in quantum and classical mechanics, Chapman and Hall/CRC, Boca Raton, FL, 2000 | MR

[17] Chen G., Chen Z., “Exact solutions of the position-dependent mass Schrödinger equation in $D$ dimensions”, Phys. Lett. A, 331 (2004), 312–315 | DOI | MR | Zbl

[18] Dong S.-H., Lozada-Cassou M., “Exact solutions of the Schrödinger equation with the position-dependent mass for a hard-core potential”, Phys. Lett. A, 337 (2005), 313–320 | DOI | Zbl

[19] Mustafa O., Mazharimousavi S. H., “$d$-dimensional generalization of the point canonical transformation for a quantum particle with position-dependent mass”, J. Phys. A: Math. Gen., 39 (2006), 10537–10547 ; math-ph/0602044 | DOI | MR | Zbl

[20] Mustafa O., Mazharimousavi S. H., “Quantum particles trapped in a position-dependent mass barrier; a $d$-dimensional recipe”, Phys. Lett. A, 358 (2006), 259–261 ; quant-ph/0603134 | DOI | Zbl

[21] Ju G.-X., Xiang Y., Ren Z.-Z., The localization of $s$-wave and quantum effective potential of a quasi-free particle with position-dependent mass, quant-ph/0601005

[22] Gönül B., Koçak M., “Explicit solutions for $N$-dimensional Schrödinger equations with position-dependent mass”, J. Math. Phys., 47 (2006), 102101, 6 pp., ages ; quant-ph/0512035 | DOI | MR | Zbl

[23] Olendski O., Mikhailovska L., “Bound-state evolution in curved waveguides and quantum wires”, Phys. Rev. B, 66 (2002), 035331, 8 pp., ages | DOI

[24] Gudmundsson V., Tang C.-S., Manolescu A., “Bound state with negative binding energy induced by coherent transport in a two-dimensional quantum wire”, Phys. Rev. B, 72 (2005), 153306, 4 pp., ages ; cond-mat/0506009 | DOI

[25] Goldstein H., Classical mechanics, Addison-Wesley, Reading, MA, 1980 | MR | Zbl

[26] Dirac P. A. M., The principles of quantum mechanics, Oxford University Press, Oxford, 1981 | MR

[27] Friš I., Mandrosov V., Smorodinsky Ya. A., Uhlir M., Winternitz P., “On higher symmetries in quantum mechanics”, Phys. Lett., 16 (1965), 354–356 | DOI | MR

[28] Winternitz P., Smorodinsky Ya. A., Uhlir M., Friš I., “Symmetry groups in classical and quantum mechanics”, Sov. J. Nucl. Phys., 4 (1967), 444–450 | MR

[29] Makharov A. A., Smorodinsky Ya. A., Valiev Kh., Winternitz P., “A systematic search for nonrelativistic systems with dynamical symmetries. Part I: the integrals of motion”, Nuovo Cimento A, 52 (1967), 1061–1084 | DOI

[30] Hietarinta J., “Direct methods for the search of the second invariant”, Phys. Rep., 147 (1987), 87–154 | DOI | MR

[31] Granovskii Ya. I., Lutzenko I. M., Zhedanov A. S., “Mutual integrability, quadratic algebras, and dynamical symmetry”, Ann. Physics, 217 (1992), 1–20 | DOI | MR

[32] Zhedanov A. S., ““Hidden symmetry” of Askey–Wilson polynomials”, Theoret. and Math. Phys., 89:2 (1991), 1146–1157 | DOI | MR | Zbl

[33] Granovskii Ya. I., Zhedanov A. S., Lutsenko I. M., “Quadratic algebras and dynamics in curved spaces. I. Oscillator”, Theoret. and Math. Phys., 91:2 (1992), 474–480 | DOI | MR

[34] Granovskii Ya. I., Zhedanov A. S., Lutsenko I. M., “Quadratic algebras and dynamics in curved spaces. II. The Kepler problem”, Theoret. and Math. Phys., 91:3 (1992), 604–612 | DOI | MR

[35] Bonatsos D., Daskaloyannis C., Kokkotas K., “Deformed oscillator algebras for two-dimensional quantum superintegrable systems”, Phys. Rev. A, 50 (1994), 3700–3709 ; hep-th/9309088 | DOI | MR

[36] Daskaloyannis C., “Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems”, J. Math. Phys., 42 (2001), 1100–1119 ; math-ph/0003017 | DOI | MR | Zbl

[37] Daskaloyannis C., Ypsilantis K., “Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold”, J. Math. Phys., 47 (2006), 042904, 38 pp., ages ; math-ph/0412055 | DOI | MR | Zbl

[38] Daskaloyannis C., Tanoudes Y., Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds, math-ph/0607058

[39] Létourneau P., Vinet L., “Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonians”, Ann. Physics, 243 (1995), 144–168 | DOI | MR | Zbl

[40] Rañada M. F., “Superintegrable $n=2$ systems, quadratic constants of motion, and potentials of Drach”, J. Math. Phys., 38 (1997), 4165–4178 | DOI | MR | Zbl

[41] Rañada M. F., Santander M., “Superintegrable systems on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$”, J. Math. Phys., 40 (1999), 5026–5057 | DOI | MR | Zbl

[42] Tempesta P., Turbiner A. V., Winternitz P., “Exact solvability of superintegrable systems”, J. Math. Phys., 42 (2001), 4248–4257 ; hep-th/0011209 | DOI | MR | Zbl

[43] Kalnins E. G., Miller W. Jr., Pogosyan G. S., “Superintegrability on the two-dimensional hyperboloid”, J. Math. Phys., 38 (1997), 5416–5433 | DOI | MR | Zbl

[44] Kalnins E. G., Miller W. Jr., Hakobyan Y. M., Pogosyan G. S., “Superintegrability on the two-dimensional hyperboloid. II”, J. Math. Phys., 40 (1999), 2291–2306 ; quant-ph/9907037 | DOI | MR | Zbl

[45] Kalnins E. G., Kress J. M., Miller W. Jr., “Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 053509, 28 pp., ages | DOI | MR | Zbl

[46] Kalnins E. G., Kress J. M., Miller W. Jr., “Second-order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform”, J. Math. Phys., 46 (2005), 053510, 15 pp., ages | DOI | MR | Zbl

[47] Kalnins E. G., Kress J. M., Miller W. Jr., “Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems”, J. Math. Phys., 47 (2006), 093501, 25 pp., ages | DOI | MR | Zbl

[48] Kalnins E. G., Kress J. M., Miller W. Jr., “Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties”, J. Phys. A: Math. Theor., 40 (2007), 3399–3411 | DOI | MR | Zbl

[49] Quesne C., “Generalized deformed parafermions, nonlinear deformations of so(3) and exactly solvable potentials”, Phys. Lett. A, 193 (1994), 245–250 | DOI | MR | Zbl

[50] Daskaloyannis C., “Generalized deformed oscillator and nonlinear algebras”, J. Phys. A: Math. Gen., 24 (1991), L789–L794 | DOI | MR | Zbl

[51] Roy B., Roy P., “Effective mass Schrödinger equation and nonlinear algebras”, Phys. Lett. A, 340 (2005), 70–73 | DOI | Zbl

[52] von Roos O., “Position-dependent effective masses in semiconductor theory”, Phys. Rev. B, 27 (1983), 7547–7552 | DOI