A large class of bound-state solutions of the Schrödinger equation via Laplace transform of the confluent hypergeometric equation. (English) Zbl 1345.81034
Summary: It is shown that analytically soluble bound states of the Schrödinger equation for a large class of systems relevant to atomic and molecular physics can be obtained by means of the Laplace transform of the confluent hypergeometric equation. It is also shown that all closed-form eigenfunctions are expressed in terms of generalized Laguerre polynomials. The generalized Morse potential is used as an illustration.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
References:
| [1] | Morse, PM, No article title, Phys. Rev., 34, 57 (1929) · JFM 55.0539.02 · doi:10.1103/PhysRev.34.57 |
| [2] | E.U. Condon, P.M. Morse, Quantum Mechanics (McGraw-Hill, New York, 1929) · JFM 55.1172.09 |
| [3] | L.D. Landau, E.M. Lifshitz, Quantum Mechanics—Non-relativistic Theory (Pergamon, New York, 1958) · Zbl 0178.57901 |
| [4] | I.I. Gol’dman, V.D. Krivchenkov, Problems in Quantum Mechanics (Pergamon, London, 1961) · Zbl 0094.23407 |
| [5] | D. ter Haar, Problems in Quantum Mechanics (Pion, London, 1975) |
| [6] | Wallace, R., No article title, Chem. Phys. Lett., 37, 115 (1976) · doi:10.1016/0009-2614(76)80173-X |
| [7] | Nieto, MM; Simmons, LM, No article title, Phys. Rev. A, 19, 438 (1979) · doi:10.1103/PhysRevA.19.438 |
| [8] | Duru, IH, No article title, Phys. Rev. D, 28, 2689 (1983) · doi:10.1103/PhysRevD.28.2689 |
| [9] | Gerry, CC, No article title, Phys. Rev. A, 33, 2207 (1986) · doi:10.1103/PhysRevA.33.2207 |
| [10] | Kais, S.; Levine, RD, No article title, Phys. Rev. A, 41, 2301 (1990) · doi:10.1103/PhysRevA.41.2301 |
| [11] | Cooper, IL, No article title, J. Phys. A, 25, 1671 (1992) · doi:10.1088/0305-4470/25/6/022 |
| [12] | S. Flügge, Practical Quantum Mechanics (Springer, Berlin, 1994) |
| [13] | Drigo Filho, E.; Ricota, RM, No article title, Phys. Lett. A, 269, 269 (2000) · Zbl 1115.81348 · doi:10.1016/S0375-9601(00)00267-X |
| [14] | Frank, A.; Rivera, AL; Wolf, KB, No article title, Phys. Rev. A, 61, 054102 (2000) · doi:10.1103/PhysRevA.61.054102 |
| [15] | Dong, S-H; Lemus, R.; Frank, A., No article title, Int. J. Quantum Chem., 86, 433 (2002) · doi:10.1002/qua.10038 |
| [16] | Dong, S-H; Sun, G-H, No article title, Phys. Lett. A, 314, 261 (2003) · Zbl 1073.81557 · doi:10.1016/S0375-9601(03)00895-8 |
| [17] | Chen, G., No article title, Phys. Lett. A, 326, 55 (2004) · Zbl 1161.34361 · doi:10.1016/j.physleta.2004.04.029 |
| [18] | Lemus, R.; Arias, JM; Gómez-Camacho, J., No article title, J. Phys. A, 37, 1805 (2004) · Zbl 1063.81101 · doi:10.1088/0305-4470/37/5/023 |
| [19] | S.-H. Dong, Factorization Method in Quantum Mechanics (Springer, Dordrect, 2007) · Zbl 1130.81001 |
| [20] | Weissman, Y.; Jortner, J., No article title, Phys. Lett. A, 81, 202 (1981) · doi:10.1016/0375-9601(81)90240-1 |
| [21] | Sage, M., No article title, Chem. Phys., 87, 431 (1984) · doi:10.1016/0301-0104(84)85123-X |
| [22] | Sage, M.; Goodisman, J., No article title, Am. J. Phys., 53, 350 (1985) · doi:10.1119/1.14408 |
| [23] | Popov, D., No article title, J. Phys. A, 34, 5283 (2001) |
| [24] | Wang, LY; Gu, X-Y; Ma, Z-Q; Dong, S-H, No article title, Found. Phys. Lett., 15, 569 (2002) · doi:10.1023/A:1023937704158 |
| [25] | B. Thaller, Advanced Visual Quantum Mechanics (Springer, New York, 2005) · Zbl 1059.81002 |
| [26] | Sever, R.; Tezcan, C.; Aktaş, M.; Yeşiltaş, O., No article title, J. Math. Chem., 43, 845 (2007) · Zbl 1151.81333 · doi:10.1007/s10910-007-9233-y |
| [27] | Ikhdair, SM; Sever, R., No article title, J. Mol. Struct., 806, 155 (2007) · doi:10.1016/j.theochem.2006.11.019 |
| [28] | Oyewumi, KJ; Akinpelu, FO; Agboola, AD, No article title, Int. J. Theor. Phys., 47, 1039 (2008) · Zbl 1140.81365 · doi:10.1007/s10773-007-9532-x |
| [29] | Ikhdair, SM; Sever, R., No article title, Cent. Eur. J. Phys., 6, 685 (2008) |
| [30] | Ikhdair, SM; Sever, R., No article title, Cent. Eur. J. Phys., 6, 697 (2008) |
| [31] | Tezcan, C.; Sever, R., No article title, Int. J. Theor. Phys., 48, 337 (2009) · Zbl 1162.81369 · doi:10.1007/s10773-008-9806-y |
| [32] | Kratzer, A., No article title, Z. Phys., 3, 289 (1920) · doi:10.1007/BF01327754 |
| [33] | Fues, E., No article title, Ann. Phys., 80, 281 (1926) · JFM 52.0965.02 · doi:10.1002/andp.19263861905 |
| [34] | Molski, M., No article title, Phys. Rev. A, 76, 022107 (2007) · doi:10.1103/PhysRevA.76.022107 |
| [35] | Barut, AO; Inomata, A.; Wilson, R., No article title, J. Math. Phys., 28, 605 (1987) · Zbl 0621.70017 · doi:10.1063/1.527646 |
| [36] | V.G. Bagrov, D.M. Gitman, Exact Solutions of Relativistic Wave Equations (Kluer, Dordrecht, 1990) · Zbl 0723.35066 · doi:10.1007/978-94-009-1854-2 |
| [37] | Chetouani, L.; Guechi, L.; Hammann, TF, No article title, Helv. Phys. Acta, 65, 1069 (1992) |
| [38] | Chetouani, L.; Guechi, L.; Hammann, TF, No article title, Czechoslov. J. Phys., 43, 13 (1993) · doi:10.1007/BF01589580 |
| [39] | Arda, A.; Sever, R., No article title, Commun. Theor. Phys., 58, 27 (2012) · Zbl 1263.81145 · doi:10.1088/0253-6102/58/1/05 |
| [40] | F. Constantinescu, E. Magyari, Problems in Quantum Mechanics (Pergamon, Oxford, 1971) |
| [41] | Ballhausen, CJ, No article title, Chem. Phys. Lett., 146, 449 (1988) · doi:10.1016/0009-2614(88)87476-1 |
| [42] | Ballhausen, CJ, No article title, Chem. Phys. Lett., 151, 428 (1988) · doi:10.1016/0009-2614(88)85162-5 |
| [43] | Palm, G.; Raff, U., No article title, Am. J. Phys., 71, 247 (2003) · doi:10.1119/1.1526131 |
| [44] | Palm, G.; Raff, U., No article title, Am. J. Phys., 71, 956 (2003) · doi:10.1119/1.1592514 |
| [45] | Dong, S-H; Louzada-Cassou, M., No article title, Int. J. Mod. Phys. B, 19, 4219 (2005) · Zbl 1079.81070 · doi:10.1142/S0217979205032735 |
| [46] | Dong, S-H; Louzada-Cassou, M.; Yu, J.; Jiménez-Ángeles, F.; Rivera, AL, No article title, Int. J. Quantum Chem., 107, 366 (2006) · doi:10.1002/qua.21103 |
| [47] | Singh, CA; Devi, OB, No article title, Int. J. Quantum Chem., 106, 415 (2006) · doi:10.1002/qua.20775 |
| [48] | Dong, S-H; Morales, D.; García-Ravelo, J., No article title, Int. J. Mod. Phys. E, 16, 189 (2007) · doi:10.1142/S0218301307005661 |
| [49] | Patil, SH; Sen, KD, No article title, Phys. Lett. A, 362, 109 (2007) · doi:10.1016/j.physleta.2006.11.045 |
| [50] | Ikhdair, SM; Sever, R., No article title, Int. J. Mod. Phys. C, 20, 361 (2009) · Zbl 1169.81012 · doi:10.1142/S0129183109013674 |
| [51] | Arda, A.; Sever, R., No article title, J. Math. Chem., 50, 971 (2012) · Zbl 1251.81035 · doi:10.1007/s10910-011-9944-y |
| [52] | Pimentel, DRM; Castro, AS, No article title, Rev. Bras. Ens. Fis., 35, 3303 (2013) · doi:10.1590/S1806-11172013000300003 |
| [53] | Hall, RL; Saad, N., No article title, J. Chem. Phys., 109, 2983 (1998) · doi:10.1063/1.476889 |
| [54] | Oyewumi, KJ, No article title, Found. Phys. Lett., 18, 75 (2005) · doi:10.1007/s10702-005-2481-9 |
| [55] | Ikhdair, SM, No article title, Chin. J. Phys., 46, 291 (2008) |
| [56] | Agboola, D., No article title, Acta Phys. Pol., 120, 371 (2011) · doi:10.12693/APhysPolA.120.371 |
| [57] | Pimentel, DRM; Castro, AS, No article title, Rev. Bras. Ens. Fis., 36, 1307 (2014) · doi:10.1590/S1806-11172014000100007 |
| [58] | Schrödinger, E., No article title, Ann. Phys., 384, 361 (1926) · JFM 52.0965.08 · doi:10.1002/andp.19263840404 |
| [59] | Englefield, MJ, No article title, J. Aust. Math. Soc., 8, 557 (1968) · doi:10.1017/S1446788700006212 |
| [60] | Swainson, RA; Drake, GWF, No article title, J. Phys. A, 24, 79 (1991) · Zbl 0747.35035 · doi:10.1088/0305-4470/24/1/019 |
| [61] | Ran, Y.; Xue, L.; Hu, S.; Su, R-K, No article title, J. Phys. A, 33, 9265 (2000) · Zbl 1119.81323 · doi:10.1088/0305-4470/33/50/310 |
| [62] | Arda, A., No article title, Commun. Theor. Phys., 58, 27 (2012) · Zbl 1263.81145 · doi:10.1088/0253-6102/58/1/05 |
| [63] | Castro, AS, No article title, Rev. Bras. Ens. Fis., 34, 4301 (2012) |
| [64] | Pimentel, DRM; Castro, AS, No article title, Eur. J. Phys., 34, 199 (2013) · Zbl 1266.81075 · doi:10.1088/0143-0807/34/1/199 |
| [65] | Arda, A.; Sever, R., No article title, J. Math. Chem., 50, 1484 (2012) · Zbl 1381.35024 · doi:10.1007/s10910-012-9984-y |
| [66] | Chen, G., No article title, Chin. Phys., 14, 1075 (2005) · doi:10.1088/1009-1963/14/6/003 |
| [67] | G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation (Springer, New York, 1974) · Zbl 0278.44001 · doi:10.1007/978-3-642-65690-3 |
| [68] | N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, 1965) · Zbl 0131.07002 |
| [69] | I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 7th edn. (Academic, New York, 2007) · Zbl 1208.65001 |
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