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Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions

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Abstract

We present analytically the exact solutions of the Schrödinger equation in the N-dimensional spaces for the pseudoharmonic oscillator potential by means of the ansatz method. The energy eigenvalues of the bound states are easily calculated from this eigenfunction ansatz. The normalized wavefunctions are also obtained. A realization of the ladder operators for the wavefunctions is studied and we deduced that these operators satisfy the commutation relations of the generators of the dynamical group SU(1,1). Some expectation values for 〈r −2〉, 〈r 2〉, 〈T〉, 〈V〉, 〈H〉, 〈p 2〉 and the virial theorem for the pseudoharmonic oscillator potential in an arbitrary number of dimensions are obtained by means of the Hellmann–Feynman theorems. Each solution obtained is dimensions and parameters dependent.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1970)

    Google Scholar 

  2. Al-Jaber, S.: Nuovo Cimento B 110, 993 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  3. Al-Jaber, S.: Nuovo Cimento B 112, 761 (1997)

    Google Scholar 

  4. Al-Jaber, S.: Int. J. Theor. Phys. 37(4), 1289 (1998)

    MATH  Google Scholar 

  5. Al-Jaber, S.: Int. J. Theor. Phys. 38(3), 919 (1999)

    MATH  Google Scholar 

  6. Al-Jaber, S.: Int. J. Theor. Phys. 42(1), 111 (2003)

    MATH  MathSciNet  Google Scholar 

  7. Al-Jaber, S., Lombard, R.J.: J. Phys. A: Math. Gen. 38, 4637 (2005)

    MATH  ADS  MathSciNet  Google Scholar 

  8. Arkani-Hammed, N.: Phys. Rev. D 65, 6003 (2002)

    Google Scholar 

  9. Avery, J.: Hyperspherical Harmonics: Applications in Quantum Theory. Kluwer Academic, Dordrecht (1989)

    MATH  Google Scholar 

  10. Avery, J.: J. Math. Chem. 24, 169 (1998)

    MATH  MathSciNet  Google Scholar 

  11. Balasubramanian, S.: Am. J. Phys. 62(12), 1116 (1994)

    ADS  MathSciNet  Google Scholar 

  12. Balasubramanian, S.: Am. J. Phys. 58(12), 1204 (1990)

    ADS  MathSciNet  Google Scholar 

  13. Brajamani Singh, S., Singh, C.A.: Am. J. Phys. 57(10), 894 (1989)

    ADS  Google Scholar 

  14. Barnea, N.: Phys. Rev. A 59(2), 1135 (1999)

    ADS  MathSciNet  Google Scholar 

  15. Barut, A.O., Ra̧czka, R.: Theory of Group Representations and Applications, 2nd edn. World Scientific, Singapore (2000)

    Google Scholar 

  16. Bateman, D.S., Boyd, C., Dutta-Roy, B.: Am. J. Phys. 60(9), 833 (1992)

    ADS  Google Scholar 

  17. Biesiada, M., Malec, B.: Phys. Rev. D 65, 3008 (2002)

    ADS  Google Scholar 

  18. Camblong, H.E., Epele, L.N., Fauchiotti, H., Canal, C.A.: Ann. Phys. 287, 57 (2001)

    MATH  ADS  Google Scholar 

  19. Cardoso, J.L., Álvarez-Nodarse, R.: J. Phys. A: Math. Gen. 36, 2055 (2003)

    MATH  ADS  Google Scholar 

  20. Casimir, H.: Proc. R. Acad. 34, 844 (1931)

    Google Scholar 

  21. Chatterjee, A.: Phys. Rep. 186, 249 (1990)

    ADS  Google Scholar 

  22. Coelho, J.L.A., Amaral, R.L.P.G.: J. Phys. A: Math. Gen. 35, 5255 (2002)

    MATH  ADS  MathSciNet  Google Scholar 

  23. Cooper, L.I.: J. Phys. A: Math. Gen. 25, 1671 (1992)

    ADS  Google Scholar 

  24. Cooper, L.I.: J. Phys. A: Math. Gen. 26, 1601 (1993)

    MATH  ADS  Google Scholar 

  25. Cordani, B.: J. Phys. A: Math. Gen. 22, 2441 (1989)

    MATH  ADS  MathSciNet  Google Scholar 

  26. Chen, G.: Chin. Phys. 14(6), 1075 (2005)

    ADS  Google Scholar 

  27. Crisan, M., Bodea, D., Grosu, I., Tifrea, I.: J. Phys. A: Math. Gen. 35, 239 (2002)

    MATH  ADS  MathSciNet  Google Scholar 

  28. Davtyan, L.S., Mardoyan, L.G., Pogosyan, G.S., Sissakian, A.N., Ter-Antonyan, V.M.: J. Phys. A: Math. Gen. 20, 6121 (1987)

    ADS  MathSciNet  Google Scholar 

  29. Dehesa, J.S., López-Rosa, S., Olmos, B.: J. Math. Phys. 47, 052104 (2006)

    ADS  MathSciNet  Google Scholar 

  30. Dong, S.H.: Found. Phys. Lett. 15(4), 385 (2002)

    MathSciNet  Google Scholar 

  31. Dong, S.H.: J. Phys. A: Math. Gen. 36, 4977 (2003)

    MATH  ADS  Google Scholar 

  32. Dong, S.S., Dong, S.H.: Int. J. Mod. Phys. E 11(4), 265 (2002)

    ADS  Google Scholar 

  33. Dong, S.S., Dong, S.H., Lozada-Cassou, M.: Phys. Scr. 73, 173 (2006)

    MATH  ADS  MathSciNet  Google Scholar 

  34. Dong, S.H., Gu, X.Y., Ma, Z.Q., Yu, J.: Int. J. Mod. Phys. E 12(4), 555 (2003)

    ADS  Google Scholar 

  35. Dong, S.H., Ma, Z.Q.: Phys. Rev. A 65, 042717 (2002)

    ADS  Google Scholar 

  36. Dong, S.H., Ma, Z.Q.: Int. J. Mod. Phys. E 11, 155 (2002)

    ADS  Google Scholar 

  37. Dong, S.H., Morales, D., García-Ravelo, J.: Int. J. Mod. Phys. E 16(1), 189 (2007)

    ADS  Google Scholar 

  38. Dong, S.H., Sun, G.H., Popov, D.: J. Math. Phys. 44(10), 4467 (2003)

    MATH  ADS  MathSciNet  Google Scholar 

  39. Dong, S.H., Lozada-Cassou, M.: Int. J. Mod. Phys. B 19(18), 4219 (2005)

    MATH  ADS  MathSciNet  Google Scholar 

  40. Epstein, J.H., Epstein, S.T.: Am. J. Phys. 30, 266 (1962)

    MATH  ADS  Google Scholar 

  41. Erdelyi, A.: Higher Transcendental Functions, vol. 2, p. 232. McGraw–Hill, New York (1953)

    Google Scholar 

  42. Erkoç, S., Sever, R.: Phys. Rev. D 30, 2117 (1984)

    ADS  MathSciNet  Google Scholar 

  43. Erkoç, S., Sever, R.: Phys. Rev. D 33, 588 (1986)

    ADS  Google Scholar 

  44. Erkoç, S., Sever, R.: Phys. Rev. A 37, 2687 (1988)

    ADS  Google Scholar 

  45. Feynman, R.P.: Phys. Rev. 56, 340 (1939)

    MATH  ADS  Google Scholar 

  46. Fukutaka, H., Kashiwa, T.: Ann. Phys. 176, 301 (1987)

    ADS  MathSciNet  Google Scholar 

  47. Gol’dman, I.I., Krivchenkov, V.D., Kogan, V.I., Galitskii, V.M.: Problems in Quantum Mechanics, p. 308. Academic Press, New York (1960)

    Google Scholar 

  48. Gönül, B., Özer, O., Koçak, M.: Unified treatment of screening Coulomb and anharmonic oscillator potentials in arbitrary dimensions. quant-ph/0106144 (2001)

  49. Gönül, B., Özer, O., Koçak, M., Tutcu, D., Cançelik, Y.: J. Phys. A: Math. Gen. 34, 8271 (2001)

    MATH  ADS  Google Scholar 

  50. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 5th edn. Academic Press, New York (1994)

    Google Scholar 

  51. Griffiths, D.J.: Introduction to Quantum Mechanics. Prentice-Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  52. Grosche, C., Steiner, F.: Z. Phys. C 36, 699 (1987)

    ADS  MathSciNet  Google Scholar 

  53. Grosche, C., Steiner, F.: J. Math. Phys. 36(5), 2354 (1998)

    ADS  MathSciNet  Google Scholar 

  54. Grosche, C., Steiner, F.: Handbook of Feynman Path Integrals, p. 99. Springer, Berlin (1998)

    MATH  Google Scholar 

  55. Gu, X.Y., Duan, B., Ma, Z.Q.: J. Math. Phys. 43(6), 2895 (2002)

    MATH  ADS  MathSciNet  Google Scholar 

  56. Gu, X.Y., Ma, Z.Q., Dong, S.H.: Int. J. Mod. Phys. E 11(4), 335 (2002)

    ADS  Google Scholar 

  57. Hellmann, G.: Einführung in die Quantenchemie. Denticke, Vienna (1937)

    Google Scholar 

  58. Ikhdair, S., Sever, R.: Exact polynomial solutions of the Mie-type potential in the D-dimensional Schrödinger equation. arXiv: quant-ph/0611065 (2006)

  59. Ikhdair, S., Sever, R.: On the solutions of the Schrödinger equation with some molecular potentials: wave function ansatz. arXiv: quant-ph/0702052 (2007)

  60. Ikhdair, S., Sever, R.: Exact solutions of the Schrödinger equation in D-dimensions for the pseudoharmonic potential plus ring-shaped potential. arXiv: quant-ph/0703131 (2007)

  61. Ikhdair, S., Sever, R.: Exact solutions of the pseudo-Coulomb potential plus ring-shaped potential in the D-dimensional Schrödinger equation by the Nikiforov–Uvarov method. arXiv: quant-ph/0703042 (2007)

  62. Ikhdair, S., Sever, R.: Exact solutions of the modified Kratzer potential plus ring-shaped potential in the D-dimensional Schrödinger equation by the Nikiforov–Uvarov method. arXiv: quant-ph/0703008 (2007)

  63. Infeld, L., Hull, T.E.: Rev. Mod. Phys. 23, 21 (1951)

    MATH  ADS  MathSciNet  Google Scholar 

  64. Kalnin, E.G., Miller Jr., W., Pogosyan, G.S.: Phys. At. Nucl. 65(6), 1086 (2002)

    Google Scholar 

  65. Kirchberg, A., Lange, J.D., Pisani, P.A.G., Wipf, A.: arXiv, hep-th/0208228 (2002)

  66. Kirchberg, A., Lange, J.D., Pisani, P.A.G., Wipf, A.: Ann. Phys. 303, 359 (2003)

    MATH  ADS  MathSciNet  Google Scholar 

  67. Lèvai, G., Kónya, B., Papp, Z.: J. Math. Phys. 39(11), 5811 (1998)

    MATH  ADS  MathSciNet  Google Scholar 

  68. Levi, D., Tremblay, S., Winternitz, P.: J. Phys. A: Math. Gen. 34, 9507 (2001)

    MATH  ADS  MathSciNet  Google Scholar 

  69. Lorente, M.: J. Phys. A: Math. Gen. 34, 569 (2001)

    MATH  ADS  MathSciNet  Google Scholar 

  70. Louck, J.D.: J. Mol. Spectrosc. 4, 298 (1960)

    ADS  Google Scholar 

  71. Marc, G., McMillan, W.G.: Adv. Chem. Phys. 58, 205 (1985)

    Google Scholar 

  72. Martínez-Y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: Phys. Lett. A 339, 258 (2005)

    ADS  Google Scholar 

  73. Martínez-Y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: J. Phys. A: Math. Gen. 38, 8579 (2005)

    MATH  ADS  Google Scholar 

  74. Martínez-Y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: Int. J. Quantum Chem. 107, 1608 (2007)

    ADS  Google Scholar 

  75. Martínez-Y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: Am. J. Phys. 75(6), 629 (2007)

    ADS  Google Scholar 

  76. Ma, Z.Q., Dong, S.H.: Int. J. Mod. Phys. E 16(1), 179 (2007)

    ADS  Google Scholar 

  77. Mavromatis, H.A.: Am. J. Phys. 64, 1074 (1996)

    ADS  Google Scholar 

  78. Mavromatis, H.A.: Rep. Math. Phys. 40(1), 17 (1997)

    MATH  MathSciNet  ADS  Google Scholar 

  79. Mavromatis, H.A.: Am. J. Phys. 66(4), 335 (1998)

    ADS  MathSciNet  Google Scholar 

  80. Merzbacher, E.: Quantum Mechanics, 3rd edn. Wiley, New York (1998)

    Google Scholar 

  81. Mukai, H.: J. Math. Phys. 43, 473 (2002)

    MATH  ADS  MathSciNet  Google Scholar 

  82. Neves, C., Wotzasek, C.: J. Phys. A: Math. Gen. 33, 6447 (2000)

    MATH  ADS  MathSciNet  Google Scholar 

  83. Oyewumi, K.J., Bangudu, E.A.: Arab. J. Sci. Eng. 28(2A), 173 (2003)

    MathSciNet  Google Scholar 

  84. Oyewumi, K.J., Ogunsola, A.W.: Global J. Pure Appl. Sci. 10(2), 343 (2004)

    Google Scholar 

  85. Oyewumi, K.J.: Found. Phys. Lett. 18, 75 (2005)

    Google Scholar 

  86. Oyewumi, K.J.: Applications of the quantum mechanical hypervirial theorems in the study of bound state problems. PhD thesis, University of Ilorin, Nigeria (2006)

    Google Scholar 

  87. Paz, G.: Eur. J. Phys. 22, 337 (2001)

    MATH  Google Scholar 

  88. Popov, D.: Cze. J. Phys. 49(8), 1121 (1998)

    ADS  Google Scholar 

  89. Popov, D.: Int. J. Quantum Chem. 69, 159 (1999)

    Google Scholar 

  90. Popov, D.: J. Phys. A: Math. Gen. 34, 5283 (2001)

    MATH  ADS  Google Scholar 

  91. Pradhan, P.: Am. J. Phys. 63, 664 (1995)

    ADS  Google Scholar 

  92. Romera, E., Sánchez-Moreno, P., Dehesa, J.S.: J. Math. Phys. 47, 103504 (2006)

    ADS  MathSciNet  Google Scholar 

  93. Rosati, S.: Hyperspherical harmonics methods for strongly interacting systems: a summary and new developments I and II. In: SMR1348, The Second European Summer School on Microscopic Quantum Many-Body Theories and Their Applications. ICTP Lecture Notes (2001)

  94. Roychoudhury, R.K., Roy, B.: J. Math. Phys. 27(1), 229 (1986)

    ADS  MathSciNet  Google Scholar 

  95. Sánchez del Rio, C.: Am. J. Phys. 50(6), 556 (1982)

    ADS  Google Scholar 

  96. Sánchez-Moreno, P., González-Férez, R., Dehesa, J.S.: New J. Phys. 8, 330 (2006)

    Google Scholar 

  97. Saelen, L., Nepstad, P., Hansen, J.P., Madsen, L.B.: J. Phys. A: Math. Gen. 40, 1097 (2002)

    ADS  MathSciNet  Google Scholar 

  98. Sage, M., Goodisman, J.: Am. J. Phys. 53(4), 350 (1984)

    ADS  Google Scholar 

  99. Sato, M., Goodisman, J.: Am. J. Phys. 33, 350 (1985)

    ADS  Google Scholar 

  100. Seze, M.L.: Chem. Phys. 87, 431 (1984)

    ADS  Google Scholar 

  101. Schrödinger, E.: Proc. R. Irish Acad. A 46, 183 (1940)

    Google Scholar 

  102. Shimakura, N.: Partial Differential Operator of Elliptic Type. American Mathematical Society, Providence (1992)

    Google Scholar 

  103. Wang, L.Y., Gu, X.Y., Ma, Z.Q., Dong, S.H.: Found. Phys. Lett. 15(6), 569 (2002)

    MathSciNet  Google Scholar 

  104. Wei-Qin, Z.: Relation between dimension and angular momentum for radially symmetric potential in N-dimensional space. quant-ph/0511086 (2005)

  105. Weissman, Y., Jortner, J.: Phys. Lett. A 70, 177 (1979)

    ADS  MathSciNet  Google Scholar 

  106. Wipf, A., Kirchberg, A., Länge, D.: Bulg. J. Phys. 33, 206 (2006)

    MATH  MathSciNet  Google Scholar 

  107. Wybourne, G.B.: Classical Groups for Physicists. Wiley–Interscience, New York (1974)

    MATH  Google Scholar 

  108. Yáñez, R.J., Assche, W.V., Dehesa, J.: Phys. Rev. A 50, 3065 (1994)

    ADS  Google Scholar 

  109. Yáñez, R.J., Assche, W.V., González-Férez, R., Dehesa, J.S.: J. Math. Phys. 40(11), 5675 (1999)

    MATH  ADS  MathSciNet  Google Scholar 

  110. Zeng, G.J., Su, K.L., Li, M.: Phys. Rev. A 50(5), 4373 (1994)

    ADS  Google Scholar 

  111. Zeng, G.J., Zhou, S.L., Ao, S.M., Jiang, F.S.: J. Phys. A: Math. Gen. 30, 1775 (1997)

    MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Theoretical Physics Section, Department of Physics, University of Ilorin, P.M.B. 1515, Ilorin, Nigeria

    K. J. Oyewumi

  2. Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, P.M.B. 4000, Ogbomoso, Nigeria

    F. O. Akinpelu & A. D. Agboọla

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  1. K. J. Oyewumi
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  2. F. O. Akinpelu
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Oyewumi, K.J., Akinpelu, F.O. & Agboọla, A.D. Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions. Int J Theor Phys 47, 1039–1057 (2008). https://doi.org/10.1007/s10773-007-9532-x

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