OJM  Vol.3 No.1 , February 2013
Relativistic Schrödinger Wave Equation for Hydrogen Atom Using Factorization Method
ABSTRACT

In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.


Cite this paper
M. Pahlavani, H. Rahbar and M. Ghezelbash, "Relativistic Schrödinger Wave Equation for Hydrogen Atom Using Factorization Method," Open Journal of Microphysics, Vol. 3 No. 1, 2013, pp. 1-7. doi: 10.4236/ojm.2013.31001.
References
[1]   W. Greiner, “Quantum Mechanics,” 3rd Edition, Springer-Verlag, Berlin, 1994.

[2]   I. T. Todorov, “Quasipotential Equation Corresponding to the Relativistic Eikonal Approximation,” Physical Review D, Vol. 3, 1971, pp. 2351-2356. doi:10.1103/PhysRevD.3.2351

[3]   E. Brezin, C. Itzykson and J. Zinn-Justin, “Relativistic Balmer Formula Including Recoil Effects,” Physical Rview D, Vol. 1, No. 8, 1970, pp. 2349-2355. doi:10.1103/PhysRevD.1.2349.

[4]   C. Itzykson and J. B. Zuber, “Quantum Field Theory,” Mc-Graw-Hill, New York, 1985.

[5]   E. Fermi and C. N. Yang, “A Relativistic Equation for Bound-State Problems,” Physical Review, Vol. 84, No. 6, 1951, pp. 1232-1242. doi:10.1103/PhysRev.84.1232

[6]   E. E. Salpeter and H. A. Bethe, “Are Mesons Elementary Particles?” Physical Review, Vol. 76, No. 12, 1949, pp. 1739-1743. doi:10.1103/PhysRev.76.1739

[7]   G. Breit, “Dirac’s Equation and the Spin-Spin Interactions of Two Electrons,” Physical Review, Vol. 39, No. 4, 1932, pp. 616-624. doi:10.1103/PhysRev.39.616

[8]   G. D. Tsibidis, “Quark-Antiquark Bound States and the Breit Equation,” Acta Physica Polonica B, Vol. 35, No. 10, 2004, pp. 2329-2365.

[9]   R. J. Duffin, “On the Characteristic Matrices of Covariant Systems,” Physical Review, Vol. 54, No. 12, 1939, p. 1114. doi:10.1103/PhysRev.54.1114

[10]   J. T. Lunardi, L. A. Manzoni and B. M. Pimentel, “Duffin-Kemmer-Petiau Theory in the Causal Approa,” International Journal of Modern Physics A, Vol. 17, No. 2, 2002, p. 205. doi:10.1142/S0217751X02005682

[11]   I. Boztosun, M. Karakus, F. Yasuk and A. Durmus, “Asymptotic Iteration Method Solutions to the Relativistic Duffin-Kemmer-Petiau Equation,” Journal of Mathematical Physics, Vol. 47, No. 6, 2006, Article ID: 062301. doi:10.1063/1.2203429

[12]   Y. Nedjadi and R. C. Barrett, “The Duffin-Kemmer-Petiau Oscillator,” Journal of Physics A: Mathematical and General, Vol. 27, No. 12, 1994, p. 4301. doi:10.1088/0305-4470/27/12/033

[13]   Y. Nedjadi and R. C. Barrett, “Solution of the Central Field Problem for a Duffin-Kemmer-Petiau Vector Boson,” Journal of Mathematical Physics, Vol. 35, No. 9, 1994, pp. 4517-4533. doi:10.1063/1.530801

[14]   Y. Nedjadi and R. C. Barrett, “On the Properties of the Duffin-Kemmer-Petiau Equation,” Journal of Physics G: Nuclear and Particle Physics, Vol. 19, No. 1, 1993, pp. 87-98. doi:10.1088/0954-3899/19/1/006

[15]   B. Boutabia-Cheraitia and T. Boudjedaa, “Solution of DKP Equation in Woods-Saxon Potential,” Physics Letters A, Vol. 338, No. 2, 2005, pp. 97-107. doi:10.1016/j.physleta.2005.02.029

[16]   V. Y. Fainberg and B. M. Pimentel, “Duffin-Kemmer-Petiau and Klein-Gordon-Fock Equations for Electromagnetic, Yang-Mills and External Gravitational Field Interactions: Proof of Equivalence,” Physics Letters A, Vol. 271, No. 1-2, 2000, pp. 16-25. doi:10.1016/S0375-9601(00)00330-3

[17]   J. T. Lunardi, P. M. Pimental and R. G. Teixeiri, “Remarks on Duffin-Kemmer-Petiau Theory and Gauge Invariance,” Physics Letters A, Vol. 268, No. 10, 2000, pp. 165-173. doi:10.1016/S0375-9601(00)00163-8

[18]   L. Chetouani, M. Merad and T. Boudjedaa, “Solution of Duffin-Kemmer-Petiau Equation for the Step Potential,” International Journal of Theoretical Physics, Vol. 43, No. 4, 2004, pp. 1147-1159. doi:10.1023/B:IJTP.0000048606.29712.13

[19]   A. Boumali, “Particule de Spin 0 dans un Potentiel d’Aharonov-Bohm,” Canadian Journal of Physics, Vol. 82, No. 1, 2004, pp. 67-74. doi:10.1139/p03-112

[20]   D. A. Kulikov, R. S. Tutik and A. P. Yaroshenko “An Alternative Model for the Duffin-Kemmer-Petiau Oscillator,” Modern Physics Letters A, Vol. 20, No. 1, 2005, pp. 43-49. doi:10.1142/S0217732305016324

[21]   N. Ogawa, “Quantum Mechanical Embedding of Spinning Particle and Induced Spin-Connection,” Modern Physics Letters A, Vol. 12, No. 21, 1997, pp. 1583-1588. doi:10.1142/S0217732397001618

[22]   H. Koura and M. Yamada, “Single-Particle Potentials for Spherical Nuclei,” Nuclear Physics A, Vol. 671, No. 1-4, 2000, pp. 96-118. doi:10.1016/S0375-9474(99)00428-5

[23]   J. Sadeghi “Superalgebras for Three Interacting Particles in an External Magnetic Field,” European Physical Journal B, Vol. 50, No. 3, 2006, pp. 453-457. doi:10.1140/epjb/e2006-00150-9

[24]   A. F. Nikiforov and V. B. Uvarov, “Special Functions of Mathematical Physics,” Birkhauser, Basle, 1988.

[25]   E. Schrodinger, “A Method of Determining Quantum-Mechanical Eigenvalues and Eigenfunctions,” Proceedings of the Royal Irish Academy, Vol. 46A, 1940, pp. 9-16.

[26]   E. Schrodinger, “The Factorization of the Hypergeometric Equation,” Proceedings of the Royal Irish Academy, Vol. 47A, 1941, pp. 53-54.

[27]   L. Infeld and T. D. Hull, “The Factorization Method,” Reviews of Modern Physics, Vol. 23, No.1, 1951, pp. 21-68. doi:10.1103/RevModPhys.23.21.

[28]   H. Nicolai, “Supersymmetry and Spin Systems,” Journal of Physics A, Vol. 9, No. 9, 1976, p. 1497. doi:10.1088/0305-4470/9/9/010

[29]   E. Witten, “Gauge Theories, Vertex Models, and Quantum Groups,” Nuclear Physics B, Vol. 380, No. 2-3, 1990, pp. 285-346.

[30]   F. Cooper and B. Freedman, “Aspects of Supersymmetric Quantum Mechanics,” Annals of Physics, Vol. 146, No. 2, 1983, pp. 262-288. doi:10.1016/0003-4916(83)90034-9

[31]   L. E. Gendenshtein, “Derivation of Exact Spectra of the Schrodinger Equation by Means of Supper Symmetry,” Letters to Jounal of Experimental and Theoretical Physics, Vol. 38, 1983, pp. 356-359.

[32]   C. X. Chuan, “Exactly solvable potentials and the concept of shape invariance,” Journal of Physics A, Vol. 24, No. 19, 2006, p. L1165. doi:10.1088/0305-4470/24/19/008

[33]   F. Cooper, A. Khare and U. Sukhatme, “Supersymmetry and Quantum Mechanics,” Physics Reports, Vol. 251, No. 5-6, 1995, pp. 267-385. doi:10.1016/0370-1573(94)00080-M

[34]   A. Balantekin, “Algebraic Approach to Shape Invariance,” Physical Review A, Vol. 57, No. 6, 1998, pp. 4188-4191. doi:10.1103/PhysRevA.57.4188

[35]   A. Balantekin, M. A. C. Ribeiro and A. N. F. Aleixo, “Algebraic Nature of Shape-Invariant and Self-Similar Potentials,” Journal of Physics A: Mathematical and General, Vol. 32, No. 15, 1999, pp. 2785-2790. doi:10.1088/0305-4470/32/15/007

[36]   J. F. Carinena and A. Ramos, “The Partnership of Potentials in Quantum Mechanics and Shape Invariance,” Modern Physics Letters A, Vol. 15, No. 16, 2000, p. 1079. doi:10.1142/S0217732300001249

[37]   H. Aoyama, M. Sato and T. Tanaka, “N-Fold Supersymmetry in Quantum Mechanics: General Formalism,” Nuclear Physics B, Vol. 619, No. 1-3, 2001, pp. 105-127. doi:10.1016/S0550-3213(01)00516-8

[38]   S. W. Qian, B. W. Huang and Z. Y. Gu, “Supersymmetry and Shape Invariance of the Effective Screened Potential,” New Journal of Physics, Vol. 4, 2002, pp. 13.1-13.6. doi:10.1088/1367-2630/4/1/313

[39]   L. Landau and E. M. Lifshitz, “Quantum Mechanics,” Pergmon, Oxford, 1979.

[40]   M. Morse, “Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels,” Physical Review, Vol. 34, No. 1, 1929, pp. 57-64. doi:10.1103/PhysRev.34.57

[41]   C. Eckart, “The Penetration of a Potential Barrier by Electrons,”Physical Review, Vol. 35, No. 11, 1930, pp. 1303-1309. doi:10.1103/PhysRev.35.1303

[42]   V. Bargmann, “On the Connection between Phase Shifts and Scattering Potential,” Reviews of Modern Physics, Vol. 21, No. 3, 1949, pp. 488-493. doi:10.1103/RevModPhys.21.488

[43]   J. Sadeghi, “Factorization Method and Solution of the Non-Central Modified Kreutzer Potential,” Acta Physica Polonica A, Vol. 112, No. 1, 2007, pp. 23-28.

[44]   M. A. Jafarizadeh and H. Fakhri, “The Embedding of Parasupersymmetry and Dynamical Symmetry intoGL(2, c) Group,” Annals of Physics, Vol. 266, No. 1, 1998, pp. 178-206. doi:10.1006/aphy.1998.5788

[45]   M. A. Jafarizadeh and H. Fakhri, “Supersymmetry and Shape Invariance in Differential Equations of Mathematical Physic,” Physics Letters A, Vol. 230, No. 3-4, 1997, pp. 164-170. doi:10.1016/S0375-9601(97)00161-8

[46]   H. Fakhri and J. Sadeghi, “Supersymmetry Approaches to the Bound States of the Generalized Woods-Saxon Potential,” Modern Physics Letters A, Vol. 19, No. 8, 2004, p. 615. doi:10.1142/S0217732304013313

 
 
Top