Novel Superpotentials for Supersymmetric Quantum Mechanics: A New Mathematical Investigation and Study

Alireza Heidari, Seyedali Vedad, Mohammadali Ghorbani

Institute for Advanced Studies, Tehran, Iran.

DOI: 10.4236/jmp.2012.34043                

Abstract

The following article has been retracted due to the investigation of complaints received against it. Mr. Mohammadali Ghorbani (corresponding author and also the last author) cheated the authors’ name: Alireza Heidari and Seyedali Vedad. The scientific community takes a very strong view on this matter and we treat all unethical behavior such as plagiarism seriously. This paper published in Vol.3 No.4 304-311, 2012, has been removed from this site.

Keywords

Superpotentials; Supersymmetric; Quantum Mechanics; Schrödinger Equation; Operator Method; Coulomb Superpotential; Klein-Gordon Energies

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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
[2]	F. Cooper, J. N. Ginocchio and A. Khare, “Relationship between Supersymmetry and Solvable Potentials,” Physical Review D, Vol. 36, No. 8, 1987, pp. 2458-2473.
[3]	R. Dutt, A. Khare and U. P. Sukhatme, “Exactness of Supersymmetric WKB Spectra for Shape-Invariant Potentials,” Physics Letters B, Vol. 181, No. 3-4, 1986, pp. 295-298. doi:10.1016/0370-2693(86)90049-3
[4]	J. W. Dabrowska, A. Khare and U. P. Sukhatme, “Explicit Wavefunctions for Shape-Invariant Potentials by Operator Techniques,” Journal of Physics A: Mathematical and General, Vol. 21, No. 4, 1988, pp. L195-L200. doi:10.1088/0305-4470/21/4/002
[5]	F. Cooper, A. Khare and U. Sukhatme, “Periodic Solutions of Nonlinear Equations Obtained by Linear Superposition,” Journal of Physics A: Mathematical and General, Vol. 35, No. 47, 2002, pp. 10085-10100. doi:10.1088/0305-4470/35/47/309
[6]	D. T. Barclay, R. Dutt, A. Gangopadhyaya, A. Khare, A. Pagnamenta and U. Sukhatme, “New Exactly Solvable Hamiltonians: Shape Invariance and Self-Similarity,” Physical Review A, Vol. 48, No. 4, 1993, pp. 2786-2797. doi:10.1103/PhysRevA.48.2786
[7]	A. Khare and U. P. Sukhatme, “Scattering Amplitudes for Supersymmetric Shape-Invariant Potentials by Operator Methods,” Journal of Physics A: Mathematical and General, Vol. 21, No. 9, 1988, p. L501. doi:10.1088/0305-4470/21/9/005
[8]	R. Adhikari, R. Dutt, A. Khare and U. P. Sukhatme, “Higher-Order WKB Approximations in Supersymmetric Quantum Mechanics,” Physical Review A, Vol. 38, No. 4, 1988, pp. 1679-1686. doi:10.1103/PhysRevA.38.1679
[9]	R. Dutt, A. Khare and U. P. Sukhatme, “Supersymmetry- Inspire WKB Approximation in Quantum Mechanics,” American Journal of Physics, Vol. 59, No. 8, 1991, pp. 723-727. doi:10.1119/1.16840
[10]	A. Khare and U. P. Sukhatme, “New Shape-Invariant Potentials in Supersymmetric Quantum Mechanics,” Journal of Physics A: Mathematical and General, Vol. 26, 1993, pp. L901-L904. doi:10.1088/0305-4470/26/18/003
[11]	U. P. Sukhatme, C. Rasinariu and A. Khare, “Cyclic Shape Invariant Potentials,” Physics Letters A, Vol. 234, No. 6, 1997, pp. 401-409. doi:10.1016/S0375-9601(97)00555-0
[12]	A. Khare and U. P. Sukhatme, “Linear Superposition in Nonlinear Equations,” Physical Review Letters, Vol. 88, No. 24, 2002, pp. 244101-244104. doi:10.1103/PhysRevLett.88.244101
[13]	W. Keung, E. Kovacs and U. P. Sukhatme, “Supersym- metry and Double-Well Potentials,” Physical Review Let- ters, Vol. 60, No. 41, 1988, pp. 41-44. doi:10.1103/PhysRevLett.60.41
[14]	S. Chaturvedi, R. Dutt, A. Gangopadhyaya, P. Panigrahi, C. Rasinariu and U. Sukhatme, “Algebraic Shape Invariant Models,” Physics Letters A, Vol. 248, No. 2-4, 1998, pp. 109-113. doi:10.1016/S0375-9601(98)00636-7
[15]	R. De, R. Dutt and U. Sukhatme, “Mapping of Shape Invariant Potentials under Point Canonical Transforma- tions,” Journal of Physics A: Mathematical and General, Vol. 25, No. 13, 1992, pp. L843-L850. doi:10.1088/0305-4470/25/13/013
[16]	A. Gangopadhyaya, A. Khare and U. P. Sukhatme, “Meth- ods for Generating Quasi-Exactly Solvable Potentials,” Physics Letters A, Vol. 208, No. 4, 1995, pp. 261-268. doi:10.1016/0375-9601(95)00824-3
[17]	M. Hruska, W. Keung and U. Sukhatme, “Accuracy of Semiclassical Methods for Shape-Invariant Potentials,” Physical Review A, Vol. 55, No. 5, 1997, pp. 3345-3350. doi:10.1103/PhysRevA.55.3345
[18]	A. Khare and U. P. Sukhatme, “Cyclic Identities Involving Jacobi Elliptic Functions,” Journal of Mathematical Physics, Vol. 43, No. 7, 2002, pp. 3798-3806. doi:10.1063/1.1484541
[19]	A. Gangopadhyaya, J. V. Mallow and U. P. Sukhatme, “Translational Shape Invariance and the Inherent Potential Algebra,” Physics Letters A, Vol. 58, No. 3, 1998, pp. 4287-4292.
[20]	F. Cooper, J. N. Ginocchio and A. Wipf, “Derivation of the S-Matrix Using Supersymmetry,” Physics Letters A, Vol. 129, No. 3, 1988, pp. 145-147. doi:10.1016/0375-9601(88)90131-4
[21]	A. Khare and U. Sukhatme, “Phase-Equivalent Potentials Obtained from Supersymmetry,” Journal of Physics A: Mathematical and General, Vol. 22, No. 14, 1989, p. 2847. doi:10.1088/0305-4470/22/14/031
[22]	J. Pappademos, U. Sukhatme and A. Pagnamenta, “Bound States in the Continuum from Supersymmetric Quantum Mechanics,” Physical Review A, Vol. 48, No. 3525, 1993, pp. 3525-3531. doi:10.1103/PhysRevA.48.3525
[23]	F. Cooper, A. Khare, B. Mihaila and A. Saxena, “Exact Solitary Wave Solutions for a Discrete λ?4 Field Theory in 1 + 1 Dimensions,” Physical Review E, Vol. 72, No. 3, 2005, Article ID 036605.
[24]	A. Gangopadhyaya and U. P. Sukhatme, “Potentials with Two Shifted Sets of Equally Spaced Eigenvalues and Their Calogero Spectrum,” Physics Letters A, Vol. 224, No. 1-2, 1996, pp. 5-14. doi:10.1016/S0375-9601(96)00807-9
[25]	R. Dutt, A. Gangopadhyaya, A. Khare, A. Pagnamenta and U. Sukhatme, “Solvable Quantum Mechanical Examples of Broken Supersymmetry,” Physics Letters A, Vol. 174, No. 5-6, 1993, pp. 363-367. doi:10.1016/0375-9601(93)90191-2
[26]	A. Gangopadhyaya, J. V. Mallow and U. P. Sukhatme, “Broken Supersymmetric Shape Invariant Systems and Their Potential Algebras,” Physics Letters A, Vol. 283, No. 5, 2001, pp. 279-284. doi:10.1016/S0375-9601(01)00266-3
[27]	A. Khare and U. Sukhatme, “Analytically Solvable PT- Invariant Periodic Potentials,” Physics Letters A, Vol. 324, 2004, pp. 406-414. doi:10.1016/j.physleta.2004.03.006
[28]	R. Dutt, A. Gangopadhyaya, A. Khare, A. Pagnamenta and U. Sukhatme, “Semiclassical Approach to Quantum- Mechanical Problems with Broken Supersymmetry,” Physical Review A, Vol. 48, No. 3, 1993, pp. 1845-1853. doi:10.1103/PhysRevA.48.1845
[29]	W. Keung, U. P. Sukhatme, Q. Wang and T. D. Imbo, “Families of Strictly Isospectral Potentials,” Journal of Physics A: Mathematical and General, Vol. 22, No. 21, 1989, pp. L987-L992. doi:10.1088/0305-4470/22/21/002
[30]	P. K. Panigrahi and U. P. Sukhatme, “Singular Superpotentials in Supersymmetric Quantum Mechanics,” Physics Letters A, Vol. 178, No. 3-4, 1993, pp. 251-257. doi:10.1016/0375-9601(93)91098-P
[31]	F. Cooper, A. Khare, R. Musto and A. Wipf, “Supersymmetry and the Dirac Equation,” Annals of Physics, Vol. 187, No. 1, 1988, pp. 1-28. doi:10.1016/0003-4916(88)90279-5
[32]	M. S. Kumar and A. Khare, “Coherent States for Isospectral Hamiltonians,” Physics Letters A, Vol. 217, No. 2-3, 1996, pp. 73-77. doi:10.1016/0375-9601(96)00332-5
[33]	F. Cooper, J. N. Ginocchio and A. Wipf, “Supersymmetry, Operator Transformations and Exactly Solvable Potentials,” Journal of Physics A: Mathematical and General, Vol. 22, No. 17, 1989, pp. 3707-3716. doi:10.1088/0305-4470/22/17/035
[34]	R. Dutt, A. Khare and Y. P. Varshni, “New Class of Conditionally Exactly Solvable Potentials in Quantum Mechanics,” Journal of Physics A: Mathematical and Gen- eral, Vol. 28, No. 3, 1995, p. L107. doi:10.1088/0305-4470/28/3/008
[35]	A. Khare, K. Rasmussen, M. Salerno, M. R. Samuelsen and A. Saxena, “Discrete Nonlinear Schr?dinger Equations with Arbitrarily High-Order Nonlinearities,” Physical Review E, Vol. 74, No. 1, 2006, Article ID 016607.
[36]	A. Khare, “Parasupersymmetric Quantum Mechanics of Arbitrary Order,” Journal of Physics A: Mathematical and General, Vol. 25, No. 12, 1992, p. L749. doi:10.1088/0305-4470/25/12/008
[37]	A. Khare and Y. P. Varshni, “Is Shape Invariance Also Necessary for Lowest Order Supersymmetric WKB to Be Exact?” Physics Letters A, Vol. 142, No. 1, 1989, pp. 1-4. doi:10.1016/0375-9601(89)90701-9
[38]	R. Dutt, A. Gangopadhyaya, C. Rasinariu and U. Sukhatme, “New Solvable Singular Potentials,” Journal of Physics A: Mathematical and General, Vol. 34, No. 19, 2001, p. 4129. doi:10.1088/0305-4470/34/19/311
[39]	E. Harikumar, V. S. Kumar and A. Khare, “Supersymmetric Quantum Mechanics on Non-Commutative Plane,” Physics Letters B, Vol. 589, No. 3-4, 2004, pp. 155-161. doi:10.1016/j.physletb.2004.03.042
[40]	A. Gangopadhyaya, P. K. Panigrahi and U. P. Sukhatme, “Supersymmetry and Tunneling in an Asymmetric Double Well,” Physical Review A, Vol. 47, No. 4, 1993, pp. 2720-2724. doi:10.1103/PhysRevA.47.2720
[41]	R. Dutt, A. Khare and U. P. Sukhatme, “Supersymmetry, Shape Invariance, and Exactly Solvable Potentials,” American Journal of Physics, Vol. 56, No. 2, 1988, p. 163. doi:10.1119/1.15697
[42]	C. X. Chuan, “Exactly Solvable Potentials and the Concept of Shape Invariance,” Journal of Physics A: Mathematical and General, Vol. 24, No. 19, 1991, p. L1165. doi:10.1088/0305-4470/24/19/008
[43]	C. X. Chuan, “Odd Potentials in Supersymmetric Quantum Mechanics,” Journal of Physics A: Mathematical and General, Vol. 23, No. 13, 1990, p. L659. doi:10.1088/0305-4470/23/13/005
[44]	C. X. Chuan, “The Theory of Coupled Differential Equations in Supersymmetric Quantum Mechanics,” Journal of Physics A: Mathematical and General, Vol. 23, No. 23, 1990, p. L1217. doi:10.1088/0305-4470/23/23/006
[45]	C. X. Chuan, “Some Solvable Eigenvalue Problems,” International Journal of Theoretical Physics, Vol. 34, No. 9, 1995, pp. 1907-1914. doi:10.1007/BF00674072
[46]	C. X. Chuan, “Alternative Approach to the Concept of Shape Invariance in Quantum Mechanics,” International Journal of Theoretical Physics, Vol. 37, No. 9, 1998, pp. 2439-2448. doi:10.1023/A:1026679311801
[47]	C. X. Chuan, “Higher Order Generation of Exactly Solvable Supersymmetric Systems,” International Journal of Theoretical Physics, Vol. 38, No. 2, 1999, pp. 745-756. doi:10.1023/A:1026671710777
[48]	T. Barakat, M. Odeh and O. Mustafa, “Perturbed Coulomb Potentials in the Klein-Gordon Equation via the Shifted-l Expansion Technique,” Journal of Physics A: Mathematical and General, Vol. 31, No. 15, 1998, p. 3469. doi:10.1088/0305-4470/31/15/012
[49]	H. Egrifes and R. Sever, “Bound-State Solutions of the Klein-Gordon Equation for the Generalized PT-Symmetric Hulthén Potential,” International Journal of Theoretical Physics, Vol. 46, No. 4, 2007, pp. 935-950. doi:10.1007/s10773-006-9251-8
[50]	T. Barakat, “Perturbed Coulomb Potentials in the Klein- Gordon Equation via the Asymptotic Iteration Method,” Annals of Physics, Vol. 324, No. 3, 2009, pp. 725-733. doi:10.1016/j.aop.2008.10.008
[51]	M. M. Panja, R. Dutt and Y. P. Varshni, “Shifted Large-N Expansion for the Power-Law Potentials in the Klein- Gordon Equation with Applications,” Journal of Physics A: Mathematical and General, Vol. 22, No. 15, 1989, p. 2291. doi:10.1088/0305-4470/22/15/015
[52]	Y. Chargui, L. Chetouani and A. Trabelsi, “Exact Solution of the One-Dimensional Klein-Gordon Equation with Scalar and Vector Linear Potentials in the Presence of a Minimal Length,” Chinese Physics B, Vol. 19, No. 2, 2010, Article ID 020305. doi:10.1088/1674-1056/19/2/020305
[53]	Y. Nedjadi, S. Ait-Tahar and R. C. Barrett, “An Extended Relativistic Quantum Oscillator for S = 1 Particles,” Journal of Physics A: Mathematical and General, Vol. 31, No. 16, 1998, p. 3867. doi:10.1088/0305-4470/31/16/014
[54]	J. B. Wang, J. F. Williams, A. T. Stelbovics, J. E. Furst and D. H. Madison, “Coherent Excitation of the Singlet- Triplet Mixed 1s4f State of Helium,” Physical Review A, Vol. 52, No. 4, 1995, pp. 2885-2900. doi:10.1103/PhysRevA.52.2885
[55]	P. F. Smith, “Coherent Neutrino Scattering—Relativistic and Nonrelativistic,” Il Nuovo Cimento A, Vol. 83, No. 3, 1984, pp. 263-274. doi:10.1007/BF02902601
[56]	G. Ni, J. Xu and W. Chen, “The Thermal Coherent State and Its Application to the One-Dimensional Field Theory,” Journal of Physics A: Mathematical and General, Vol. 18, No. 1, 1985, pp. 149-164. doi:10.1088/0305-4470/18/1/026
[57]	A. Messina and F. Persico, “Spontaneous Coherent Phonons in the Ground State of Strong Spin-Phonon Interaction,” Journal of Physics C: Solid State Physics, Vol. 6, No. 24, 1973, pp. 3557-3570.