Darboux Transformations and Isospectral Potentials in Quantum Mechanics

Transcription

1 Apeiron, Vol. 9, No. 3, July 00 0 Darboux Transformations an Isospectral Potentials in Quantum Mechanics J. López-Bonilla, J. Morales an G. Ovano. Sección e estuios e Posgrao e Investigación Escuela Superior e Ingeniería Mecánica y Eléctrica Instituto Politécnico Nacional, Eif.. Z-4, 3er Piso Col. Linavista, 7738 México D.F. lopezbjl@hotmail.com. ea e Física AMA, CBI Universia Autónoma Metropolitana-Azc. Apo. Postal 6-306, 000 México D.F. gaoz@correo.azc.uam.mx We show that the Darboux transformations allow to construct generalize isospectral potentials for a given stanar potential, this in the frame of the Schöinger equation. An application is mae for the Hulthén interaction. PACS: Db Keywors: Darboux transformations, isospectral potentials, Hulthén interaction. 00 C. Roy Keys Inc.

2 .- Introuction Apeiron, Vol. 9, No. 3, July 00 In the one-imensional stationary case, the Schröinger equation is given by []: ψ + u( x) ψ = λψ x which is written in natural units taking ħ / m =. The values of λ represent the energy spectrum allowe for etermine bounary conitions an corresponing to the stanar potential u (x). In this work, we show the very useful proceure of Darboux [-4] as to generalize any specific stanar potential an to generate in this way new interaction moels with the same energy levels. After the proceure, equation () is transforme to ϕ + U ( x) ϕ = λϕ () x an we say that the potential U (x) is isospectral to u (x). The Darboux transformation (DT) (explaine in Sec. ) is relate to the Sturm-Liouville theory [5,6], an it is easy to see the implicit presence of DT in supersymmetric quantum mechanics [,4,7-]. Finally, in Sec. 3 we use DT to construct generalize isospectral potentials for the Hulthén interaction [,3-6]..- Darboux transformations We suppose that () permits the particular solution ψ for the eigenvalue λ : ψ ' + u ( x) ψ = λ, (3) ' ψ then we employ ψ as a ``see'' to construct the DT [-4,7]: 00 C. Roy Keys Inc. ()

3 Apeiron, Vol. 9, No. 3, July 00 ϕ( x ) = ψ ' σ( x) ψ, σ = lnψ, (4) x then () aopts the structure () with the generalize potential: U ( x) = u( x) σ (5) x an the same energy spectrum. That is, the Schröinger equation is covariant with respect to the DT. Selecting other ``see functions'' we generate many DT an therefore a great family of generalize interactions isospectral to u (x). The DT is a mathematical technique that applie in quantum mechanics can be interprete as a supersymmetry [,4,7-,8,9]. In [0-6] it is employe the Riccati equation [5,7,8] to construct generalize potentials isospectral to the free particle, harmonic oscillator, hyrogenic, Morse [,9,,9-33] an Hulthén [,3-6,34] interactions. It is possible to obtain all these generalize potentials via an aequate DT, which show the power of the Darboux's proceure. As an example, in the next Section we apply the DT to Hulthén moel, as an alternative metho to the Riccati equation, in the search of new quantum mechanical potentials for a given spectrum. 3.- Generalize Hulthén potentials. The Hulthén potential [3] is a useful interaction moel that has been use extensively in ifferent areas of Physics, incluing nuclear [35] an atomic physics [36], ue to the fact that it yiels to close analytic solutions for the s waves [5,37]. Its expression is given by [5,6]: V0 u ( r) = (6) e 00 C. Roy Keys Inc.

4 Apeiron, Vol. 9, No. 3, July 00 3 where A an V 0 are positive constants such that V 0 > A. It is clear that [,0] the Schröinger equation for the raial wave function R (r) takes the form () with R = r ψ in the case l = 0. Accoring with equation (4) the DT epens on the function ψ selecte, so that now we will show two options verifying (3): a) ψ is the usual wave function [5,37] for the groun state associate to (6): kr ( e ) e λ = ψ ( k (7) r) =, where k = ( V0 A ) > 0. A Employing (7), the relations (4) an (5) lea to the generalize potential of Hulthén (Darboux potential): 00 C. Roy Keys Inc. V0 A e U m = + (8) e ( e ) equivalent to (36) of [6], which is isospectral to (6). b) Another possibility is to utilize: kr ( ) ( ) kr e ψ ( r) = e e γ + b r (9) b e satisfying (3) with λ = k, γ an b are arbitrary constants. Expressions (4), (5) an (9) imply the following generalize Hulthén interaction: U b b + k ρ ρ e g = Um A, (0) kr with ρ = b( e ) e ψ, corresponing to () of [6] obtaine via Riccati equation.

5 Apeiron, Vol. 9, No. 3, July 00 4 Finally, we conclue by remarking that the Darboux proceure use to generalize a stanar potential is a straightforwar metho, which is far simpler than equivalent approaches employe to fin new families of isospectral known potentials [38]. References. O.L. e Lange an R.E. Raab, Operators methos in quantum mechanics, Clarenon Press, Oxfor (99) Chap.. G. Darboux, Compt. Ren. Aca. Sc. (Paris) 94 (88) A. Khare an U. Sukhatme, J. Phys. A: Math. Gen. (989) V.B. Matveev an M.A. Salle, Darboux transformations an solitons, Springer- Verlag, Berlin (99) 5. C. Lanczos, Linear ifferential operators, D. van Nostran, Lonon (96) 6. J.B. Seaborn, Hypergeometric functions an their applications, Springer- Verlag, Berlin (99) 7. A.A. Anrianov, N.V. Borisov, an M.J. Ioffe, Theor. Math. Fiz. 6() (984) 7 an 6() (984) A.A. Anrianov, N.V. Borisov, an M.J. Ioffe, Phys. Lett. B8 (986) 4 9. R.W. Haymaker, an A.R.P. Rau. Am. J. Phys. 54 (986) F. Schwabl, Quantum mechanics, Springer-Verlag, Berlin (99). F. Cooper, A. Khare, an U. Sukhatme, Phys. Rep. 5 (995) 67. H.R. Hauslin, Helv Phys. Acta 6 (988) L. Hulthén, k. Mat. Astron. Fys. 8 (94) 5 4. P. Matthys, an H. DeMeyer, Phys. Rev. A38 (988) O.L. e Lange, Am. J. Phys. 59 (99) 5 6. R.L. Hall, J. Phys. A.: Math. Gen. 5 (99) M. Crum, Quat. J. Math. 6 (955) 8. D. Yi-Bing Ding, J. Phys. A: Math. Gen. 0 (987) H. Rosu, an J.R. Guzmán, Nuovo Cim. (997) B. Mielnik, J. Math. Phys. 5 (984) D.J. Fernánez, Lett. Math. Phys. 8 (984) 337. Zhu Dongpei, J. Phys. A: Math. Gen. 0 (987) C. Roy Keys Inc.

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