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Redalyc.Exact Spectrum and Wave Functions of the Hyperbolic Scarf Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Alvarez–Castillo, D.E.; Kirchbach, M. Exact spectrum and wave functions of the hyperbolic Scarf potential in terms of finite Romanovski polynomials Revista Mexicana de Física, vol. 53, núm. 2, diciembre, 2007, pp. 143-154 Sociedad Mexicana de Física A.C. Distrito Federal, México Disponible en: http://www.redalyc.org/articulo.oa?id=57028298003 Cómo citar el artículo Número completo Sistema de Información Científica Más información del artículo Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Página de la revista en redalyc.org Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto ENSENANZA˜ REVISTA MEXICANA DE FISICA´ E53 (2) 143–154 DICIEMBRE 2007 Exact spectrum and wave functions of the hyperbolic Scarf potential in terms of finite Romanovski polynomials D.E. Alvarez-Castillo and M. Kirchbach Instituto de F´ısica, Universidad Autonoma´ de San Luis Potos´ı, Av. Manuel Nava 6, San Luis Potos´ı, S.L.P. 78290, Mexico. Recibido el 1 de septiembre de 2006; aceptado el 24 de febrero de 2007 The Schrodinger¨ equation with the hyperbolic Scarf potential reported so far in the literature is somewhat artificially manipulated into the form of the Jacobi equation with an imaginary argument and parameters that are complex conjugate to each other. Instead we here solve the former equation anew and make the case that it reduces straightforward to a particular form of the generalized real hypergeometric equation whose solutions are referred to in the mathematics literature as the finite Romanovski polynomials, in reference to the observation that for any parameter set only a finite number of such polynomials appear to be orthogonal. This is a qualitatively new integral property that does not copy any of the features of the Jacobi polynomials. In this manner, the finite number of bound states within the hyperbolic Scarf potential is brought into correspondence with a finite system of a new class of orthogonal polynomials. This work adds a new example to the circle of the problems on the Schrodinger¨ equation. The techniques used by us extend the teachings on the Sturm-Liouville theory of ordinary differential equations beyond their standard presentation in the textbooks on mathematical methods in physics. Keywords: Schrodinger equation; Scarf potentials; Romanovski polynomials. La solucion´ a la ecuacion´ de Schrodinger¨ con el potencial de Scarf hiperbolico´ reportada hasta ahora en la literatura f´ısica esta´ manipulada artificialmente para obtenerla en la forma de los polinomios de Jacobi con argumentos imaginarios y parametros´ que son complejos conju- gados entre ellos. En lugar de eso, nosotros resolvimos la nueva ecuacion´ obtenida y desarrollamos el caso en el que realmente se reduce a una forma particular de la ecuacion´ hipergeometrica´ generalizada real, cuyas soluciones se refieren en la literatura matematica´ como los polinomios finitos de Romanovski. La notacion´ de finito se refiere a que, para cualquier parametro´ fijo, solo un numero´ finito de dichos poli- nomios son ortogonales. Esta es una nueva propiedad cualitativa de la integral que no surge como copia de ninguna de las caracter´ısticas de los polinomios de Jacobi. De esta manera, el numero´ finito de estados en el potencial de Scarf hiperbolico´ es consistente en correspondencia a un sistema finito de polinomios ortogonales de una nueva clase. Descriptores: Ecuacion´ de Schrodinger; potenciales de Scarf; polinomios de Romanovski. PACS: 02.30.Gp; 03.65.Ge; 12.60.Jv 1. Introduction then employing the relevant group algebra in order to con- struct the solutions as the group representation spaces [8, 9]. The exactly solvable Schrodinger¨ equations occupy a pole Finally, there is also the most recent and powerful method position in quantum mechanics insofar as most of them re- of super-symmetric quantum mechanics (SUSYQM) which late directly to physical systems. Suffices to mention as considers the special class of Schrodinger¨ equations (in units prominent examples the quantum Kepler or Coulomb prob- of ~ = 1 = 2m) that allow for a factorization according lem and its importance in the description of the discrete spec- to [10-12], trum of the hydrogen atom [1], the harmonic-oscillator, the µ 2 ¶ Hulthen, and the Morse potentials with their relevance to vi- d (H(z) ¡ en) Ãn(z) = ¡ 2 + v(z) ¡ en Ãn(z) = 0 ; brational spectra [2, 3]. Another good example is given by dz the Poschl-Teller¨ potential [4] which appears as an effective + ¡ H(z) = A (z)A (z) + e0 ; mean field in many-body systems with ±-interactions [5]. In µ ¶ d terms of path integrals, the criteria for exact resolvability of A§(z) = § + U(z) : (1) the Schrodinger¨ equation is the existence of exactly solvable dz corresponding path integrals [6]. H(z) There are various methods of finding the exact solutions Here, stands for the (one-dimensional) Hamiltonian, U(z) A§(z) of the Schrodinger¨ equation (SE) for the bound states, an is- is the so called super-potential, and are the lad- der operators connecting neighboring solutions. The super- sue on which we shall focus in the present work. The tra- potential allows us to recover the ground state wave function, ditional method, to be pursued by us here, consists in re- Ãgst(z), as ducing SE by an appropriate change of the variables to that R ¡ z U(y)dy very form of the generalized hypergeometric equation [7] Ãgst(z) » e : (2) whose solutions are polynomials, the majority of them be- ing well known. The second method suggests to first un- The excited states are then built up on top of Ãgst(z) through veiling the dynamical symmetry of the potential problem and the repeated action of the A+(z) operators. 144 D.E. ALVAREZ-CASTILLO AND M. KIRCHBACH FIGURE 2. The hyperbolic Scarf potential (Scarf II) for the set of FIGURE 1. The trigonometric Scarf potential (Scarf I) for the set parameters, a = 10, b = 5, and ® = 1. The horizontal lines of parameters, a = 10, b = 5, and ® = 1. The horizontal lines represent the energies, en, of the bound states. represent the discrete levels. Scarf I is transformed into the so-called hyperbolic Scarf po- (a;b) 1.1. The trigonometric Scarf potential tential (Scarf II), here denoted by vh (z) and displayed in Fig. 2, Super-symmetric quantum mechanics manages a family of (a;b) 2 2 2 2 exactly solvable potentials (see Refs. 11 to 13 for details), vh (z) = a + (b ¡ a ¡ a®)sech ®z one of which is the so called trigonometric Scarf potential + b(2a + ®)sech®z tanh ®z : (7) (Scarf I) [14], here denoted by vt(z) and given by The latter potential has also been found independently within v(a;b)(z) = ¡ a2 + (a2 + b2 ¡ a®) sec2 ®z t the framework of super-symmetric quantum mechanics while ¡ b(2a + ®) tan ®z sec ®z: (3) exploring the super-potential [11, 13, 15]: It has been used in the construction of a periodic potential and U(z) = a tanh ®z + bsech®z : (8) employed in one-dimensional crystal models in solid state physics. Upon the above substitutions, and taking ® = 1 for simplic- The exact solution of the Schrodinger¨ equation with the ity, the energy changes to trigonometric Scarf potential (displayed in Fig. 1) is well ² = e ¡ a2 = ¡(a ¡ n)2 ; n = 0; 1; 2; ::: < a : (9) known [11, 13] and given in terms of the Jacobi polynomi- n n als, P ¯;®(x), as n It is important to notice that, while the trigonometric q 1 1 Scarf potential allows for an infinite number of bound states, γ ± γ¡ 2 ;±¡ 2 Ãn(x) = (1 ¡ x) (1 + x) Pn (x); the number of discrete levels within the hyperbolic Scarf po- tential is finite, a difference that will be explained in Sec. 3 x = sin ®z; below. Yet the most violent changes seem to be suffered by γ¡ 1 ;±¡ 1 γ¡ 1 ±¡ 1 w 2 2 (x) = (1 ¡ x) 2 (1 + x) 2 ; the Jacobi weight function in Eq. (4) and are due to the open- ing of the finite interval [¡1; +1] toward infinity 1 1 γ = (a ¡ b); ± = (a + b) : (4) ® ® x= sin ®z 2 [¡1; 1] ¡! x= sinh ®z 2 [¡1; +1] : (10) γ¡ 1 ;±¡ 1 Here, w 2 2 (x) stands for the weight function from 1 1 In this case, the wave functions become [11, 16, 17], γ¡ 2 ;±¡ 2 which the Jacobi polynomials Pn (x) are obtained via a ¡1 ¤ 2 ¡ 2 ¡b tan x ´ ;´ the Rodrigues formula. Ãn(¡ix) = (1 + x ) e cnPn (¡ix) ; The corresponding energy spectrum is obtained as 1 ´ = ib ¡ a ¡ : (11) 2 2 2 ²n = en + a = (a + n®) : (5) Here, cn is some state dependent complex phase to be fixed 1.2. The hyperbolic Scarf potential later on. The latter equation gives the impression that the ex- act solutions of the hyperbolic Scarf potential rely exclusively By means of the substitutions upon those peculiar Jacobi polynomials with imaginary argu- ments and complex indices. We here draw attention to the a ¡! ia ; ® ¡! ¡i® ; b ¡! b ; (6) fact that this need not be so. Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 EXACT SPECTRUM AND WAVE FUNCTIONS OF THE HYPERBOLIC SCARF POTENTIAL IN TERMS. 145 1.3. The goal physics, Scarf II finds application in studies of the non- perturbative sector of gauge theories by means of toy mod- The goal of this work is to solve the Schrodinger¨ els such as the scalar field theory in (1+1) space-time di- equation with the hyperbolic Scarf potential mensions. Here, one encounters the so called “kink-like” anew and to make the case that it reduces in solutions which are nothing more than static solitons.
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