Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions

In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function; the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x ² and g(x)~x ²/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.

• Canonical Transformation
• Schrödinger-Like Equation
• Hypergeometric DE
• Exactly-Solvable Potentials

1. Introduction

Exactly and quasi-exactly solvable potential models are important in practically any field of theoretical quantum chemistry and physics for two principal reasons: first, they are useful to understand the behavior of quantum systems and second, can be used as a basis to study problems that can only be treated using perturbative and nonperturbative procedures. In spite of the above, the exactly solvable Schrödinger equations are rather scanty and in their research different analytical or operational approaches have been used. Also, the well known exactly- solvable Scarf, Eckart, Rosen-Morse I and II, Poschl-Teller I and II as well as Hulthén, Manning-Rosen and Woods-Saxon potentials, all they have, as common feature, hypergeometric wavefunctions. Similarly, the harmonic oscillator, Morse, Coulomb or Kratzer potential models have confluent hypergeometric solutions. Consequently, it becomes clear that the exact solution for the Schrödinger equation is reduced to the study of hypergeometric and/or confluent hypergeometric Differential Equations (DE). At this regard, many efforts have been conducted to find the intermapping between different solvable potentials [1] [2] with the aim to give a unified treatment of partner potentials [3] . For example, in the case of potentials with the hypergeometric wavefunctions, the hexagonal diagram proposed by Cooper et al. [4] is very useful to show how all the shape invariant potentials are inter-related. Also, it has been proposed a pre-potential approach to study of Eckart-type potentials [5] and a five-parameter exponential-type potential to unify the treatment of exactly solvable trigonometric potential models [6] -[8] . Furthermore, to find exactly solvable Schrödinger equations different methods based on Supersymetric Quantum Mechanics (SUSY-QM) [9] , Asymptotic Iteration Method (AIM) [10] and on the transformation of a Schrödinger equation into a hypergeometric-type DE by the Nikiforov-Uvarov (NU) [11] approach, have been used. In this work, we present a proposal that can be considered inverse to the NU method. However, instead of transforming a DE into a Schrödinger equation we consider the transformation of a general homogeneous linear second order DE to their canonical form. Obviously, the general DE has as particular cases the hypergeometric and confluent hypergeometric DE. For that reason, this work aims at finding solvable potentials with hypergeometric wavefunctions leaving the treatment of models with confluent hypergeometric solutions elsewhere [12] . Thus, the proposed approach to transform a general DE into a Schrödinger-like equation is given in next section by means of the canonical transformation method given in the Appendix. The application of the present proposal is given in Section 3, where we consider the hypergeometric DE by means of an auxiliary function, defined in Section 2, that indicates the required transformation. That is, as will be shown, each possibility of leads to different generalized potentials which are reduced to well known particular cases. Finally, in Section 4 we presents the concluding remarks emphasizing that our proposition is general for which can be directly applied to other important DE.

2. Transformation of a General DE into a Schrödinger-Like Equation

According to the proposition given in Equation (A5), the generalized canonical transformation of Equation (A1) becomes

that can be rewritten as

where

and

that will be referred as auxiliary function hereafter. Consequently, from the above, the variable r is given by

in such a way that Equation (A5) is feasible on condition to have the inverse function

Furthermore, the general transform given in Equation (A6) can be rewritten as

where we have used the auxiliary function and the fact that

such that

Consequently, by substituting Equation (7) into Equation (2), the generalized canonical form of Equation (A1) will be

At this point, it should be noticed that above equation can be identified with a Schrödinger-like equation where the wavefunction can be obtained from Equation (7) on condition that coefficients and the solution of Equation (A1) are known. Besides, the potential and energy spectra are given in general by

being and a set of parameters that come from Equation (A1) and Equation (5) respectively.

In short, it is worth noting that is involved in the construction of potential as well as in its corresponding solutions given by Equation (7) through the function. Also, according to Equation (4), there will be different transformations depending on the selection of the auxiliary function which means the existence of various exactly-solvable potentials having the same wavefunctions of the former DE.

3. Application to the Hypergeometric DE

To show the usefulness of the approach given in above section, in the search of exactly-solvable Schrödinger equations, let us apply the above results to the hypergeometric DE

whose solution is [13]

where is the Pochhammer symbol and parameters a and b are constant with That is, in this case Equation (12) matches with Equation (A1) provided that and

As mentioned before, depending on the choice of, or their corresponding auxiliary function, our proposal leads to different exactly-solvable potentials. Consequently, the different options for can be obtained by Equation (3) given by

where we have used the explicit form of function. So, in order to have at least one constant term associated to the eigenvalue of Equation (11), one option is to propose the following cases:

and So, by using Equation (4), the first option leads to, while the second implies and the third option gives Another possibility arises when which means

since. All these options are considered explicitly in what follows by using instead of.

3.1. Generalized Potentials from g(x) Constant

Let us consider the identity

to rewrite Equation (5) as

with the purpose to use the integral [14]

That is, the above integral and the choice of as constant given by

allows to evaluate the integral of Equation (16) as

Consequently, in this case the corresponding transformation is

leading to the particular cases + and − specified by

Thus, according to Equation (3), becomes

which means

and

Thus, as mentioned before, the potential function and the eigenvalue can be identified from Equation (11) as

leading to the corresponding generalized potentials

with eigenvalues

and eigenfunctions

Also, in order to have physically acceptable wavefunctions, the original parameters a, b and c are redefined as

with, for which the potentials of Equation (26) become

as well as their respective wavefunctions

with energy spectra

At this point, is important to notice that particular cases

and

are identified with the well known trigonometric and hyperbolic Scarf potentials having as eigenfunctions

and

respectively, where is proportional to the Jacobi polynomials. As a consequence the following question arises: From where comes the Posch-Teller potential? The answer to that question follows from the negative of given in Equation (18) but now along with the integral [14]

As can be proved, the plus (+) case leads to given in Equation (33). So, we are going to consider only the minus (−) case. That is, now Equation (16) gives rise to

which indicates that the corresponding transformation is

Then, according to Equation (3), in this case will be

leading, from Equation (11), to the potential

with eigenvalue

and eigenfunction

Finally, using the conditions for physically acceptable wavefunctions one obtains

that leads to the potential

with eigenfunctions

and energy spectra

that coincides with the well known Posch-Teller potential.

3.2. Generalized V(r) from g(x) Proportional to P(x)

Similarly to the cases considered in above section, the choice of

can be worked by means of the integrals

In fact, the use of Equation (49) lets write Equation (16) as

leading to the corresponding transformation

So, according to Equation (3)

Thus, the potential function V(r) and the eigenvalue E can be identified from Equation (11) as

to obtain the corresponding generalized potentials

with eigenvalue

and wavefunction

As before, with the aim to have physically acceptable solutions it becomes necessary to redefine the original parameters a, b and c as follow

with and This fact, leads to

with their corresponding wavefunctions

and energy spectra

where

At this point, is important to notice that potential given by

corresponds to the Rosen-Morse II hyperbolic potential and

to the exactly solvable Eckart potential. Also, we want to point out that and are isospectral, accordingly to Equation (60), with wavefunctions

and

respectively. Similarly to the cases analyzed in section III.1), the question is now; How can obtain the Rosen-Morse II trigonometric potential? To answer that question, we are going to use the same choice of but now proportional to the negative. That is,

that together with Equation (49) in Equation (16) permit us to obtain

Thus, with the aid of the identity

the corresponding transformation is

Consequently, according to Equation (3)

such that

leading to

with eigenvalue

and eigenfunction

Similarly, to have physically acceptable wavefunctions the original parameters a, b and c are defined as

where and p a new real parameter. Consequently, the potentials are corrected as

with energy spectra

and eigenfunctions

where

Specifically, the and are given by

and

with their respective wavefunctions

and

with the same energy spectra. At this point, we want to notice that potential corresponds, as expected, to the exactly solvable Rosen-Morse II trigonometric potential and seems to be a new exactly solvable potential.

3.3. A class of Multiparameter Exponential-Type Potential

a). Option with.

According to the third option for in this case one have

In fact, by choosing the minus sign in Equation (5) and the down integral limit as a new parameter q, one obtains

from which

being an additional set of new parameters. In consequence, from Equation (3), is given by

leading, from Equation (11), to

Thus, by defining

and

it becomes possible to identify the potential

the eigenvalue

and, from Equation (7), the eigenfunction

In what follow, the parameters a,b and c will be calculated by considering the boundary conditions of the system in order to have physically acceptable wavefunctions. For example, by combining Equation (89) and Equation (90) one obtains

with

Thus, to have a node in is necessary to apply the condition in Equation (93) with the singular point of, which is achieved if

and

On the other hand, by using Equation (94) in Equation (89) we obtain

and

Also, since the hypergeometric function of Equation (93) is an infinite series, the condition

leads to a polynomial of n degree in the variable. Besides, when indicates

that in the first exponent of Equation (93) which gives the number of states for the system

These assumptions on the original parameters {a, b, c} assure that boundary conditions are fulfilled, leading to a physically acceptable wavefunction for the Schrödinger equation under consideration. Additionally, the energy spectra for the potential is obtained by using Equation (99) in Equation (92). That is

with the corresponding wavefunctions

It should be pointed out that in this case the potential V(r) has a minimum value

in

on condition that

since in this case the argument of the logarithm function, given in Equation (105), is always positive. Consequently, Equations (104)-(106) ensures that potential V(r) is attractive with a infinite wall in. In short, the potential given in Equation (91) is general and contains as particular cases exactly-solvable potentials for specific

values of A, B, and C. For example, the choice of and lead to the exactly-solvable potential

which is identified with the standard Hulthén potential with eigenvalues [15] [16]

and eigenfunctions

In a similar way, the selection and permit us to obtain the supersymmetric or generalized

Hulthén potential already given by Morales et al., [17] . Also, the selection

and gives rise to the Manning-Rosen potential [18]

with energy spectra

and wavefunctions

Another important exactly-solvable exponential potential that have hypergeometric wavefunctions is the Woods-Saxon potential which has been worked using the Numerov method for the standard model [19] or by means of the Nikiforov-Uvarov procedure for the generalized case [20] . According to our results, one can show that the generalized Woods-Saxon potential is obtained as particular case of our proposal when choosing and, where, namely [21] ,

However, due to the fact that the approach considered until now in this work is consistent with, in the case some conditions have to be considered affecting wavefunctions and energy spectra as well as coordinates. That is, for q negative, it must be necessary to change in the transformation, which means that r is no longer the radial variable. Also, the conditions on the original parameters a, b and c, given in Eqs.(94-99), will be corrected to, that is,

with

and

as well as

Thus, by using Equation (114) and the negative of Equation (89) we obtain

and

with the condition for Thus, with these new parameters, the energy spectra and wavefunctions of the generalized Wood-Saxon potential are respectively

and

b). Option with.

Similarly to the above case, in this new situation one have

that leads to the transformation

where and

that, from Equation (3), lets write

In this new situation, by considering the identities

and

as well as

one find the multiparameter exponential-type potential

with eigenvalue

and eigenfunction

At first glance, the potential of Equation (129) can be considered as a new one and however it should be noticed that it can be written as

That is, the above has the same structure of potential given in Equation (91) which means that leads to similar results to those shown in Section 3.3.a). Also, thinking in molecular physics applications, the usefulness of the proposed exactly solvable multiparameter exponential-type potential given in Equation (91) is focused to obtain vibrational modes of diatomic molecules. At this regard, for the study of bound states solutions, one option is a redefinition of parameter B in order to include the centrifugal term and then to apply some of the

approximation schemes proposed to. For example, one can consider and with and so on; see for example Falaye et al. [22] . Moreover, in the same way,

other particular radial potentials can be derived from our proposal as, for example, the recently considered Manning-Rosen potential [23] which, as explained before, comes from Equation (91).

4. Concluding Remarks

In this paper, we present a method to obtain the general canonical form of second order differential equations on the field of theoretical physics. The procedure is similar to the method proposed by Levai [24] and however we consider a general DE to be converted into a Schrödinger-like equation. The algorithm we propose, makes use of an auxiliary function to find the needed transformation to obtain the exactly-solvable potentials with the same wavefunctions of the original DE. To show the usefulness of the proposed method, their application to the hypergeometric DE along with different forms of is given explicitly. Thus, for each we found a generalized exactly-solvable potential from where well known potential models are obtained as particular cases. For example, the choice gives place to the trigonometric and hyperbolic Scarf potentials as well as to the Posh-Teller potential, while the choice of proportional to leads to the Eckart and to the trigonometric and hyperbolic Rosen-Morse potentials. Similarly, with the choice of we obtain a generalized exactly-solvable multiparameter (and k) exponential-type potential that contains, depending on the parameters selection, the particular cases of the Hulthén, Manning-Rosen and Wood-Saxon potentials, standard and generalized, among others. Besides, the proposed multiparameter exponential-type potential can be used in the search of bound state solutions of specific potential models of diatomic molecules in the frame of different approaches to the centrifual term [25] . In short, with the application given in this work we have shown that it is possible to give a unified treatment of exactly-solvable potentials having hypergeometric wavefunctions as common feature. Finally, we want to notice that the method is general for which can be applied to other DE as for example on the position-dependent mass Schrödinger equation. Furthermore, advantageously, our proposal avoids the use of specialized solution methods applied to particular potential models as has been usually done.

Acknowledgements

This work was partially supported by the projects UAM-A-CBI-2232004 and 009. JGR thanks to the Instituto Politécnico Nacional for the financial support given through the COFAA-IPN project SIP-200150019. JGM acknowledges to the IPN-ESFM, for the hospitality during his PhD studies in Science and Technology.

Appendix

The general second order DE

is transformed into their standard canonical form [26]

by means of the transformation

i.e.

where apostrophes indicates derivative with respect to the argument.

In order to have a generalized canonical transformation of Equation A1, we consider the variable change

to obtain the general transformation [27]

Consequently, one have

being

where

It will be noticed that, for the particular case Equation (A8) reduces to the expression given in the square parenthesis of Equation (A4).

References

1. Sukumar, C.V. (1985) Supersymmetric Quantum Mechanics of One-Dimensional Systems. Journal of Physics A: Mathematical and General, 18, 2917-2936. http://dx.doi.org/10.1088/0305-4470/18/15/020

2. De, R., Dutt, R. and Sukhatme, U. (1992) Mapping of Shape Invariant Potentials under Point Canonical Transformations. Journal of Physics A: Mathematical and General, 25, L843-L850. http://iopscience.iop.org/0305-4470/25/13/013

3. Setare, M.R. and Karimi, E. (2008) Mapping of Shape Invariant Potentials by the Point Canonical Transformation. International Journal of Theoretical Physics, 47, 891-897.

4. Cooper, F., Khare, A. and Sukhatme, U. (1995) Supersymmetry and Quantum Mechanics. Physics Reports, 251 267-385. http://dx.doi.org/10.1016/0370-1573(94)00080-M

5. Ho, C.-L. (2009) Simple Unified Derivation and Solution of Coulomb, Eckart and Rosen-Morse Potentials in Prepotential Approach. Annals of Physics, 324, 1095-1104. http://dx.doi.org/10.1016/j.aop.2008.10.004

6. Jia, C.-S., Diao, Y.-F., Li, M., Yang, Q.-B., Sun, L.-T. and Huang, R.-Y. (2004) Mapping of the Five-Parameter Exponential-Type Potential Model into Trigonometric-Type Potentials. Journal of Physics A: Mathematical and General, 37 11275-11284. http://dx.doi.org/10.1088/0305-4470/37/46/012

7. Jia, C.-S., Yi, L.-Z., Zhao, X.-Q., Liu, J.-Y. and Sun, L.-T. (2005) Systematic Study of Exactly Solvable Trigonometric Potentials with Symmetry. Modern Physics Letters A, 20, 1753-1762. http://dx.doi.org/10.1142/S0217732305017081

8. Jia, C.-S., Liu, J.-Y., Sun, Y., He, S. and Sun, L.-T. (2006) A Unified Treatment of Exactly Solvable Trigonometric Potential Models. Physica Scripta, 73, 164-168. http://iopscience.iop.org/1402-4896/73/2/006

9. Cooper, F., Khare A. and Sukhatme, U. (2001) Supersymmetry in Quantum Mechanics. World Scientific Publishing Co Pte Ltd.

10. Ciftci, H., Hall, R.L. and Saad, N. (2003) Asymptotic Iteration Method for Eigenvalue Problems. Journal of Physics A: Mathematical and General, 36, 11807-11816. http://dx.doi.org/10.1088/0305-4470/36/47/008

11. Nikiforov, A.F. and Uvarov, V.B. (1988) Special Functions of Mathematical Physics. Birkhauser, Basel. http://dx.doi.org/10.1007/978-1-4757-1595-8

12. Peña, J.J., Morales, J., García-Martínez, J. and García-Ravelo, J. (2008) Exactly Solvable Quantum Potentials with Special Functions Solutions. International Journal of Quantum Chemistry, 108, 1750-1757. http://dx.doi.org/10.1002/qua.21611

13. Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Wiley and Sons, New York.

14. Granville, W.A., Smith, P.F. and Longley, W.R. (1941) Elements of the Differential and Integral Calculus. Ginn & Co., Boston.

15. Hulthén L. ,et al. (1942)On the Characteristic Solutions of the Schrñdinger Deuteron Equation Arkiv fñr Matematik Astronomi och Fysik A 28, art 5: 1-12.

16. Chen, G. (2004) Shape Invariance and the Supersymmetric WKB Approximation for the Generalized Hulthén Potential. Physica Scripta, 69, 257-259. http://dx.doi.org/10.1238/physica.regular.069a00257

17. Morales, J., Peña, J.J. and Morales-Guzman, J.D. (2000) The Generalized Hulthén Potential. Theoretical Chemistry Accounts, 104, 179-182. http://dx.doi.org/10.1007/s002140000130

18. Ahmed, S.A.S. and Buragohain, L. (2010) Exactly Solved Potentials Generated from the Manning-Rosen Potential Using Extended Transformation Method. Electronic Journal of Theoretical Physics, 7, 145-154.

19. Fatah A.H. ,et al. (2012)Calculation of the Eigenvalues for Wood-Saxon’s Potential by Using Numerov Method Advances in Theoretical and Applied Mechanics 5, 23-31.

20. Berkdemir, C., Berkdemir, A. and Sever, R. (2005) Polynomial Solutions of the Schrñdinger Equation for the Generalized Woods-Saxon Potential. Physical Review C, 72, 027001-1-027001-4.

21. Gñnül, B. and Kñksal, K. (2007) Solutions for a Generalized Woods-Saxon Potential. Physica Scripta, 76, 565-570. http://dx.doi.org/10.1088/0031-8949/76/5/026

22. Falaye, B.J., Oyewumi, K.J., Ibrahim, T.T., Punyasena, M.A. and Onate, C.A. (2013) Bound State Solutions of the Manning-Rosen Potential. Canadian Journal of Physics, 91, 98-104. http://dx.doi.org/10.1139/cjp-2012-0330

23. Nasser, I., Abdelmonem, M.S. and Abdel-Hady, A. (2013) The Manning-Rosen Potentials Using J-Matrix Approach. Molecular Physics, 111, 1-8. http://dx.doi.org/10.1080/00268976.2012.698026

24. Lévai, G. (1989) A Search for Shape-Invariant Solvable Potentials. Journal of Physics A: Mathematical and General, 22, 689-702. http://dx.doi.org/10.1088/0305-4470/22/6/020

25. Peña, J.J., García-Martínez, J., García-Ravelo, J. and Morales, J. (2014) l-State Solutions of Multiparameter Exponential-Type Potentials. Journal of Physics: Conference Series, 490, 012199. http://dx.doi.org/10.1088/1742-6596/490/1/012199

26. Arfken, G. and Weber, H. (2005) Mathematical Methods for Physicists. 6th Edition, Elsevier AP, Boston.

27. Polyani, A.D. and Zaistev, V.F. (2003) Handbook of Exact Solutions for Ordinary Differential Equations. 2nd Edition, Chapman& Hall/CRC, Boca Raton, New York.