Advances in Mathematical Physics

Volume 2016 (2016), Article ID 3693572, 6 pages

http://dx.doi.org/10.1155/2016/3693572

## Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

^{1}Department of Material Sciences, Faculty of Science, University of M’sila, 28000 M’sila, Algeria^{2}Faculty of Technology, University of Ferhat Abbas Sétif-1, 19000 Sétif, Algeria^{3}Laboratory of Optoelectronics and Compounds (LOC), Department of Physics, Faculty of Science, University of Ferhat Abbas Sétif-1, 19000 Sétif, Algeria^{4}Department of Radiologic Technology, Daegu Health College, Daegu 41453, Republic of Korea

Received 27 October 2015; Accepted 28 March 2016

Academic Editor: Yao-Zhong Zhang

Copyright © 2016 Salim Medjber et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

#### 1. Introduction

The quantum dynamics of central potential systems that is originally developed in the context of stationary Hamiltonian systems [1–3] can be extended to more complicated systems that involve time-dependent parameters. Time-dependent Hamiltonians are useful for describing nonstationary quantum dynamical systems which are ubiquitous in the physics world. Among many time-dependent Hamiltonian systems, central potential systems with time-dependent parameters have attracted considerable interest in the literature because they provide solvable models for various physical systems. As a consequence, active researches have been carried out in this context, accompanying lots of reports so far [4–16]. For instance, a dissipative harmonic oscillator perturbed by an inverse quadratic potential described with time-dependent parameters has been studied [7]. Quantum properties of harmonic motion of a charged particle that have time-dependent effective mass under time-dependent singular potential have been investigated [16]. However, the topics of most of these researches were limited to the case of one- or two-dimensional systems due to the difficulty in developing mathematical procedures for three- or higher-dimensional systems that involve time-dependent parameters.

For a full understanding of profound physical properties of such systems, quantum analytical solutions of corresponding Schrödinger equations, as well as classical solutions, are necessary. Various mathematical tools and methods for solving Schrödinger equations of complicated time-dependent Hamiltonian systems have been suggested and developed up to now. They are the invariant method [17], the method based on path integral formulation [18], the trial function method [19], the canonical/unitary transformation approach [20], and so on. In particular, the invariant method which was devised by Lewis and Riesenfeld [21, 22] is very useful for unfolding fundamental theories of quantum dynamics for the case where the Hamiltonian of the given system explicitly depends on time. Among diverse nonstationary quantum systems, the time-dependent quantum system with a harmonic plus an inverse harmonic potential in one or two dimensions is one of the interesting problems that has attracted permanent interest on account of its emerging applications in actual physical systems [9, 16, 23, 24]. For instance, exact quantum solutions for many body systems [2, 3] and polyatomic molecules [23] and the quantum behavior of a charged particle in electromagnetic fields [24] can be investigated through this model. These studies for nonstationary quantum systems may be possibly extended to more complicated systems, such as the time-dependent quantum systems with spherical harmonic plus an inverse harmonic potential in three dimensions.

Stimulated by this trend for recent researches, we investigate exact quantum solutions of the time-dependent harmonic potential system perturbed by an inverse harmonic potential in three dimensions in the present work. To this end, several particular mathematical procedures will be employed in order to overcome the mathematical difficulty for unfolding quantum theory under the consideration of time-dependent parameters in three dimensions. Typically, the Schrödinger solutions or wave functions for a time-dependent Hamiltonian system are represented in terms of the eigenfunctions of an invariant operator of the system. Such eigenfunctions are obtained by solving the eigenvalue equation of the invariant operator. Because the invariant operator is represented in terms of some time functions, the direct evaluation of the eigenvalue equation is somewhat difficult. This difficulty can be resolved by introducing the unitary transformation method and the Nikiforov-Uvarov (NU) method together. The eigenvalue equation can be simplified by unitary transformation with an appropriate unitary operator. The NU method enables us to solve such arbitrary-type second-order differential equations.

#### 2. Hamiltonian and Invariant Operator

As is well known, the study of quantum dynamical properties of a system requires Schrödinger solutions as a preliminary step. While quantum features of time-dependent harmonic plus an inverse harmonic potential in one or two dimensions are extensively studied in the literature, the same problems in three dimensions are not extensively studied yet as far as we know. As a solvable dynamical system in three dimensions, we consider a time-dependent harmonic plus an inverse harmonic potential with a linearly increasing mass [25, 26]. In this case, the Hamiltonian has the formwhere , , , , and are real constants. Notice that the linearly increasing mass appears in several dynamical systems such as the expanding shells with mass accretion in a plasma [25] and the mass-accreting Algol-type eclipsing binaries [26]. Simple classical modeling of a system with a linearly increasing mass has been carried out by Kunkel and Harrington [27]. A pail-rain model of a linearly mass-accreting system is described in [28, 29]. Here, we consider only a slowly time-varying function for convenience, which is viable under the levy of a condition .

The momentum operator in three dimensions is defined aswhereand is the total angular momentum of the formNotice that the position and the momentum operators satisfy the commutation relation .

To investigate quantum features of the system, it is necessary to solve the Schrödinger equation in the first place. Let us express the Schrödinger equation for this system in the formBecause this equation is a three-dimensional form that involves time-dependent parameters, it may not be an easy task to solve it. Hence, special mathematical techniques are necessary. To solve this equation, we first use the dynamical invariant method devised for evaluating quantum solutions of time-dependent Hamiltonian systems by Lewis and Riesenfeld [22]. This method can be developed through an establishment of a nontrivial Hermitian operator which satisfies the equationIn case that such invariant exists for a system and if it does not contain time derivative operators, the solutions of the Schrödinger equation (5) take the formwhere is an eigenfunction of that entails a time-independent eigenvalue and is a phase of the wave function that can be derived from

According to the Lewis-Riesenfeld method which is available for deriving quantum solutions, it is necessary to construct an invariant operator of the system. By solving (6), we have the following invariant operator [30]: where is a unitary operator of the formand is another invariant quantity that is given byHere, is the solution of an auxiliary equation which iswhere and

Now, we try to solve (12). The transformation leads to the Yermakov’s differential equation [31] which isThrough the aid of the solution of this equation, we obtain the solution for to bewhere and are arbitrary real constants, and and are given bywhere and are arbitrary constants.

#### 3. Eigenvalues and Eigenfunctions of the Invariant

Let us express the eigenvalue equation for the invariant operator aswhere is a time-independent eigenvalue. Because the wave functions for the time-dependent Hamiltonian system are represented in terms of the eigenfunction of the invariant operator, we now solve this equation in order to derive the eigenfunction . Through the use of the unitary relation with , (17) can be mapped, in terms of given in (11), intowhere is the eigenfunction of . Notice that the relation between and is given by

Using (11), the eigenvalue equation given in (18) can be written aswhere , , , and , , are the Cartesian coordinates.

By taking the function to beand substituting this into (20), the differential equations for the radial and angular parts are separated aswhere is the orbital momentum number. The solution of (23) is just the spherical harmonics:By taking the transformation , the radial equation leads to the following differential equation:We use the NU method [32] in order to solve this equation in the next section.

#### 4. The Nikiforov-Uvarov Method

The NU method was proposed to treat complicated second-order differential equations in a convenient way by reducing it to a hypergeometric-type equation through an appropriate coordinate transformation . To illustrate the general theory of this, let us consider a differential equation which iswhere and are polynomials that are allowed up to the second order of and is a polynomial allowed up to the first order of . If we take the factorizationwhere is defined through a logarithmic derivative of the formthen, according to [33], the function required for this method is given bywherewhile the other variable is the hypergeometric-type function whose polynomial solutions are given by the Rodrigues relation:where are normalizing constants and the weight function must satisfy the conditionwith

For the case of a general form of the Schrödinger equation, which is given bythe value given in (30) can be represented as [33]wherewithAgain on the basis of the evaluation of [33], the general form of solution for is given bywhere while, for , the general solution is expressed aswhere

By comparing (34) with (25), we get At this stage, we should determine which kind of wave solutions is physically acceptable among the two given in (38) and (40), respectively. Because the wave solutions not only should be finite at but also should converge to zero at , we can confirm that (38) is the only physically suitable kind of wave solutions among them. Then, after a minor evaluation regarding this, the solutions of (25) are given in terms of generalized Laguerre polynomials [34] such thatwhere , and the eigenvalue of the invariant is expressed in the formThe complete normalized eigenfunctions of the invariant can be constructed by taking account of normalizing (21) with the use of the following relation [35]:Then, by gathering (19), (24), and (43) and introducing an appropriate factor into these functions for the purpose of ensuring the normalization condition, we obtainwhere the normalization constants are given byHence, the exact solutions of the original Schrödinger equation given in (5), which is associated with the central harmonic plus an inverse harmonic potential, can now be obtained in the form where the global phases are given byThese wave functions are useful for investigating quantum properties of the system. For example, the uncertainty relations between a Cartesian coordinate and its conjugate momentum are obtained through the use of these wave functions, such thatThus, the ground state uncertainty product under the time-dependent situation is . If the time-dependence of the Hamiltonian disappears, the wave functions reduce to the familiar ones known for the time-independent harmonic oscillator perturbed by the inverse harmonic potential and the uncertainty relations become as expected.

#### 5. Conclusion

Quantum solutions of the central time-dependent harmonic plus an inverse harmonic potential system in three dimensions were investigated using several mathematical techniques. According to the theory of Lewis and Risenfeld, the wave functions of the system are represented in terms of the eigenfunctions of the invariant operator. An exact invariant is established on the basis of its fundamental definition via the use of its unitary relation with the unitary operator given in (10). The invariant operator is represented in terms of a time function , while its overall time derivative results in zero. The eigenvalue equation for has been written in terms of a second-order differential equation. To derive the solutions of such equation, the unitary transformation approach and the NU method are used together. The NU method enabled us to reduce the second-order differential equation to the hypergeometric equation. From these procedures, we have obtained the eigenfunctions and they are represented in terms of the Laguerre polynomials. The eigenfunctions of the invariant operator derived in this way are necessary for representing the Schrödinger solutions, that is, the wave functions. The finally obtained wave functions are expressed in terms of these eigenfunctions and time-dependent phase factors, given in (49), that correspond to global phases.

The difference of this research from the previous one, performed by Ferkous et al. [30], is that we used the NU method in order to derive exact quantum solutions of the system whereas Ferkous et al. did not. Quantum wave functions that we derived in this work were represented in terms of spherical harmonics , while those of Ferkous et al. were represented in terms of the Jacobi polynomials. Our results for the wave functions given in (48) are useful for investigating diverse fundamental quantum properties of the system. For example, they can be used to derive fluctuations of canonical variables, uncertainty relations, energy eigenvalues, Wigner functions, and so on. For a particular case where the time-dependence of parameters disappears, the wave functions (48) reduce to the familiar ones that are well known in quantum mechanics of the stationary system.

#### Competing Interests

The authors declare that they have no competing interests.

#### Acknowledgments

This research was supported by the Algerian Ministry of Higher Education and Scientific Research (Grant no. CNEPRU/D01220120010) and was supported by the Basic Science Research Program of the year 2015 through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant no. NRF-2013R1A1A2062907).

#### References

- F. Calogero, “Solution of a three-body problem in one dimension,”
*Journal of Mathematical Physics*, vol. 10, no. 12, pp. 2191–2196, 1969. View at Publisher · View at Google Scholar · View at Scopus - F. Calogero, “Solution of the one-dimensional
*N*-body problems with quadratic and/or inversely quadratic pair potentials,”*Journal of Mathematical Physics*, vol. 12, no. 3, pp. 419–436, 1971. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Sutherland, “Quantum many-body problem in one dimension: ground state,”
*Journal of Mathematical Physics*, vol. 12, no. 2, pp. 246–250, 1971. View at Publisher · View at Google Scholar · View at Scopus - V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, “Green function and excitation of a singular oscillator,”
*Physics Letters A*, vol. 39, no. 5, pp. 377–378, 1972. View at Google Scholar · View at MathSciNet - V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, “Excitations of a nonstationary asymmetrical singular oscillator,”
*Il Nuovo Cimento B*, vol. 24, no. 1, pp. 46–56, 1974. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. A. Markov, Ed.,
*Invariants and the Evolution of Nonstationary Quantum Systems*, Nova Science, New York, NY, USA, 1989. - J. R. Choi and B. H. Kweon, “Operator method for a nonconservative harmonic oscillator with and without singular perturbation,”
*International Journal of Modern Physics B*, vol. 16, no. 31, pp. 4733–4742, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - I. A. Pedrosa, G. P. Serra, and I. Guedes, “Wave functions of a time-dependent harmonic oscillator with and without a singular perturbation,”
*Physical Review A*, vol. 56, no. 5, pp. 4300–4303, 1997. View at Publisher · View at Google Scholar · View at Scopus - V. V. Dodonov, V. I. Man'ko, and L. Rosa, “Quantum singular oscillator as a model of a two-ion trap: an amplification of transition probabilities due to small-time variations of the binding potential,”
*Physical Review A*, vol. 57, no. 4, pp. 2851–2858, 1998. View at Publisher · View at Google Scholar · View at Scopus - D. A. Trifonov, “Exact solutions for the general nonstationary oscillator with a singular perturbation,”
*Journal of Physics A: Mathematical and General*, vol. 32, no. 19, pp. 3649–3661, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J.-R. Choi, “Exact quantum state for the time-dependent repulsive oscillator with a driving force and with a singular perturbation,”
*Physica Scripta*, vol. 70, no. 5, pp. 271–275, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - C.-I. Um, K.-H. Yeon, and T. F. George, “The quantum damped harmonic oscillator,”
*Physics Reports*, vol. 362, no. 2-3, pp. 63–192, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J.-R. Choi, “Quantum states with continuous spectrum for a general time-dependent oscillator,”
*Pramana—Journal of Physics*, vol. 65, no. 2, pp. 165–176, 2005. View at Publisher · View at Google Scholar · View at Scopus - J.-R. Choi, “Thermal state of general time-dependent harmonic oscillator with a singular perturbation,”
*Physica Scripta*, vol. 68, no. 1, pp. 36–40, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. R. Choi, “Exact wave functions of time-dependent Hamiltonian systems involving quadratic, inverse quadratic, and $(1/\widehat{x})\widehat{p}+\widehat{p}(1/\widehat{x})$ terms,”
*International Journal of Theoretical Physics*, vol. 42, no. 4, pp. 853–861, 2003. View at Publisher · View at Google Scholar - S. Menouar and J. R. Choi, “A hybrid approach for quantizing complicated motion of a charged particle in time-varying magnetic field,”
*Annals of Physics*, vol. 353, pp. 307–316, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - G. Lu, W. Hai, and L. Cai, “Near coherent states of an electron in a time-dependent linear potential,”
*Physics Letters A*, vol. 357, no. 3, pp. 181–187, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. S. Child,
*Semiclassical Mechanics with Molecular Applications*, Oxford University Press, Oxford, UK, 1991. - O. Éboli, R. Jackiw, and S.-Y. Pi, “Quantum fields out of thermal equilibrium,”
*Physical Review D*, vol. 37, no. 12, pp. 3557–3581, 1988. View at Publisher · View at Google Scholar · View at Scopus - J. Bauer, “Comment on ‘Solution of the Schrödinger equation for the time-dependent linear potential’,”
*Physical Review A*, vol. 65, no. 3, Article ID 036101, 2002. View at Google Scholar · View at Scopus - H. R. Lewis Jr., “Classical and quantum systems with time-dependent harmonic-oscillator-type hamiltonians,”
*Physical Review Letters*, vol. 18, no. 13, pp. 510–512, 1967. View at Publisher · View at Google Scholar · View at Scopus - H. R. Lewis Jr. and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,”
*Journal of Mathematical Physics*, vol. 10, no. 8, pp. 1458–1473, 1969. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. M. Chumakov, V. V. Dodonov, and V. I. Man'ko, “Correlation functions of the nonstationary quantum singular oscillator,”
*Journal of Physics A: Mathematical and General*, vol. 19, no. 16, pp. 3229–3239, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Yuce, “Exact solution of the time-dependent harmonic plus an inverse harmonic potential with a time-dependent electromagnetic field,”
*Annals of Physics*, vol. 308, no. 2, pp. 599–604, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. V. Bulanov and F. Pegoraro, “Stability of a mass accreting shell expanding in a plasma,”
*Physical Review E*, vol. 65, no. 6, Article ID 066405, 2002. View at Publisher · View at Google Scholar · View at Scopus - D. Mkrtichian, S.-L. Kim, A. V. Kusakin et al., “A search for pulsating, mass-accreting components in algol-type eclipsing binaries,” in
*Close Binaries in the 21st Century: New Opportunities and Challenges*, pp. 167–169, Springer, Amsterdam, The Netherlands, 2006. View at Google Scholar - W. Kunkel and R. Harrington, “Modeling the motion of an increasing mass system,”
*The Physics Teacher*, vol. 48, no. 4, pp. 243–245, 2010. View at Publisher · View at Google Scholar - H. Dekker, “Classical and quantum mechanics of the damped harmonic oscillator,”
*Physics Reports*, vol. 80, no. 1, pp. 1–110, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. R. Ray, “Dissipation and quantum theory,”
*Lettere al Nuovo Cimento*, vol. 25, no. 2, pp. 47–50, 1979. View at Publisher · View at Google Scholar · View at Scopus - N. Ferkous, A. Bounames, and M. Maamache, “Time-dependent Schrödinger equation with non-central potentials,”
*Physica Scripta*, vol. 88, no. 3, Article ID 035001, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - V. F. Zaitsev and A. D. Polyanin,
*Handbook of Exact Solution for Ordinary Differential Equations*, CRC Press, Boca Raton, Fla, USA, 2002. - A. F. Nikiforov and V. B. Uvarov,
*Special Function of Mathematical Physics*, Birkhäuser, Basel, Switzerland, 1988. - C. Tezcan and R. Sever, “A general approach for the exact solution of the Schrödinger equation,”
*International Journal of Theoretical Physics*, vol. 48, no. 2, pp. 337–350, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. X. Wang and D. R. Guo,
*Special Functions*, World Scientific, River Edge, NJ, USA, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - I. S. Gradshteyn and I. M. Ryzhik,
*Table of Integrals, Series, and Products*, Academic Press, San Diego, Calif, USA, 8th edition, 2015.