## Abstract

Quantum states of a particle subjected to time‐dependent singular potentials in one‐dimension are investigated using invariant operator method and the Nikiforov‐Uvarov method. We consider the case that the system is governed by two singular potentials which are the Coulomb potential and the inverse quadratic potential. An invariant operator that is a function of time has been constructed via a fundamental mechanics. This invariant operator is transformed to a simple one using a unitary operator, which is a time‐independent invariant operator. By solving the Schrödinger equation in the transformed system, analytical forms of exact eigenvalues and eigenfunctions of the invariant operator are evaluated in a simple elegant manner with the help of the Nikiforov‐Uvarov method. Eventually, the full wave functions in the original system (untransformed system) are obtained through an inverse unitary transformation from the wave functions in the transformed system. Quantum characteristics of the system associated with the wave functions are addressed in detail.

### Keywords

- time‐dependent Hamiltonian systems
- singular potentials
- unitary transformation
- wave function
- Schrödinger equation

## 1. Introduction

After a seminal work of Lewis [1] for a quantum time‐dependent harmonic oscillator, much attention has been paid to investigating quantum properties of time‐dependent Hamiltonian systems (TDHSs). Any type of the time‐dependent harmonic oscillator is a good example of TDHSs, and the study of its analytical quantum solutions requires particular mathematical techniques. The research topic of TDHSs has been gradually extended to more complicated systems beyond the one‐dimensional time‐dependent harmonic oscillators which are relatively simple. The analytical forms of quantum wave functions of the time‐dependent coupled oscillators have been reported by several researchers [2–4]. The system associated with a class of time‐dependent singular potentials was investigated [5–10] and some of the corresponding results were applied to study the problem of a two‐ion trap within a binding potential [7]. A TDHS that is described by a Hamiltonian that involves

In this chapter, quantum features of a time‐dependent singular potential system [14] will be investigated. The singular potentials that will be considered here are the combination of the inverse quadratic potential and the Coulomb potential. Singular potentials not only can be applied for describing many actual physical systems but can also serve as mathematical models for quantum field theory and elementary particle theory. The research interest for singular potentials was first shown in a context of relativistic mechanics. Various applications of the singular potentials include interatomic or intermolecular descriptions of a molecular force, the scattering problem of elementary particles, and the interaction of relativistic particles such as quark‐antiquark bound states [14–16].

It was reported by Plesset [17] that there is a difficulty in the derivation of a physically accepted solution for a relativistic Coulomb‐like singular potential. To overcome such difficulty, the invariant operator method together with a unitary transformation method will be used in this chapter. These methods are useful for investigating the mechanics of TDHSs, like the case that will be represented here. For a TDHS, the eigenstates of the invariant operator are the same as the Schrödinger solutions of the system when we neglect the phase factors of the wave functions [18]. The unitary transformation with a suitable operator allows us to manage a certain complicated system in a transformed space that requires relatively simple mathematical treatments for the system.

## 2. Singular potential system

Let us consider a one‐dimensional quantum system that is described by a time‐dependent Hamiltonian of the form

where

As is well known, a useful method for a quantum mechanical treatment of the system in the situation where there exist time‐dependent parameters is to use an invariant operator method [1, 18]. An invariant of the system that is described by a time‐dependent Hamiltonian

As represented in this equation, the whole time derivative of the invariant operator

where

The substitution of Eqs. (1) and (3) into the Liouville‐von Neumann equation represented in Eq. (2) gives the following equations for the coefficients:

By solving these equations, it is possible to determine the time‐dependent coefficients. Hence, as a result of a minor evaluation, we have

where

with an auxiliary condition for the solvability of the system, which is that the time dependence of

Now, notice that Eq. (3), with the coefficients given in Eqs. (5–8), is the exact invariant operator. If we express the eigenvalue equation of the invariant operator as

By denoting the wave functions as

Considering the Schrödinger equation, we can easily verify that

Hence, if the eigenstates of the invariant operator,

The strategy of our manipulation for deriving exact quantum solutions of the system is that we transform the operator

For this purpose, let us first perform a unitary transformation of the eigenstates such that

where

The transformation of the invariant operator using this operator can be performed in a straightforward way with Eq. (3),

Here, the transformed invariant operator

If we put

Then, the eigenvalue equation given in Eq. (15) becomes

where

with the condition that

## 3. Spectrum of quantized solutions

In this section, we consider the solvable case that

By comparing this equation with Eq. (A1) in the NU method of Appendix A, we get

For further development of the theory, we introduce a function

where

where

For the polynomial of

with

For the case of

Now, let us equate Eq. (25) with Eq. (26) such that

Then, by inserting the first and the second relations in Eq. (18) into the above equation, we easily confirm that the eigenvalues are given in the form

where

where

Finally, regarding Eq. (A2) in Appendix A for bound states, the eigenfunctions of the invariant

where

the corresponding normalized wave functions are found to be

Because the eigenstates of

There still remains the problem of finding the phases

Then, with the help of Eq. (28), this equation can be easily evaluated and, consequently, we obtain the phase factors in the form

Now, by substituting Eq. (36) into Eq. (34), we find the exact *n*th‐order solutions of the Schrödinger equation associated to the Hamiltonian

Let us see for a particular case that

where

In this formula, we have used

Besides, Eq. (10) becomes

If we choose

within the time interval

Considering the relation given in Eq. (39), we have plotted the phase given in Eq. (36) in Figures 1 and 2 as a function of time. From Figure 1, the increment of

## 4. Conclusion

The invariant operator method and unitary transformation method were used in order to derive the quantum solutions of a time‐dependent singular potential system that is described by the Hamiltonian given in Eq. (1). The quadratic invariant operator of the system has been determined from the use of its definition as shown in Eq. (3). The wave functions that satisfy the Schrödinger equation are given by multiplying the eigenstates

During the derivation of quantum solutions of the system, no approximation or perturbation methods were used. In fact, the merit of the invariant operator method for investigating quantization problem of TDHSs is that the corresponding quantum results are exact [3, 4]. Several methods for numerical treatment of time‐dependent Schrödinger equations are known. If we enumerate some of them, they are the finite difference time domain (FDTD) method [26–31], the discretization method that takes advantage of the asymptotic behavior correspondence (ABC) [32, 33], and the discrete local discontinuous Galerkin method [34]. In particular, the FDTD method has been widely applied to obtain numerical solutions of mechanical problems of dynamical systems including Maxwell‐Schrödinger equations for electromagnetic fields [30, 31]. If the methods for deriving numerical solutions of the Schrödinger equation for singular potential systems would be known in the future, it will be possible to compare our results developed in this chapter with them, leading to deepen the knowledge on quantum characteristics of relevant systems.

In this appendix, we introduce a useful method for solving Eq. (17) in the text, which is known as the NU method. This is useful for deriving the solutions of the Schrödinger‐like second‐order differential equations that play central roles in studying many important problems of theoretical physics. We first start from an appropriate coordinate transformation

where

where

where

Notice that the derivative of

In terms of the weight function

the hypergeometric‐type function

where

The relationship between

where