**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

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 (1/) + (1/) term, which in fact is necessary for the description of radial equation for a central force system, was also studied [11– 13].

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 mathemat‐ ical treatments for the system.
