Exact wavefunctions for a time-dependent Coulomb potential

Abstract

The one-dimensional Schrödinger equation associated with a time-dependent Coulomb potential is studied. The invariant operator method (Lewis and Riesenfeld) and unitary transformation approach are employed to derive quantum solutions of the system. We obtain an ordinary second-order differential equation whose analytical exact solution has been unknown. It is confirmed that the form of this equation is similar to the radial Schrödinger equation for the hydrogen atom in a (arbitrary) strong magnetic field. The qualitative properties for the eigenstates spectrum are described separately for the different values of the parameter ω0 appearing in the x2 term, x being the position, i.e., ω0 > 0, ω0 < 0 and ω0 = 0. For the ω0 = 0 case, the eigenvalue equation of invariant operator reduces to a solvable form and, consequently, we have provided exact eigenstates of the time-dependent Hamiltonian system.

Time-independent approximations for time-dependent optical potentials

“…The Lewis and Riesenfeld method of invariants [18] is one of them. This scheme has been applied successfully to many models and scenarios, such as the harmonic oscillator with time-dependent mass and frequency in one [19] and two dimensions [20,21], the damped harmonic oscillator [22], a time-dependent Coulomb potential [23], a time-dependent Hamiltonians given in terms of linear combinations of SU(1,1) and SU(2) generators [24,25,26], in the inverse construction of time-dependent Hamiltonian [27,28], for systems on noncommutative spaces in time-dependent backgrounds [29], time-dependent non-Hermitian Hamiltonian systems [30,31,32,33] and other specific systems.…”

Quantization of time-dependent singular potential systems in one-dimension by using the Nikiforov-Uvarov method

“…A TDHS whose Hamiltonian involves a (1/q)p + p(1/q) term, which gives the radial equation for a system obeying a Hydrogen-like force, was investigated [6,7]. Quantum solutions for other types of TDHSs have also become topics of active research in this context [8][9][10][11][12].…”

Quantum features of molecular interactions associated with time-dependent non-central potentials

“…Quantum problem of molecular interactions such as the Coulomb-type and harmonic interactions can be described in terms of central [36][37][38] or noncentral [7,[39][40][41] time-dependent Hamiltonians. Attractive or repulsive forces between various molecules including non-bonded atoms are responsible for a specific formation of molecular structure and its change.…”

“…To attain accurate results when we study molecular systems, it is necessary to introduce an exact Hamiltonian that yields actual time dependence of molecule behaviors. If we consider the convention that time-varying factors have usually been neglected on most studies of dynamical systems, the recent tendency [3,[36][37][38][39][40][41][42][43][44][45][46][47][48] for considering time dependence of physical parameters in this field may open up a new trend in the analysis of molecular interactions.…”