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Edited by: Francisco Gomez, Centro Nacional de Investigación y Desarrollo Tecnológico, Mexico

Reviewed by: Silvestro Fassari, Università degli Studi Guglielmo Marconi, Italy; Fabio Rinaldi, Università degli Studi Guglielmo Marconi, Italy

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We explore how to apply perturbation theory to complicated time-dependent Hamiltonian systems that involve complex potentials. To do this, we introduce a generalized time-dependent oscillator to which the complex potentials are connected through a weak coupling strength. We regard the complex potentials in the Hamiltonian as the perturbed terms. Quantum characteristics of the system, such as wave functions and expectation values of the Hamiltonian, are investigated on the basis of the perturbation theory. We apply our theory to particular systems with explicit choices of time-dependent parameters. Through such applications, the time behavior of the quantum wave packets and the spectrum of expectation values of the Hamiltonian are analyzed in detail. We confirm that the imaginary parts of expectation values of the Hamiltonian are not zero but very small, whereas the real parts deviate slightly from those of the unperturbed system.

In the case that we are unable to derive exact quantum solutions for a perturbed system, perturbation theory is a useful tool for obtaining approximate quantum solutions. When we apply perturbation theory, it should be supposed that the scale of the perturbed terms in the potential is relatively small. The perturbation theory is valid only when the quantum solutions of the system in which the perturbation potentials have been removed are exactly known or derivable. The perturbation theory was originally developed for Hermitian systems in which the potential is real. Hence, in conventional quantum mechanics, the perturbation theory has, in large, been developed for the systems in which the potentials are real Hermitian that allows only the spectrum of real expectation values for quantum observables. Whereas the eigenvalue problem and the effects of perturbations on stationary Hamiltonian systems are well known in non-relativistic quantum physics, the perturbation techniques for the time-dependent Hamiltonian systems (TDHSs) with complex potentials have been investigated much less. This is partly due to the difficulty of mathematical procedures when we apply perturbation techniques in TDHSs.

Time-

On account of recent attention to the quantum problem of physical systems characterized by complex potentials, the necessity for the extension of perturbation theory to complex potential systems has gradually emerged. The data in some of the processes of elastic scattering such as nucleus-nucleus scattering [

According to such a research trend, we will investigate, in this work, how to apply perturbation theory to complex potential systems. We will adopt a general Hamiltonian that corresponds to a time-dependent harmonic oscillator to which complex potentials are coupled as perturbation terms. We will consider adiabatic evolution [

In this section, we outline the quantum structure for a TDHS which involves complex potentials as perturbations to the system. The equation of motion for lots of actual physical models in quantum mechanics are described not only by linear terms which allow us to have exact mathematical solutions, but also by nonlinear terms associated with interactions of the system with various subsystems. If the coupling parameters of the nonlinear parts are sufficiently small compared to others, we can treat them as perturbation terms. Regarding this, we consider the system of which Hamiltonian is represented as

where Ĥ is a time-dependent quadratic Hamiltonian and Ĥ_{p} is a perturbing Hamiltonian, while ϵ is a small coupling parameter. In this case, the perturbation theory is available only when the second term on the right hand side of Equation (1) is very small relative to the first term. We assume that Ĥ and Ĥ_{p} are given by

where _{R, j}(_{I, l}(_{0} is the mass and ω(_{0} as the initial mass. Notice that the additional term Ĥ_{p} in the Hamiltonian makes the system be an anharmonic oscillator.

The Schrödinger solutions of the system are different depending on the choice of the time functions in the Hamiltonian. If we consider a discrete spectrum of solutions, the Schrödinger equation for the overall Hamiltonian

To apply the perturbation theory at this stage, it is necessary to know complete quantum solutions associated with the unperturbed Hamiltonian Ĥ as mentioned in the introductory part. For this reason, we first derive exact solutions of the Schrödinger equation for Ĥ from

Let us take a moment to review the theory for solving this equation exactly. Because Ĥ involves time functions, it is very difficult to solve this equation relying on the conventional method. For such TDHSs, it is known based on the invariant operator theory that the Schrödinger solutions are expressed in terms of the solutions of a classical equation associated with the system [

where

Then, we can define the annihilation operator of the system in terms of ρ(

where Ω is an arbitrary real positive constant, and

Of course, the Hermitian adjoint of Equation (8), â^{†}, is the creation operator. These ladder operators obey the commutation relation [â, â^{†}] = 1. The time evolution of â(^{−i[η(t)−η(0)]}, where

It is also possible to define an invariant operator using the ladder operators, such that [

We can check that the direct differentiation of Î with respect to time results in zero, which means that Î(_{n}(

This is analogous to the time-_{n}(

where

while _{n}(

We can easily check that the ladder operators yield

The quantum theory for a TDHS represented above will be used in the subsequent section in order to develop the perturbation theory of the system associated with the complex potentials.

Although perturbation theory does not give the spectrum of exact analytical quantum solutions for a dynamical system, it enables us to solve difficult problems in quantum mechanics from a series of routine calculations. If we regard that we cannot derive exact Schrödinger solutions in many cases of quantum mechanical problems, perturbation theory is quite useful in quantum mechanics. We will show how to manage perturbation theory for the case where the Hamiltonian involves time-dependent complex potentials on the basis of the associated theory represented in the previous section.

By expressing Equation (8) and its Hermitian adjoint â^{†} inversely, we obtain the representation of canonical variables ^{†}. Then we can easily derive the expectation values of Ĥ using Equations (13), (15), and (16), leading to

where

The purpose of using perturbation techniques is to find the effects of small perturbed potentials on the whole system. In general, perturbation expansion gives a power series representation of a resultant quantity with respect to the perturbing parameter ϵ. The results of a calculation considering perturbations agree quite well with experimental data, but entails an infinite number of minor terms. However, the higher orders of such minor terms can, in general, be negligible from the perturbation corrections because their numeric scales are small enough.

Our theory is based on time-

where 〈_{n}〉 are eigenstates and θ_{n}(

where 〈_{n}〉 = ϕ_{n}(

with

By perturbation theory, the EVH are represented as

where

Up until now, we have developed perturbation theory for the general time-dependent Hamiltonian system which involves complex potentials. In order to see the applicability of our theory to particular systems, let us consider the case where _{R, 1}(_{I, 3}(_{R, j}(_{I, l}(

Now, from Equations (20)–(22), the corresponding wave functions are derived to be

where,

with

Because

On the other hand, the EVH become

If large-scale perturbation expansions regarding higher order terms are required in the solutions, it is necessary to take the aid of computer algebra [

Now let's turn to a special case where the unperturbed Hamiltonian corresponds to the Caldirola-Kanai oscillator [

which can be obtained from the choice of time functions as ^{−1}(^{βt}, _{0}, where β and ω_{0} are real positive constants. In addition, we choose

where

provided that _{1} and _{2} are arbitrary real positive constants.

We have depicted the probability density _{1} = _{2} = 1 in this figure, ρ(

The probability densities ^{st} and 2^{nd} in the figure legends mean that the corresponding figure has been plotted with the consideration up to the first order and up to the second order of ϵ, respectively. The parameters are chosen as ℏ = 1, β = 0.3, ω_{0} = 1, _{0} = 1, _{1} = _{2} = 1, ϵ = 0.0035, and

_{0} = 1, _{0} = 1, _{1} = _{2} = 1, ϵ = 0.0001, and

Let us show that the results, Equations (30)–(35), become well-known ones for a particular case. For this purpose, we put β = 0 from Equation (36) and

where δ = ϵ_{1} = _{2} = 1, we have ρ = 1, _{0},

where

with _{1}≠1 and/or _{2}≠1, the quantum solutions for the unperturbed Hamiltonian Ĥ are somewhat different from the standard ones and the result given in Equation (41) becomes different; however, the quantum solutions for Ĥ still satisfy the Schrödinger equation in that case and the corresponding solutions approximated by the perturbation theory for the entire Hamiltonian are also allowed. Although we have chosen a simple reduced Hamiltonian, Equation (39), in order to show that our results become well-known ones for a particular case, the quantum solutions of the system described by this Hamiltonian can also be derived completely without using perturbation theory.

Now let us see another special case where the unperturbed Hamiltonian is given by

This corresponds to the case where the time functions are given by

where

with

provided that _{1, 0} and ρ_{2, 0} are real positive constants, _{ν} and _{ν} are Bessel functions of the first and second kind, respectively, while ν = 1/6.

In _{R, 1}(

_{0} = 1, _{0} = 1, _{1, 0} = ρ_{2, 0} = 1, α = 1,

The properties of the quantum states for perturbed Hamiltonian systems involving complex potentials were investigated. We have considered time-dependence of the imaginary part of the perturbing potentials as well as that of the real part, which are both coupled to a generalized harmonic oscillator through a weak coupling constant. The solutions of the Schrödinger equation of the system in the Fock state have been obtained using the perturbation theory. The effects of the perturbation on time behavior of the system have been analyzed. The perturbation corrections on the wave functions and on the EVH have been analytically investigated. Because we have considered complicated time-dependent Hamiltonian systems as a generalization, the EVH for each particular case is nonconservative. However, when we remove the time dependence of the Hamiltonian, our quantum solutions reduce to stationary states where the corresponding energy spectrum is conservative.

We have applied our theory to particular cases, such as the perturbed systems of which the unperturbed part is described by the Caldirola-Kanai Hamiltonian, and by a potential in which the angular frequency increases in proportion to the square of time. We see that the deviation of the probability densities and the EVH from those of the unperturbed system is, in large, not significant due to the weakness of the coupling. However, the deviation of the probability densities from those of the unperturbed system becomes large with the lapse of time. The imaginary part of the EVH is very small, but not zero, whereas the real part deviates slightly from that of the unperturbed system. In the case where the unperturbed part of the Hamiltonian corresponds to the Caldirola-Kanai oscillator, such deviation increases as the damping factor β grows. We have confirmed that the EVH for the system whose angular frequency is proportional to the square of

The development of the perturbation theory represented here may also be possibly applied to diverse different quantum systems which contain time-dependent complex perturbation potentials beyond those studied in the text. Some insights for characterizing complex potential systems are necessary in the light of the fact that a complex potential is receiving due attention in analyzing actual physical states, such as elastic scattering processes [

The datasets generated for this study are available on request to the corresponding author.

The author confirms being the sole contributor of this work and has approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The eigenstates of the invariant operator for the system are obtained by directly solving Equation (12) or by using the properties of Equations (15) and (16) [

_{2} given in Equation (22) is composed of two terms, let us express

where

The second term _{n}|Ĥ_{p}|ϕ_{n}〉 which are involved in it are zero. On the other hand, a rigorous evaluation of the first term using Equation (23) gives