## Abstract

A Gaussian wave packet of the inverted oscillator is investigated using the invariant operator method together with the unitary transformation method. A simple wave packet directly derived from the eigenstates of the invariant operator of the system corresponds to a plane wave that is fully delocalized. However, we can construct a weighted wave packet in terms of such plane waves, which corresponds to a Gaussian wave. This wave packet is associated with the generalized coherent state, which can be crucially utilized for investigating the classical limit of quantum wave mechanics. Various quantum properties of the system, such as fluctuations of the canonical variables, the uncertainty product, and the motion of the wave packet or quantum particle, are analyzed by means of this wave packet. We have confirmed that the time behavior of such a wave packet is very similar to the counterpart classical state. The wave packet runs away from the origin in the positive or negative direction in the 1D coordinate depending on the condition of the initial state. We have confirmed that this wave packet not only moves acceleratively but also spreads out during its propagation.

## Introduction

Quantum solutions can only be exactly derived for a very few of the numerous mechanical systems. The inverted oscillator, the potential of which exerts a repulsive force on the particle, belongs to a rare type of system that is solvable from both the classical and quantum-mechanical points of view. This oscillator can be obtained by changing the normal angular frequency of the simple harmonic oscillator to a purely imaginary one: $$\omega \rightarrow i\omega $$. For more than a decade, the quantum characteristics of the inverted oscillator have attracted noticeable interest from the theoretical physics community and there have been many associated reports [1–9]. The wave packets of the inverted oscillator do not oscillate over time and their physical properties are very different from those of the simple harmonic oscillator. It is well known that the classical solutions of the inverted oscillator in phase space diverge exponentially over time. The energy spectrum of the inverted oscillator is not discrete but continuous, like a free particle. Moreover, zero-point energy does not exist in this system [1] because the system is unbound.

As well as the ordinary harmonic oscillator, the inverted oscillator can also be applied to various physical systems. For example, it can be used to describe the physical mechanism of matter-wave bright solitons [10], a particular scalar field model in the inflationary universe [11], superfluorescence phenomena [12], and a cosmological model with a negative potential [13]. Tunneling effects and sojourn time in the unstable equilibrium position for the wave packet in the inverted oscillator have been studied by Barton [2]. Combescure and Combescure proposed a method for evaluating the quantum fidelity of the inverted oscillator with a singular perturbation, which is an important concept in the field of quantum computation and quantum chaos [14]. The factorization method with supersymmetric quantum mechanics is used by Bermudez and Fernández in order to study the general algebraic structure of the inverted oscillator [15].

Stimulated by these trends in this research field, we analyze, in this paper, the time behavior of displaced wave packets accompanying the generalized coherent state (GCS) for the inverted oscillator. Because the wave packets of the inverted oscillator are unbound, its mathematical treatment is somewhat difficult and cumbersome. Hence, it is necessary to find mathematical tools that can be potentially applicable to this situation. A convenient mathematical tool for solving the Schrödinger equation in this case is the invariant operator method [16]. There are linear and quadratic invariants for a dynamical system.

We argue that the general solutions of the Schrödinger equation for the inverted oscillator can be derived by utilizing a linear Hermitian invariant operator (LHIO). In this work, we reexamine and interpret the general solution of the Schrödinger equation for the inverted oscillator on the basis of the Lewis–Riesenfeld framework [16]. Not only for a free particle [17] but also for the inverted oscillator [5], Bagrov et al. proposed non-Hermitian time-dependent linear invariants expressed in terms of momentum $$p$$ and position operator $$q$$. We shall show that the general quantum solutions of the inverted oscillator can be formulated through the use of the LHIO instead of such non-Hermitian linear invariants. Notice that the previously adopted methods for the procedure of obtaining quantum solutions lead to “a constraint which is that the non-Hermiticity assumption should be taken in order to get physical solutions of the system” (see, e.g., Refs. [5, 17]). However, this assumption is actually unnecessary.

In the context of the quantum inverted harmonic oscillator, the eigenstates of the time-dependent linear Hermitian invariant are associated with the plane wave solutions of the system and are not normalizable. The plane waves relevant to such states are fully delocalized. However, it is customary to expand any normalizable solutions of a time-dependent Schrödinger equation for an unbound system in terms of plane waves, using the Fourier transform. This means that we can build general quantum solutions that correspond to localized wave packets through the manipulation of the plane wave solutions. Gaussian wave packets, known as typically localized wave packets, are the simplest examples of coherent states. The well known coherent states for the case of the simple harmonic oscillator are those that were originally obtained by Schrödinger [18]. The expectation values of the position and momentum operators for such Gaussian wave packets are the same as those of the corresponding classical solutions.

The LHIO that we will employ in this work is useful for studying coherent states for diverse dynamical systems [19–21]. The Schrödinger solutions of the inverted oscillator can be represented in terms of the eigenstates of the LHIO. This is the reason why the LHIO plays an important role in this research. In order to simplify the eigenvalue equation of the original LHIO, we will use a unitary transformation method by introducing a suitable unitary operator. This may help us to derive the eigenstates of the invariant operator and the corresponding wave functions.

Meanwhile, a GCS may also be derived by utilizing other kinds of invariant operators. For instance, Bagrov et al. have introduced a non-Hermitian linear invariant operator in order to obtain GCSs of the inverted oscillator considering the separability property of the Hilbert space [5]. Notice that the method of Bagrov et al. requires the imposition of several constraints on the parameters. However, our method, based on LHIO, does not require such constraints.

## The linear invariant and its unitary transformation

In order to analyze the quantum motion of the inverted oscillator, the solution of the Schrödinger equation, $$i\hbar \partial \Psi (q,t)/ \partial t -H \Psi (q,t) =0$$, is necessary. For this system, the Hamiltonian is given by

Hence, it is necessary to employ an alternate method in order to investigate the time behavior of the Gaussian wave packet for the system. Invariant operator methods based on LHIO are useful in this case, because the eigenstates for the LHIO enable us to derive the GCS for a localized wave packet, as mentioned in Sect. 1. We suppose that there exists a linear Hermitian invariant operator $$I(t)$$ for the system, which has the form

The eigenstates $$\varphi _{\lambda }(q,t)$$ of the LHIO are the solutions of the equation

## Preliminary quantum solution

Notice that the eigenvalue equation (10) can be mapped into that of the transformed system, $$I'(t)\varphi _{\lambda }'( q,t) =\lambda \varphi _{\lambda }'(q,t)$$. Using Eq. (13), this can be rewritten as

On the other hand, by substituting Eq. (17) into the Schrödinger equation $$i\hbar \partial \psi _\lambda ( q,t)/{\partial t} -H \psi _\lambda (q,t)=0$$, we have [22–24]

## Generalized coherent state

The general Schrödinger state for a localized wave packet can be described by a linear combination of the solutions given in Eq. (21):

where $$a$$, $$a_{0}$$, and $$I_{0}$$ are positive real constants. For a simple case with $$I_0 =0$$ and $$a_0 = 0$$, this reduces to that in Refs. [3] and [4]. Substituting Eqs. (21) and (23) into Eq. (22) and accomplishing the integration after changing the integration variable $$\lambda \rightarrow \lambda +I_{0}$$ without loss of generality, we obtain the normalized Gaussian solution as

It is well known that the position uncertainty can be derived from $$\Delta q =\sqrt {\langle q^{2}\rangle -\langle q\rangle ^{2}}$$. Then, using Eqs. (A2) and (A5) in Appendix A, we have

For the case of a free particle ($$\omega =0$$), $$\Delta q$$ increases linearly with time while $$\Delta p$$ is always constant. Hence, the uncertainty product of a free particle increases linearly with time. These analyses for $$\omega =0$$ are totally in agreement with the results of Ref. [25].

In terms of $$\Delta q$$ obtained here, we have the following obvious relations:

If we set $$\omega \rightarrow 0$$, the system converts to a free particle. In the case of a free particle, the quantity $$I_0$$, where $$I_{0}=a_{0}\beta _{0}$$, can be written as

## Time evolution of the Gaussian wave packet

From the absolute square of Eq. (31), we derive the probability density of the displaced wave packet in the form

If we use the integral formula given in Eq. (26), we can easily confirm from Eqs. (A2) and (A3) in Appendix A that the analytic forms of $$\langle q\rangle $$ and $$\langle p\rangle $$ can be rewritten such that

It is well known that the introduction of annihilation operator and its Hermitian conjugate operator (creation operator) is very useful when we investigate the quantum features of the simple harmonic oscillator. There also exist such annihilation and creation operators for the inverted oscillator. For detailed method of constructing the annihilation operator for the inverted oscillator and of obtaining its corresponding eigenvalue $$z$$, you can refer to Ref. [5]. The coherent state which is associated with the Gaussian wave packet can be represented in terms of $$z$$. In order to investigate the time behavior of the Gaussian wave packet derived in the previous section in more detail, we introduce the displacement unitary operator $$D(z,t)$$, which is

## Conclusion

The quantum characteristics of the Gaussian wave packet for the inverted oscillator have been studied with emphasis on its time behavior. For this purpose, we used the invariant operator and unitary transformation methods. The linear invariant operator of the system was constructed from the Liouville–von Neumann equation. The eigenstates $$\varphi _\lambda (q,t)$$ of the invariant operator were evaluated using the unitary transformation method and are given in Eq. (16). Schrödinger solutions of the system have been derived by making use of the eigenstates $$\varphi _\lambda (q,t)$$. The simple Schrödinger solutions $$\psi _\lambda (q,t)$$ given in Eq. (21) are represented in terms of the eigenstates $$\varphi _\lambda (q,t)$$. From Eq. (17), we see that the only difference between $$\psi _\lambda (q,t)$$ and $$\varphi _\lambda (q,t)$$ is that $$\psi _\lambda (q,t)$$ has a phase factor while $$\varphi _\lambda (q,t)$$ does not. Notice that $$\psi _\lambda (q,t)$$ is the plane wave. While the solutions $$\psi _\lambda (q,t)$$ are associated with a delocalized wave packet, the Schrödinger solution $$\Psi (q,t)$$ weighted by $$g(\lambda )$$, Eq. (31), corresponds to a Gaussian wave packet that is localized. The wave function $$\Psi (q,t)$$ is in fact the GCS of the system

To investigate the quantum characteristics of $$\Psi (q,t)$$, we have introduced the displacement operator $$D(z, t)$$. It is shown that the wave function $$\Psi (q,t)$$ can also be obtained by displacing the ground-state wave function $$\Psi _0(x,t)$$ with $$D(z, t)$$. The properties of this Gaussian wave packet are very similar to the classical ones. The trajectory of the centroid of the wave packet is exactly the same as the classical trajectory. Note that $$\Psi (x,t)$$ is displaced as much as $$\langle q \rangle $$ from the origin and runs away from the origin in a positive $$q$$ or negative $$q$$ direction depending on the initial condition. Moreover, from Fig. 1(A), we can confirm that the width $$\Delta q$$ of $$\Psi (q,t)$$ spreads over time. For a simple case with the condition $$\omega \rightarrow 0$$, the wave packet reduces to that of a free particle and our corresponding results coincide with the previous research into a free-particle wave packet given in Ref. [25].

## Acknowledgments

The research of J.R.C. 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).

### Appendix A.

In this appendix, we derive the expectation values of the canonical variables $$q$$ and $$p$$ and their squares. The expectation value of an arbitrary operator $$O$$ can be evaluated from

## References

*Mathematica*(Wolfram Media, 2003), 5th ed., http://www.wolfram.com