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

(1)
$$H=\frac{p^{2}}{2m}-\frac{1}{2}m\,\omega^{2}q^{2}.$$
From Hamiltonian dynamics, we can easily verify that the classical solution for this system is given by
(2)
\begin{align} q &= A \cosh \omega t + B \sinh \omega t, \end{align}

(3)
\begin{align} p &= m\omega (A \sinh \omega t + B \cosh \omega t), \end{align}
where $$A$$ and $$B$$ are arbitrary real constants. In the subsequent sections, we will analyze the quantum motion of the wave packet and examine whether the trajectory of the packet follows this classical one given in Eq. (2). The conventional procedure for obtaining quantum solutions based on the Schrödinger equation leads to plane wave solutions that cannot be normalized. The plane wave solution is in general delocalized at a position. For this reason, we cannot describe the Gaussian wave packet of the system from such a plane wave solution, because the Gaussian wave packet corresponds to the coherent state for a localized wave.

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

(4)
$$I(t)=\alpha (t)p+\beta (t)q+\gamma (t),$$
where $$\alpha (t),$$$$\beta (t)$$, and $$\gamma (t)$$ are time-dependent coefficients that will be determined afterwards. From the definition of the invariant operator, $$I(t)$$ should satisfy the Liouville–von Neumann equation as
(5)
$$\frac{dI}{dt}=\frac{\partial I}{\partial t}+\frac{1}{i\hbar }\left[ I,H \right] =0.$$
By inserting Eqs. (1) and (4) into Eq. (5), we obtain first-order linear differential equations for the coefficients such that
(6)
$$\dot{\alpha} =-\frac{\beta }{m},\quad \dot{\beta} =-\alpha m\omega ^{2},\quad \dot{\gamma} =0.$$
The general solutions of these equations are given by
(7)
\begin{align} \alpha(t) &= \alpha _{0}\cosh \omega t+\frac{\beta _{0}}{m\omega }\sinh \omega t , \end{align}

(8)
\begin{align} \beta(t) &= -\alpha _{0}m\omega \sinh \omega t-\beta _{0}\cosh \omega t , \end{align}

(9)
\begin{align} \gamma(t) &= \gamma _{0}, \end{align}
where $$\alpha _{0}$$, $$\beta _{0}$$, and $$\gamma _{0}$$ are arbitrary real constants. Now, we can confirm that Eq. (4) with Eqs. (7)–(9) are the LHIO of the system. Notice that this operator is a Hermitian, $$I=I^{\dagger }$$, as we initially supposed.

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

(10)
$$I(t)\varphi_{\lambda}(q,t) =\lambda \varphi _{\lambda }(q,t),$$
where $$\lambda$$ is an eigenvalue that does not depend on time. To facilitate the evaluation of this equation, let us consider a unitary transformation of the form
(11)
$$\varphi _{\lambda }^{\prime }( q,t) =U\varphi _{\lambda }(q,t),$$
where the unitary operator $$U$$ is given by
(12)
$$U=\exp \left[ -\frac{i}{2\hbar }\,\frac{\beta }{\alpha }q^{2}\right]\!.$$
Then, the invariant in the transformed system is evaluated as
(13)
$$I^{\prime }=\textit{UIU}^{\dagger} =\alpha p+\gamma _{0}.$$
This formula is much simpler than the original invariant operator given in Eq. (4). Hence, it is better to solve the eigenvalue equation in the transformed system rather than solving the original one, Eq. (10). In the next section, we will derive the eigenstate $$\varphi _{\lambda }^{\prime }$$ of $$I'$$ in the transformed system. Once $$\varphi _{\lambda }^{\prime }$$ is derived, its inverse transformation gives the eigenstate $$\varphi _{\lambda }$$ of $$I$$, which belongs to the original system (untransformed system). By utilizing $$\varphi _{\lambda }$$, we can further derive the wave function of the GCS.

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

(14)
$$\left[ \frac{\partial }{\partial q}+\frac{\lambda }{i\hbar \alpha }-\frac{ \gamma _{0}}{i\hbar \alpha }\right] \varphi _{\lambda }^{\prime }(q,t) =0.$$
We can easily show that the solutions of Eq. (14) are of the form
(15)
$$\varphi _{\lambda }^{\prime }(q,t) =\exp \left[ \frac{i}{\hbar }\, \frac{\lambda -\gamma _{0}}{\alpha } q\right]\!.$$
Using the inverse of Eq. (11), we can see that the eigenstates of $$I(t)$$ with the eigenvalue $$\lambda$$ are given by
(16)
$$\varphi _{\lambda }(q,t) =\exp \left[ \frac{i}{\hbar }\left( \frac{\lambda -\gamma _{0}}{\alpha } q-\frac{\beta }{2\alpha } q^{2}\right) \right]\!.$$
Because the difference between the eigenstates $$\varphi _{\lambda }$$ and the Schrödinger solutions $$\psi _{\lambda }$$ is only a multiplication of a phase factor $$\exp [ i\delta _{\lambda } (t)]$$ according to the invariant operator theory [16], we can write the solutions of the Schrödinger equation for the Hamiltonian, Eq. (1), as
(17)
$$\psi _{\lambda }(q,t) =\frac{1}{\sqrt{2\pi \hbar \alpha _{0}}} \exp \left[ i\delta_{\lambda } (t)\right] \varphi _{\lambda }(q,t),$$
where the factor $${1}/{\sqrt {2\pi \hbar \alpha _{0}}}$$ is a normalization constant.

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]

(18)
$$\hbar \dot{\delta}_{\lambda }(t)= \int d \lambda' \left\langle \varphi_{\lambda '}\big\vert \left[ i\hbar \frac{ \partial }{\partial t}-H\right] \big\vert \varphi _{\lambda}\right\rangle.$$
A little evaluation after the substitution of the Hamiltonian, Eq. (1), and the eigenstates, Eq. (16), into this equation results in
(19)
$$\hbar \dot{\delta}_{\lambda }(t)= -\frac{i\hbar }{2m}\, \frac{\beta }{\alpha }-\frac{1}{2m\hbar }\, \frac{\left( \lambda -\gamma _{0}\right)^{2}}{\alpha^{2}}.$$
Then, from the integration with respect to time, we obtain the phase $$\delta _{\lambda } (t)$$ as
(20)
$$\delta_{\lambda } (t) = \delta_{\lambda } (0)-\frac{\left( \lambda -\gamma _{0}\right) ^{2}}{ 2m\hbar }\int_{0}^{t}\frac{d\tau }{\alpha ^{2}(\tau) }+i\ln\sqrt{\frac{\alpha ( t) }{\alpha ( 0) }}.$$
In the evaluation of the last term in this equation, we have used the first relation in Eq. (6). For convenience, let us choose $$\delta _{\lambda } (0)=0$$. Then, the physical orthogonal wave functions $$\psi _{\lambda }(q,t)$$, which are the solutions of the Schrödinger equation for the system, are given by
(21)
$$\psi _{\lambda }(q,t) = \frac{1}{\sqrt{2\pi \hbar \alpha (t) }}\exp \left[-i\!\int_{0}^{t}\!\frac{\left( \lambda -\gamma _{0}\right)^{2}d\tau }{2m\hbar \alpha ^{2}(\tau)}\right] \exp\!\left[ \frac{i}{\hbar }\left( \frac{\lambda -\gamma _{0}}{\alpha } q-\frac{\beta }{2\alpha }q^{2}\right) \right]\!.$$
Notice that these Schrödinger solutions correspond to those of the plane waves. They describe delocalized wave packets that cannot be normalized in a general way. Localized quantum wave solutions will be evaluated in the next section on the basis of these wave solutions.

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):

(22)
$$\Psi (q,t) =\int_{-\infty }^{\infty }g( \lambda ) \psi _{\lambda }(q,t) d\lambda ,$$
where $$g( \lambda )$$ is a weight function that determines the state of the system in such a way that $$\Psi ( q,t)$$ can be square integrable. Any suitable choice of $$g( \lambda )$$ yields a conventional solution as the Gaussian wave-packet function. Let us now choose the weight function as a Gaussian form too:
(23)
$$g(\lambda) = \sqrt{\frac{2\sqrt{a}}{\sqrt{2\pi }}}\exp \left[-a\!\left( \lambda -I_{0}\right) ^{2}\right] \exp \left[ -\frac{i}{\hbar } \frac{a_{0}}{\alpha _{0}}\left( \lambda -\frac{I_{0}}{2}\right) \right]\!,$$

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

(24)
\begin{align} \Psi (q,t) &= \sqrt{\frac{\sqrt{a}}{\hbar \alpha (t) \sqrt{2\pi }\left( a+i\int_{0}^{t}\frac{d\tau }{2m\hbar \alpha ^{2}(\tau) }\right) }}\exp \left[ -i\int_{0}^{t}\frac{\left( \gamma _{0}-I_{0}\right) ^{2}}{2m\hbar \alpha ^{2}(\tau) }d\tau \right] \nonumber \\ &\quad \times\exp \left[ -\frac{i}{\hbar }\frac{I_{0}}{2}\frac{a_{0}}{\alpha _{0}}\right] \exp \left\{ \frac{i}{\hbar }\left[ -\frac{ \gamma _{0}-I_{0} }{\alpha }q-\frac{\beta (t) }{2\alpha (t) }q^{2} \right] \right\} \nonumber \\ &\quad \times\exp \left\{ -\frac{\left[ q- \alpha (t) \left( \frac{a_{0}}{\alpha _{0}}-\int_{0}^{t}\frac{ \gamma _{0}-I_{0} }{m\alpha ^{2}(\tau) }d\tau \right) \right] ^{2}}{4\hbar ^{2}\alpha ^{2}(\tau) \left( a+i\int_{0}^{t}\frac{1}{2m\hbar \alpha ^{2}(\tau) }d\tau \right) }\right\}\!. \end{align}
This wave function plays a central role in the quantum mechanics of the inverted oscillator and can be used to investigate various quantum properties of the system. For instance, we can derive expectation values of canonical variables and their squares (see Appendix A).

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

(25)
$$\Delta q =\frac{\hbar \alpha (t) }{\sqrt{a}}\left[ a^{2}+\left(\frac{\Gamma(t)}{2m\hbar } \right) ^{2}\right]^{1/2},$$
where $$\Gamma (t)$$ is defined in Eq. (A4) in Appendix A. By noting that $$\alpha (t)$$ is given in Eq. (7), the time function $$\Gamma (t)$$ in this equation can be evaluated as
(26)
$$\Gamma(t) = \int_{0}^{t}\frac{ 1 }{\alpha^{2} ( \tau ) }d\tau = \frac{m}{\alpha_0 \beta_0 + m \alpha_0^2 \omega \coth \omega t}.$$
Similarly, from Eqs. (A3) and (A6), we also have the momentum uncertainty such that
(27)
$$\Delta p= \frac{\hbar}{2} \left[ \left( \frac{2\beta(t)}{\hbar\alpha(t)} \Delta q- \frac{\Gamma(t)}{2ma\hbar \Delta q} \right)^2+ (\Delta q)^{-2}\right]^{1/2}.$$
By multiplying Eqs. (25) and (27), the uncertainty product for canonical variables is given by
(28)
$$\Delta q \Delta p = \frac{\hbar}{2}\left[ 1+\left( \frac{2\beta(t)}{\hbar\alpha(t)} (\Delta q)^2- \frac{\Gamma(t)}{2ma\hbar } \right)^2 \right]^{1/2}.$$
This is always larger than $$\hbar /2$$ which is the quantum-mechanically allowed minimum value for the uncertainty product. We have plotted the uncertainty product, as well as $$\Delta q$$ and $$\Delta p$$, as a function of time in Fig. 1 for several different values of $$\omega$$. From Fig. 1(A), we see that $$\Delta q$$ always increases with time provided that it moves. From Fig. 1(B), we can confirm that $$\Delta p$$ also increases with time. As a consequence, the uncertainty product increases. This phenomenon is interesting if we think that the uncertainty product of the normal oscillator is constant.

Fig. 1.

Time evolution of the uncertainty $$\Delta q$$ (A) and $$\Delta p$$ (B), and the corresponding uncertainty product $$\Delta q \Delta p$$ (C) for different values of $$\omega$$. The values of $$\omega$$ are $$\omega \rightarrow 0.0$$ for the solid red line, $$\omega = 2.5$$ for the dashed blue line, and $$\omega = 5.0$$ for the dotted green line. We have used $$\alpha _0=1$$, $$\beta _0=1$$, $$\gamma _0=1$$, $$I_0=5$$, $$a=1$$, $$a_0=2$$, $$m=1$$, and $$\hbar =1$$.

Fig. 1.

Time evolution of the uncertainty $$\Delta q$$ (A) and $$\Delta p$$ (B), and the corresponding uncertainty product $$\Delta q \Delta p$$ (C) for different values of $$\omega$$. The values of $$\omega$$ are $$\omega \rightarrow 0.0$$ for the solid red line, $$\omega = 2.5$$ for the dashed blue line, and $$\omega = 5.0$$ for the dotted green line. We have used $$\alpha _0=1$$, $$\beta _0=1$$, $$\gamma _0=1$$, $$I_0=5$$, $$a=1$$, $$a_0=2$$, $$m=1$$, and $$\hbar =1$$.

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:

(29)
\begin{align} -\left(\gamma _{0}-I_{0}\right) &= \alpha (t) \langle p\rangle +\beta (t) \langle q\rangle , \end{align}

(30)
\begin{align} \int_{0}^{t}\frac{ \gamma _{0}-I_{0} }{m\alpha ^{2}(\tau) }d\tau &= \frac{a_{0}}{\alpha _{0}}-\frac{\left\langle q\right\rangle }{\alpha (t)}. \end{align}
Then, by making use of these relations, we can rewrite the general Schrödinger state (Eq. (24)) as follows
(31)
\begin{align} \Psi (q,t) &= \sqrt{\frac{\hbar \alpha (t) \left( a-i\int_{0}^{t}\frac{d\tau }{2m\hbar \alpha ^{2}(\tau) }\right) }{\sqrt{2\pi a}(\Delta q)^{2}}}\exp \left[ -\frac{i}{2\hbar }\left( \left\langle p\right\rangle \left\langle q\right\rangle +\frac{\gamma _{0}a_{0}}{\alpha _{0}}\right) \right] \nonumber \\ &\quad\times\exp \left[ \frac{i}{\hbar }\left\langle p\right\rangle q\right] \exp \left\{ -\left( \frac{i}{\hbar }\frac{\beta (t) }{2\alpha (t) }+\frac{ a-i\int_{0}^{t}\frac{1}{2m\hbar \alpha ^{2}(\tau) }d\tau }{4a(\Delta q)^{2}}\right) \left[ q-\left\langle q\right\rangle \right] ^{2}\right\}\!. \end{align}

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

(32)
\begin{align} I_{0}-\alpha (t) \langle p\rangle -\beta (t) \langle q\rangle &= I_{0}+\left( \alpha _{0}-\frac{ \beta _{0}}{m}t\right) \frac{\gamma _{0}}{\alpha _{0}}-\beta _{0}\left( a_{0}-\frac{\gamma _{0}}{m\alpha _{0}}t\right) \nonumber \\ &= \gamma _{0}. \end{align}
For this case, Eq. (31) reduces to the wave solution of a free particle, which is
(33)
\begin{align} \Psi_{\rm free\ particle}\ (q,t) &=\sqrt{\frac{\hbar \alpha (t) \left( a-i\int_{0}^{t}\frac{d\tau }{2m\hbar \alpha ^{2}(\tau) }\right) }{\sqrt{2\pi a}(\Delta q)^{2}}}\exp \left[ -\frac{i}{2\hbar }\left( \left\langle p\right\rangle \left\langle q\right\rangle +\frac{I_{0} \gamma_{0}}{\alpha _{0}\beta _{0}}\right) \right] \nonumber \\ &\quad \times\exp \left[ \frac{i}{\hbar }\langle p\rangle q\right] \exp \left\{-\left( \frac{i}{\hbar }\frac{\beta (t) }{2\alpha (t) } +\frac{ a-i\int_{0}^{t}\frac{1}{2m\hbar \alpha ^{2}(\tau) }d\tau }{4a(\Delta q)^{2}}\right) \left[ q-\langle q\rangle \right]^{2}\right\}\!. \end{align}
This coincides with the results of Ref. [25].

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

(34)
$$\left\vert \Psi (q,t) \right\vert^{2}=\frac{1}{\sqrt{2\pi } \Delta q}\exp \left( -\frac{\left( q-\left\langle q\right\rangle \right) ^{2}}{2\left(\Delta q\right)^{2}}\right)\!.$$
Because this equation is a Gaussian form, we can readily conclude that $$\Psi (q,t)$$ is the Gaussian wave packet. This packet is displaced as much as $$\langle q\rangle$$ from the origin and spreads as the packet propagates because $$\Delta q$$ increases in time, as shown in Fig. 1. We can analyze the time behavior of this packet from its density plot, given in Fig. 2. The wave packet in this figure moves in the positive $$q$$ direction. We can confirm that the speed of the wave-packet-propagation accelerates over time, provided that $$\omega \neq 0$$. From a comparison of this figure with Fig. 3 which shows the corresponding classical solutions, we see that the centroid of the wave packet follows the counterpart classical trajectory. This result is natural if we think that $$\vert \Psi ( q,t) \vert ^{2}$$ is represented in terms of $$\langle q\rangle$$, which is expected to follow the classical solution. Figure 4 is the same as Fig. 2, but with a somewhat different choice of the parameter $$a_0$$. In this case, the wave packet or particle moves in the negative $$q$$ direction with different initial conditions.

Fig. 2.

The time evolution of the probability density $$|\Psi (q,t)|^2$$ with $$\omega \rightarrow 0.0$$ for (A), $$\omega =2.5$$ for (B), and $$\omega =5.0$$ for (C). We have used $$\alpha _0=1$$, $$\beta _0=1$$, $$\gamma _0=1$$, $$I_0=5$$, $$a=1$$, $$a_0=2$$, $$m=1$$, and $$\hbar =1$$. This figure is a density plot produced using the Mathematica program (Wolfram Research) [26] with the choice of the command “PlotRange $$\rightarrow$$$$\{$$0, 0.3$$\}$$”.

Fig. 2.

The time evolution of the probability density $$|\Psi (q,t)|^2$$ with $$\omega \rightarrow 0.0$$ for (A), $$\omega =2.5$$ for (B), and $$\omega =5.0$$ for (C). We have used $$\alpha _0=1$$, $$\beta _0=1$$, $$\gamma _0=1$$, $$I_0=5$$, $$a=1$$, $$a_0=2$$, $$m=1$$, and $$\hbar =1$$. This figure is a density plot produced using the Mathematica program (Wolfram Research) [26] with the choice of the command “PlotRange $$\rightarrow$$$$\{$$0, 0.3$$\}$$”.

Fig. 3.

The time behavior of the classical solution $$q$$ given in Eq. (2) with $$\omega \rightarrow 0.0$$ for the solid red line, $$\omega =2.5$$ for the dashed blue line, and $$\omega =5.0$$ for the dotted green line. We have used $$A=2$$ and $$B=5/\omega$$.

Fig. 3.

The time behavior of the classical solution $$q$$ given in Eq. (2) with $$\omega \rightarrow 0.0$$ for the solid red line, $$\omega =2.5$$ for the dashed blue line, and $$\omega =5.0$$ for the dotted green line. We have used $$A=2$$ and $$B=5/\omega$$.

Fig. 4.

The same as Fig. 2, but using a different value of $$a_0$$. (Here, $$a_0 =-12$$.)

Fig. 4.

The same as Fig. 2, but using a different value of $$a_0$$. (Here, $$a_0 =-12$$.)

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

(35)
\begin{align} \langle q\rangle &= \frac{1}{\alpha_0 m\omega} \big[a_0\alpha_0 m\omega \cosh \omega t + (a_0\beta_0 -\gamma_0 +I_0)\sinh \omega t\big], \end{align}

(36)
\begin{align} \langle p\rangle &= \frac{1}{\alpha_0 } \big[a_0\alpha_0 m \omega \sinh \omega t + (a_0\beta_0 -\gamma_0 +I_0)\cosh \omega t\big]. \end{align}
By rescaling the coefficients as $$a_0 \equiv A$$ and $$(a_0\beta _0-\gamma _0+I_0)/(\alpha _0 m\omega ) \equiv B$$, we see that the above equations become the same as those of the classical solutions given in Eqs. (2) and (3), respectively. Hence, the centroids of the quantum trajectories for the canonical variables are exactly the same as those of the classical ones. Based on these consequences for the inverted oscillator, we can deduce precise physical meanings associated with Eq. (34). Although the most probable trajectory of the wave packet is a classical one, it does not always follow it because of the uncertainty of $$\Delta q$$. If $$\Delta q$$ is large, it is difficult to predict the position of the quantum particle at a certain instant. While it is known that the Gaussian wave packet for a harmonic oscillator oscillates back and forth [19, 27], this packet does not oscillate. For a wave packet that initially stays at a certain position with $$p=0$$ typically runs away from the origin at a later time.

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

(37)
$$D(z,t)=\exp \left[ \frac{i}{\hbar }( \langle p\rangle q-\langle q\rangle p) \right]\!.$$
By means of this operator, we can convert coordinate and momentum operators so that they are displaced as
(38)
\begin{align} D^{\dagger}(z,t)qD(z,t) &= q+\langle q\rangle =q+q_{\rm cl}, \end{align}

(39)
\begin{align} D^{\dagger}(z,t)pD(z,t) &= p+\langle p\rangle =p+p_{\rm cl}, \end{align}
where $$q_{\rm cl}$$ and $$p_{\rm cl}$$ are the classical coordinate and momentum, respectively. We can construct the GCSs as follows:
(40)
$$D(z,t)\Psi _{0}(q,t) =\exp \left[ -\frac{\vert z\vert ^{2}}{2}\right] \sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!}}\Psi _{n}(q,t) ,$$
where $$\Psi _{n}(q,t)$$ are $$n$$th-order Fock-state wave functions [5]. It is interesting to note that the general expression of the Gaussian wave $$\Psi ( q,t)$$ (Eq. (31)) corresponds to the standard coherent state obtained by applying the displacement operator, $$D(z,t)$$, to the ground state, $$\Psi _{0}( q,t)$$. Notice that $$\Psi _{0}( q,t)$$ corresponds to a particle at rest, $$\langle p\rangle =0$$, at the origin, $$\langle q\rangle =0$$:
(41)
$$\Psi_0 (q,t) = \Psi (q,t)|_{\langle q\rangle =0, \langle p\rangle =0}.$$
Now we can easily confirm that the wave function given in Eq. (31) can be rewritten as follows:
(42)
$$\Psi (q,t) =D(z,t)\Psi _{0}(q,t) .$$
Hence, the GCS associated with the quantum motion of the inverted oscillator can be obtained by displacing the ground-state wave packet.

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

(A1)
$$\langle O\rangle =\langle \Psi (t) | O | \Psi (t) \rangle ,$$
where $$| \Psi (t) \rangle$$ is the wave function whose configuration representation is given in Eq. (24). For the case of $$q$$ and $$p$$, the expectation values in the state $$\Psi ( q,t)$$ are easily evaluated using the above equation as
(A2)
\begin{align} \langle q\rangle &= \alpha (t) \left( \frac{a_{0}}{ \alpha _{0}}-\frac{ \gamma _{0}-I_{0} }{m }\Gamma(t)\right)\!, \end{align}

(A3)
\begin{align} \langle p\rangle &= -\frac{ \gamma _{0}-I_{0} }{\alpha(t) }-\beta (t) \left( \frac{a_{0}}{\alpha _{0}}-\frac{ \gamma _{0}-I_{0} }{m } \Gamma(t)\right)\!, \end{align}
where
(A4)
$$\Gamma(t) = \int_{0}^{t}\frac{ 1 }{\alpha^{2}(\tau) }d\tau .$$
In a similar way, the square of the canonical variables can also be evaluated as
(A5)
\begin{align} \langle q^{2}\rangle &= \frac{\hbar ^{2}\alpha ^{2}\left( t \right) }{a}\left[ a^{2}+\left(\frac{\Gamma(t)}{2m\hbar } \right) ^{2}\right] +\alpha ^{2}(t) \left( \frac{a_{0}}{\alpha _{0}} -\frac{ \gamma _{0}-I_{0} }{m }\Gamma(t) \right) ^{2}, \end{align}

(A6)
\begin{align} \langle p^{2}\rangle &= \langle p\rangle^2- \frac{\beta(t)\Gamma(t)}{2ma\alpha(t)}+ \frac{\beta^2(t)}{\alpha^2(t)}(\Delta q)^2 + \left[ \frac{\hbar^2}{4}+ \frac{\Gamma^2(t)}{16m^2 a^2} \right] \frac{1}{(\Delta q)^2}. \end{align}
We can see that $$\big \langle p^{2}\big \rangle$$ is represented in terms of the variance of $$q$$, which is defined in the form $$\Delta q =[\langle q^{2}\rangle - \langle q\rangle ^{2}]^{1/2}$$. For more details of $$\Delta q$$, see Eq. (25).

References

1
Yuce
C.
,
Kilic
A.
, and
Coruh
A.
,
Phys. Scripta

74
,
114
(
2006
).
()
2
Barton
G.
,
Ann. Phys.

166
,
322
(
1986
).
()
3
Pedrosa
I. A.
,
de Lima
A. L.
, and
Carvalho
A. M. de M.
,
Can. J. Phys.

93
,
841
(
2015
).
()
4
de Lima
A. L.
,
Rosas
A.
, and
Pedrosa
I. A.
,
Ann. Phys.

323
,
2253
(
2008
).
()
5
Bagrov
V. G.
,
Gitman
D. M.
,
Macedo
E. S.
, and
Pereira
A. S.
,
J. Phys. A

46
,
325305
(
2013
).
()
6
Bagrov
V. G.
,
Gitman
D. M.
, and
Pereira
A. S.
,
Braz. J. Phys.

45
,
369
(
2015
).
()
7
R. K.
,
Khare
A.
,
Reimann
S. M.
, and
Tomusiak
E. L.
,
Ann. Phys.

254
,
25
(
1997
).
()
8
Lewis
Z.
and
Takeuchi
T.
,
Phys. Rev. D

84
,
105029
(
2011
).
()
9
Kim
K. K.
and
Kim
S. P.
,
J. Korean Phys. Soc.

56
,
1055
(
2010
).
()
10
Chong
G.
and
Hai
W.
,
J. Phys. B

40
,
211
(
2007
).
()
11
Guth
A. H.
and
Pi
S.-Y.
,
Phys. Rev. D

32
,
1899
(
1985
).
()
12
Glauber
R. J.
and
Haake
F.
,
Phys. Lett. A

68
,
29
(
1978
).
()
13
Felder
G.
,
Frolov
A.
,
Kofman
L.
, and
Linde
A.
,
Phys. Rev. D

66
,
023507
(
2002
).
()
14
Combescure
M.
and
Combescure
A.
,
J. Math. Anal. Appl.

326
,
908
(
2007
).
()
15
Bermudez
D.
and
Fernández C.
D. J.
,
Ann. Phys.

333
,
290
(
2013
).
()
16
Lewis
H. R.
Jr.
and
Riesenfeld
W. B.
,
J. Math. Phys.

10
,
1458
(
1969
).
()
17
Bagrov
V. G.
,
Gitman
D. M.
, and
Pereira
A. S.
,
Phys.-Usp.

57
,
891
(
2014
).
()
18
Schrödinger
E.
,
Naturwissenschaften

14
,
664
(
1926
).
()
19
Choi
J. R.
,
Chin. Phys. C

35
,
233
(
2011
).
()
20
Pedrosa
I. A.
,
C.
, and
Rosas
A.
,
Phys. Lett. B

651
,
384
(
2007
).
()
21
Choi
J. R.
,
Phys. Rev. A

82
,
055803
(
2010
).
()
22
Maamache
M.
, and
Y.
,
Phys. Rev. Lett.

101
,
150407
(
2008
).
()
23
Maamache
M.
, and
Y.
,
Phys. Rev. A

78
,
052109
(
2008
).
()
24
Y.
, and
Maamache
M.
,
Phys. Lett. A

376
,
1328
(
2012
).
()
25
Maamache
M.
,
Khatir
A.
,
Lakehal
H.
, and
Choi
J. R.
,
unpublished
.
26
Wolfram
S.
,
Mathematica (Wolfram Media, 2003), 5th ed., http://www.wolfram.com
.
27
Choi
J. R.
,
Sci. Rep.

4
,
6880
(
2014
).
()
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.