*Article* **Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems †**

**Qiliang Zhao <sup>1</sup> , Pengming Zhang <sup>1</sup> and Peter A. Horvathy 2,\***


**Abstract:** The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping "Eisenhart–Duval" lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

**Keywords:** quantum mechanics; semiclassical theories and applications; classical general relativity

**PACS:** 03.65.-w quantum mechanics; 03.65.Sq semiclassical theories and applications; 04.20.-q classical general relativity

#### **1. Introduction**

A nonrelativistic quantum particle with unit mass in *d* + 1 spacetime dimensions with coordinates *x*, *t* is given by the natural Lagrangian *L* = <sup>1</sup> 2 *x*˙ <sup>2</sup> <sup>−</sup> *<sup>V</sup>*(*x*, *<sup>t</sup>*). The wave function is expressed in terms of the propagator,

$$
\psi(\mathbf{x}'',\mathbf{t}'') = \int \mathcal{K}(\mathbf{x}'',\mathbf{t}''|\mathbf{x}',\mathbf{t}')\psi(\mathbf{x}',\mathbf{t}')d\mathbf{x}' \tag{1}
$$

which, following Feynman's intuitive proposal [1], is obtained as,

$$K(\mathbf{x}^{\prime\prime}, t^{\prime\prime}|\mathbf{x}^{\prime}, t^{\prime}) = \int \exp\left[\frac{i}{\hbar}\mathcal{A}(\gamma)\right] \mathcal{D}\bigcirc \,, \tag{2}$$

where the (symbolic) integration is over all paths *γ*(*t*) = *x*(*t*), *t* that link the spacetime point (*x* 0 , *t* 0 ) to (*x* 00 , *t* 00) and where:

$$\mathcal{A}(\gamma) = \int\_{t'}^{t''} \mathcal{L}\left(\gamma(t), \dot{\gamma}(t), t\right) dt \tag{3}$$

is the classical action calculated along *γ*(*t*) [1–3].

The rigorous definition and calculation of (2) are beyond our scope here. However, the *semiclassical approximation* leads to the van Vleck–Pauli formula [2–5],

$$\mathcal{K}(\mathbf{x}^{\prime\prime}, t^{\prime\prime}|\mathbf{x}^{\prime}, t^{\prime}) = \left[\frac{i}{2\pi\hbar}\frac{\partial^2 \mathcal{A}}{\partial \mathbf{x}^{\prime} \partial \mathbf{x}^{\prime\prime}}\right]^{1/2} \exp\left[\frac{i}{\hbar}\mathcal{A}(\mathbf{x}^{\prime\prime}, t^{\prime\prime}|\mathbf{x}^{\prime}, t^{\prime})\right],\tag{4}$$

**Citation:** Zhao, Q.; Zhang, P.; Horvathy, P.A. Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems. *Symmetry* **2021**, *13*, 1866. https:// doi.org/10.3390/sym13101866

Academic Editor: Georg Junker

Received: 7 July 2021 Accepted: 17 August 2021 Published: 3 October 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

where <sup>A</sup>¯(*<sup>x</sup>* 00 , *t* <sup>00</sup>|*x* 0 , *t* 0 ) = <sup>Z</sup> *<sup>t</sup>* 00 *t* 0 *L*(*γ*¯(*t*), *γ*¯˙(*t*), *t*)*dt* is the classical action calculated along the (supposedly unique (This condition is satisfied away from caustics [2,3,6]. Moreover, (5) and (8) are valid only for 0 < *T* <sup>00</sup> − *T* 0 and for 0 < *t* <sup>00</sup> − *t* 0 < *π*, respectively, as discussed in Section 4.)) classical path *γ*¯(*τ*) from (*x* 0 , *t* 0 ) and (*x* 00 , *t* 00). This expression involves data of the classical motion only. We note here also the van Vleck determinant *<sup>∂</sup>* 2A¯ *∂x* 0*∂x* <sup>00</sup> in the prefactor [4,5].

Equation (4) is exact for a quadratic-in-the-position potentials in 1 + 1 dimension *V*(*x*, *t*) = <sup>1</sup> 2*ω*<sup>2</sup> (*t*) *x* 2 that we consider henceforth.

For *ω* ≡ 0, i.e., for a free nonrelativistic particle of unit mass in 1 + 1 dimensions with coordinates *X* and *T*, the result is [1–3],

$$K\_{free}(X'', T''|X', T') = \left[\frac{1}{2\pi i\hbar(T'' - T')}\right]^{1/2} \exp\left\{\frac{i}{\hbar} \frac{(X'' - X')^2}{2(T'' - T')}\right\}.\tag{5}$$

A harmonic oscillator with dissipation is in turn described by the Caldirola–Kanai (CK) Lagrangian and the equation of motion, respectively [7,8]. For constant damping and a harmonic frequency, we have,

$$L\_{\mathbb{C}K} = \frac{1}{2} e^{\lambda\_0 t} \left( \left( \frac{d\mathbf{x}}{dt} \right)^2 - \omega\_0^2 \mathbf{x}^2 \right), \tag{6}$$

$$\frac{d^2\mathbf{x}}{dt^2} + \lambda\_0 \frac{d\mathbf{x}}{dt} + \omega\_0^2 \mathbf{x} = \mathbf{0} \tag{7}$$

with *λ*<sup>0</sup> = const. > 0 and *ω*<sup>0</sup> = const.. A lengthy calculation then yields the exact propagator [2,3,9–11]:

$$\mathcal{K}\_{\rm CK}(\mathbf{x}^{\prime\prime},t^{\prime\prime}|\mathbf{x}^{\prime},t^{\prime}) = \left[\frac{\Omega\_{0}e^{\frac{\lambda}{2}(t^{\prime}+t^{\prime})}}{2\pi i\hbar\sin\left[\Omega\_{0}(t^{\prime\prime}-t^{\prime})\right]}\right]^{\frac{1}{2}} \quad \times \tag{8}$$

$$\exp\left\{\frac{i\Omega\_{0}}{2\hbar\sin\left[\Omega\_{0}(t^{\prime\prime}-t^{\prime})\right]}\left[\left(\mathbf{x}^{\prime\prime2}e^{\lambda\_{0}t^{\prime\prime}} + \mathbf{x}^{\prime2}e^{\lambda\_{0}t^{\prime}}\right)\cos\left[\Omega\_{0}(t^{\prime\prime}-t^{\prime})\right] - 2\mathbf{x}^{\prime\prime}\mathbf{x}^{\prime}e^{\lambda\_{0}\frac{t^{\prime}-t^{\prime}}{2}}\right]\right\},$$

$$\Omega\_{0}^{2} = \omega\_{0}^{2} - \frac{1}{4}\lambda\_{0}^{2},\tag{9}$$

where an irrelevant phase factor was dropped.

Inomata and his collaborators [12–15] generalized (9) to a time-dependent frequency by redefining time, *t* → *τ*, which allowed them to transform the time-dependent problem to one with a constant frequency (see Section 2). Then, they followed by what they called a "*time-dependent conformal transformation* " (*x*, *t*) → (*X*, *T*) such that:

$$\mathbf{x} = f(T)\,\mathbf{X}(T)\exp\left[\frac{1}{2}\lambda\_0 T\right], \quad t = g(T), \quad \text{where} \quad f^2(T) = \frac{d\mathbf{g}}{dT},\tag{10}$$

which allowed them to derive the propagator from the free expression (5). When spelled out, (10) boils down to a generalized version, (22), of the correspondence found by Niederer [16].

It is legitimate to wonder: *in what sense are these transformations "conformal"* ? In Section 3, we explain that, in fact, *both* mappings can be interpreted in the Eisenhart– Duval (E-D) framework as conformal transformations between two appropriate Bargmann spaces [17–21]. Moreover, the change of variables *x*, *t* → *X*, *T* is a special case of the one put forward by Arnold [22,23] and is shown to be convenient to study time-dependent systems explicitly.

A bonus is the extension to the arbitrary time-dependent frequency *ω*(*t*) of the Maslov phase correction [2,4–6,19,24–28] even when no explicit solutions are available (see Section 4).

In Section 5.2, we illustrate our theory by the time-dependent Mathieu profile *ω*<sup>2</sup> (*t*) = *a* − 2*q* cos 2*t* , *a*, *b* const., the direct analytic treatment of which is complicated.

#### **2. The Junker–Inomata Derivation of the Propagator**

Starting with a general quadratic Lagrangian in 1 + 1 spacetime dimensions with coordinates *x*˜ and *t*, Junker and Inomata derived the equation of motion [12]:

$$
\ddot{\mathfrak{X}} + \dot{\lambda}(t)\mathfrak{X} + \omega^2(t)\mathfrak{X} = F(t) \, , \tag{11}
$$

which describes a nonrelativistic particle of unit mass with dissipation *λ*(*t*). The driving force *F*(*t*) can be eliminated by subtracting a particular solution *h*(*t*) of (11), *x*(*t*) = *x*˜(*t*) − *h*(*t*), in terms of which (11) becomes homogeneous,

$$
\ddot{\mathfrak{x}} + \dot{\lambda}(t)\dot{\mathfrak{x}} + \omega^2(t)\mathfrak{x} = \mathbf{0}.\tag{12}
$$

This equation can be obtained from the time-dependent generalization of (6),

$$L\_{\mathbb{C}K} = \frac{1}{2} e^{\lambda(t)} [\dot{\mathfrak{x}}^2 - \omega^2(t)\mathfrak{x}^2]. \tag{13}$$

The friction can be eliminated by setting *x*(*t*) = *y*(*t*)*e* <sup>−</sup>*λ*(*t*)/2, which yields a harmonic oscillator with no friction, but with a shifted frequency [29–31],

$$
\ddot{y} + \Omega^2(t)y = 0 \quad \text{where} \quad \Omega^2(t) = \omega^2(t) - \frac{\dot{\lambda}^2(t)}{4} - \frac{\ddot{\lambda}(t)}{2} \,. \tag{14}
$$

For *λ*(*t*) = *λ*0*t* and *ω* = *ω*<sup>0</sup> = const., for example, we obtain the usual harmonic oscillator with a constant shifted frequency, Ω<sup>2</sup> = *ω*<sup>2</sup> <sup>0</sup> − *λ* 2 0 /4 = const.

The frequency is in general time-dependent, though Ω = Ω(*t*); therefore, (14) is a *Sturm–Liouville equation* that can be solved analytically only in exceptional cases.

Junker and Inomata [12] followed another, more subtle path. Equation (12) is a linear equation with time-dependent coefficients, the solution of which can be searched for within the ansatz (A similar transcription was used also by Rezende [28].):

$$\mathbf{x}(t) = \rho(t) \left( A e^{i\tilde{\omega}\tau(t)} + B e^{-i\tilde{\omega}\tau(t)} \right),\tag{15}$$

where *A*, *B*, and *ω*¯ are constants and *ρ*(*t*) and *τ*(*t*) functions to be found. Inserting (15) into (12), putting the coefficients of the exponentials to zero, separating the real and imaginary parts, and absorbing a new integration constant into *A*, *B* provide us with the coupled system for *ρ*(*t*) and *τ*(*t*),

$$
\ddot{\rho} + \dot{\lambda}\dot{\rho} + (\omega^2(t) - \bar{\omega}^2 \dot{\tau}^2)\rho = 0,\tag{16}
$$

$$
\dot{\pi}(t)\,\rho^2(t)\,e^{\lambda(t)} = 1\,\. \tag{17}
$$

Manifestly, *τ*˙ > 0. Inserting *τ*˙ into (16) then yields the *Ermakov–Milne–Pinney* (EMP) equation [32–34] with time-dependent coefficients,

$$
\ddot{\rho} + \dot{\lambda}\dot{\rho} + \omega^2(t)\rho = \frac{e^{-2\lambda(t)}\bar{\omega}^2}{\rho^3}.\tag{18}
$$

We note for later use that eliminating *ρ* would yield instead:

$$
\omega^2 = \frac{1}{\tau^2} \left( \omega^2(t) - \frac{1}{2}\frac{\dddot{\overline{\tau}}}{t} + \frac{3}{4} \left( \frac{\ddot{\overline{\tau}}}{t} \right)^2 - \frac{\ddot{\lambda}}{2} - \frac{\dot{\lambda}^2}{4} \right). \tag{19}
$$

Conversely, the constancy of the r.h.s. here can be verified using Equation (17). Equivalently, starting with the Junker–Inomata condition (10),

$$
\omega^2(t) = \frac{f}{f} - 2\frac{f^2}{f^2} + \frac{\dot{\lambda}^2}{4} + \frac{\ddot{\lambda}}{2} \,. \tag{20}
$$

To sum up, the strategy to follow is [12,35,36]:


$$
\pi(t) = \int \frac{e^{-\lambda(u)}}{\rho^2(u)} \, du \, \,. \tag{21}
$$

Then, the trajectory is given by (15).

Junker and Inomata showed, moreover, that substituting into (13) the new coordinates:

$$T = \frac{\tan\left[\bar{\omega}\,\tau(t)\right]}{\bar{\omega}}\,,\qquad X = \varkappa e^{\frac{\lambda(t)}{2}}\,\tau(t)^{\frac{1}{2}}\sec\left[\bar{\omega}\,\tau(t)\right]\,,\tag{22}$$

allows us to present the Caldirola–Kanai action as (Surface terms do not change the classical equations of motion and multiply the propagator by an unobservable phase factor, and are therefore dropped.),

$$\mathcal{A}\_{\mathbb{C}K} = \int\_{I'}^{t''} \mathcal{L}\_{\mathbb{C}K} dt = \int\_{T'}^{T''} \frac{1}{2} \left(\frac{dX}{dT}\right)^2 dT \,\,\,\tag{23}$$

where we recognize the *action of a free particle* of unit mass. One checks also directly that *X*, *T* satisfy the free equation, as they should. The conditions (10) are readily verified.

The coordinates *X* and *T* describe a free particle; therefore, the propagator is (5) (as anticipated by our notation). The clue of Junker and Inomata [12] is that, conversely, trading *X* and *T* in (5) for *x* and *t* allows deriving the propagator for the CK oscillator (see also [11], Section 5.1) (The extension of (24) from 0 < *ω*¯(*τ* <sup>00</sup> − *τ* 0 ) < *π* to all *t* [2,3,6,11] is discussed in Section 4.),

$$\mathbf{K}\_{\rm osc}(\mathbf{x}'',t'|\mathbf{x}',t') = \left[\frac{\varepsilon\sigma^{\frac{\lambda''+\lambda'}{2}}(t''\mathbf{t}')^{\frac{1}{2}}}{2\pi i\hbar\sin[\bar{\omega}(\tau''-\tau')]}\right]^{\frac{1}{2}} \times \tag{24}$$

$$\exp\left\{\frac{i\bar{\omega}}{2\hbar\sin[\bar{\omega}(\tau''-\tau')]} \left[(\mathbf{x}'^{\prime\prime}\mathbf{e}^{\lambda''}\mathbf{t}''+\mathbf{x}'^{\prime}\mathbf{e}^{\lambda'}\mathbf{t}')\cos[\bar{\omega}(\tau''-\tau')]-2\mathbf{x}'^{\prime}\mathbf{x}'e^{\frac{\lambda''+\lambda'}{2}}(\mathbf{t}''\mathbf{t}')^{\frac{1}{2}}\right]\right\},$$

where we used the shorthands *λ* 00 = *λ*(*t* 00), *τ* 00 = *τ*(*t* 00), etc.

This remarkable formula says that in terms of "redefined time", *τ*, the problem is essentially one with a constant frequency. Equation (24) is still implicit, though, as it requires solving first the coupled system (17), which we can do only in particular cases.

• When *λ*(*t*) = *λ*0*t* where *λ*<sup>0</sup> = const. ≥ 0, Equation (12) describes a time-dependent oscillator with constant friction,

$$
\pounds + \lambda\_0 \,\pounds + \omega^2(t)\mathfrak{x} = 0. \tag{25}
$$

Then, setting *R*(*t*) = *ρ*(*t*)*e <sup>λ</sup>*0*t*/2, Equation (17) provide us with the EMP equation for *R*, cf. (18),

$$
\ddot{\mathcal{R}} + \Omega^2(t)\mathcal{R} - \frac{\omega^2}{R^3} = 0, \quad \text{where} \quad \Omega^2(t) = \omega^2(t) - \frac{\lambda\_0^2}{4}; \tag{26}
$$

• If, in addition, the *frequency* is *constant ω*(*t*) = *ω*<sup>0</sup> = const., then Equation (26) is solved algebraically by:

$$
\omega^2 = \omega\_0^2 - \lambda\_0^2/4, \quad \mathcal{R} = 1 \quad \Rightarrow \quad \rho(t) = e^{-\lambda\_0 t/2}, \quad \tau(t) = t. \tag{27}
$$

Thus, *x*(*t*) is a linear combination of *e* − 1 2 *λ*0*t* sin *ω*¯ *t* and *e* − 1 2 *λ*0*t* cos *ω*¯ *t*. The spacetime coordinate transformation of (*x*, *t*) → (*X*, *T*) in (22) simplifies to the friction-generalized form of that of Niederer [16],

$$T = \frac{\tan(\omega t)}{\omega}, \quad X = \text{x} \exp\left(\frac{1}{2}\lambda\_0 t\right) \text{sec}(\omega t), \tag{28}$$

for which the general expression (24) reduces to (9) when *λ*<sup>0</sup> = 0;

• When the oscillator is turned off, *ω*<sup>0</sup> = 0, but *λ*<sup>0</sup> > 0, we have motion in a dissipative medium. The coordinate transformation propagator (22) and (24) become:

$$X = \frac{2\mathbf{x}}{1 + \exp(-\lambda\_0 t)}, \qquad T = \frac{2}{\lambda\_0} \frac{1 - \exp(-\lambda\_0 t)}{1 + \exp(-\lambda\_0 t)}\tag{29}$$

and:

$$\begin{split} K\_{\rm diss}(\mathbf{x}^{\prime\prime}, t^{\prime\prime}|\mathbf{x}^{\prime}, t^{\prime}) &= \left[ \frac{\lambda\_0}{2\pi i \hbar [\exp(-\lambda\_0 t^{\prime}) - \exp(-\lambda\_0 t^{\prime\prime})]} \right]^{\frac{1}{2}} \\ &\times \exp\left\{ \frac{i\lambda\_0}{2\hbar} \frac{(\mathbf{x}^{\prime\prime} - \mathbf{x}^{\prime})^2}{\exp(-\lambda\_0 t^{\prime}) - \exp(-\lambda\_0 t^{\prime\prime})} \right\}, \end{split} \tag{30}$$

respectively. A driving force *F*<sup>0</sup> (e.g., terrestrial gravitation) could be added and then removed by *x* → *x* + (*F*0/*λ*0)*t*.

Further examples can be found in [13–15]. An explicitly time-dependent example is presented in Section 5.2.

#### **3. The Eisenhart–Duval Lift**

Further insight can be gained by "Eisenhart–Duval (E-D) lifting" the system to one higher dimension to what is called a "Bargmann space" [17–21]. The latter is a *d* + 1 + 1 dimensional manifold endowed with a Lorentz metric, the general form of which is:

$$g\_{\mu\nu}dx^{\mu}dx^{\nu} = g\_{i\bar{j}}(\mathbf{x},t)dx^{i}dx^{j} + 2dtds - \mathcal{D}V(\mathbf{x},t)dt^{2},\tag{31}$$

which carries a covariantly constant null Killing vector *∂<sup>s</sup>* . Then:

**Theorem 1** ([18,20])**.** *Factoring out the foliation generated by ∂<sup>s</sup> yields a nonrelativistic spacetime in d* + 1 *dimensions. Moreover, the null geodesics of the Bargmann metric gµν project to ordinary spacetime, consistent with Newton's equations. Conversely, if* (*γ*(*t*), *t*) *is a solution of the nonrelativistic equations of motion, then its null lifts to Bargmann space are:*

$$\left(\gamma(t), t, s(t)\right), \quad s(t) = s\_0 - \mathcal{A}(\gamma) = s\_0 - \int\_{}^{t} \mathcal{L}(\gamma(r), r) dr \tag{32}$$

*where s*<sup>0</sup> *is an arbitrary initial value.*

Let us consider, for example, a particle of unit mass with the Lagrangian of:

$$L = \frac{1}{2\alpha(t)} g\_{i\bar{j}}(\mathbf{x}^k) \dot{\mathbf{x}}^i \dot{\mathbf{x}}^j - \beta(t) V(\mathbf{x}^i, t), \tag{33}$$

where *gij*(*x k* )*dxidx<sup>j</sup>* is a positive metric on a curved configuration space *Q* with local coordinates *x i* , *i* = 1, . . . , *d*. The coefficients *α*(*t*) and *β*(*t*) may depend on time *t*, and *V*(*x i* , *t*) is some (possibly time-dependent) scalar potential. The associated equations of motion are:

$$\frac{d^2\mathbf{x}^i}{dt^2} + \Gamma^i\_{jk}\frac{d\mathbf{x}^j}{dt}\frac{d\mathbf{x}^k}{dt} - \frac{\hbar}{a}\frac{d\mathbf{x}^i}{dt} = -a\beta\mathbf{g}^{ij}\partial\_j V\,.\tag{34}$$

where the Γ *i jk* are the Christoffel symbols of the metric *gij*. For *d* = 1, *gij* = *δij* and *V* = <sup>1</sup> 2*ω*<sup>2</sup> (*t*)*x* 2 for *α* = *β* = 1, resp. for *α* = *β* <sup>−</sup><sup>1</sup> = *e* −*λ*(*t*) , we obtain a (possible timedependent) 1d oscillator without, resp. with, friction, Equation (7) [7–9,29–31].

Equation (34) can also be obtained by projecting a null-geodesic of *d* +1+1-dimensional Bargmann spacetime with coordinates (*x µ* ) = (*x i* , *t*,*s*), whose metric is:

$$g\_{\mu\nu}dx^{\mu}dx^{\nu} = \frac{1}{a}g\_{ij}dx^{i}dx^{j} + 2dtds - 2\beta Vdt^{2}.\tag{35}$$

For *α* = *β* <sup>−</sup><sup>1</sup> = *e* −*λ*(*t*) , we recover (12).

Choosing *λ*(*t*) = ln *m*(*t*) would describe motion with a time-dependent mass *m*(*t*). The friction can be removed by the conformal rescaling *x* → *y* = √ *m x*, and the null geodesics of the rescaled metric describe, consistent with (14), an oscillator with no friction, but with a time-dependent frequency, <sup>Ω</sup><sup>2</sup> <sup>=</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*¨ <sup>2</sup>*<sup>m</sup>* + *m*˙ 2*m* ) 2 [37].

The friction term −(*α*˙ /*α*)*x*˙ *i* in (34) can be removed also by introducing a new time parameter ˜*t*, defined by *<sup>d</sup>*˜*<sup>t</sup>* <sup>=</sup> *<sup>α</sup> dt* [21]. For *<sup>λ</sup>*(*t*) = *<sup>λ</sup>*0*t*, for example, putting ˜*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*<sup>e</sup>* <sup>−</sup>*λ*0*t*/*λ*<sup>0</sup> eliminates the friction, but it does this at the price of obtaining a manifestly time-dependent frequency [38,39]:

$$\frac{d^2\mathbf{x}}{d\mathbf{f}^2} + \boldsymbol{\Omega}^2(\mathbf{f})\mathbf{x} = \mathbf{0}, \qquad \boldsymbol{\Omega}^2(\mathbf{f}) = \frac{\omega^2}{\vec{F}^2 \lambda\_0^2}. \tag{36}$$

#### *3.1. The Junker–Inomata Ansatz as a Conformal Transformation*

The approach outlined in Section 2 admits a Bargmannian interpretation. For simplicity, we only consider the frictionless case *λ* = 0.

**Theorem 2.** *The Junker–Inomata method of converting the time-dependent system into one with a constant frequency by switching from "real" to "fake time",*

$$t \to \pi(t), \qquad \mathfrak{F} = \mathfrak{k}\,\mathfrak{x} \tag{37}$$

*induces a conformal transformation between the Bargmann metrics:*

$$\int d\mathbf{x}^2 + 2dt ds - \omega^2(t)\mathbf{x}^2 dt^2 \qquad \qquad \qquad \text{frequency} \qquad \omega^2(t) \tag{38a}$$

$$d\xi^2 + 2d\tau d\sigma - \bar{\omega}^2 \xi^2 d\tau^2 \, , \qquad \qquad \text{frequency} \qquad \bar{\omega} = \text{const.} \tag{38b}$$

$$d\xi^2 + 2d\tau d\sigma - \bar{\omega}^2 \xi^2 d\tau^2 = \mathfrak{r}(t) \left( d\mathbf{x}^2 + 2dt ds - \omega^2(t) \mathbf{x}^2 dt^2 \right). \tag{39}$$

**Proof.** Putting *µ* = ln *τ*˙ allows us to present the constant-frequency *ω*¯ (19) as:

$$
\omega^2 = \mathfrak{t}^{-2} \left( \omega^2(t) - \frac{1}{2}\vec{\mu} + \frac{1}{4}\vec{\mu}^2 \right). \tag{40}
$$

Then, with the notation ◦ *ξ*= *dξ*/*dτ*, we find,

$$\stackrel{\circ}{\xi}{}^2 = \mathfrak{t}^{-1} \left[ \mathfrak{x}^2 + \frac{1}{4} \dot{\mu}^2 \mathfrak{x}^2 - \frac{1}{2} \dot{\mu} \mathfrak{x}^2 + \frac{d}{dt} \left( \frac{1}{2} \dot{\mu} \mathfrak{x}^2 \right) \right].$$

Let us now recall that the null lift to the Bargmann space of a spacetime curve is obtained by subtracting the classical action as the vertical coordinate,

$$d\sigma = -L(\check{\xi}, \stackrel{\circ}{\xi}, \tau)d\tau = -\frac{1}{2}(\stackrel{\circ}{\xi}^2 - \bar{\omega}^2 \tilde{\xi}^2)d\tau. \tag{41}$$

Setting here *ξ* = *xτ*˙ 1/2 and dropping surface terms yield, using the same procedure for the time-dependent-frequency case,

$$d\sigma = ds = -\frac{1}{2} \left( \dot{\mathbf{x}}^2 - \omega^2(t)\mathbf{x}^2 \right) dt \tag{42}$$

up to surface terms. Then, inserting all our formulae into (38a) and (38b) yields (39), as stated. In Junker–Inomata language (10), *f*(*t*) = *τ*˙ 1/2 sec(*ωτ*¯ ), *g*(*t*) = (*ω*¯) −1 tan(*ωτ*¯ ).

Our investigation has so far concerned classical aspects. Now, we consider what happens quantum mechanically. Restricting our attention at *d* = 1 space dimensions as before (In *d* > 2, conformal invariance requires adding a scalar curvature term to the Laplacian.), we posit that the E-D lift *ψ*e of a wave function *ψ* is equivariant,

$$
\tilde{\psi}(\mathbf{x},t,\mathbf{s}) = e^{\frac{i}{\hbar}s} \psi(\mathbf{x},t) \quad \Rightarrow \quad \partial\_{\mathbf{s}} \tilde{\psi} = \frac{i}{\hbar} \tilde{\psi} \,. \tag{43}
$$

Then, the massless Klein–Gordon equation for *ψ*e associated with the 1 + 1 + 1 = 3 d Barmann metric implies the Schrödinger equation in 1+1 d,

$$i\Delta\_{\mathcal{S}}\tilde{\psi} = 0 \quad \Rightarrow \quad i\partial\_t \psi = \left[ -\frac{\hbar^2}{2}\Delta\_{\mathcal{X}} + V(\mathfrak{x}, t) \right] \psi \tag{44}$$

where ∆*<sup>g</sup>* is the Laplace–Beltrami operator associated with the metric. In *d* = 1, it is of course ∆*<sup>x</sup>* = *∂* 2 *x* .

A conformal diffeomorphism (*X*, *<sup>T</sup>*, *<sup>S</sup>*) <sup>→</sup> ˜ *f*(*X*, *T*, *S*) = (*x*, *t*,*s*) with conformal factor *σ* 2 *f* , ˜ *f* <sup>∗</sup>*gµν* = *σ* 2 *f gµν*, projects to a spacetime transformation (*X*, *T*) → *f*(*X*, *T*) = (*x*, *t*). It is implemented on a wave function lifted to the Bargmann space as:

$$
\tilde{\psi}(\mathbf{x}, t, \mathbf{s}) = \sigma\_f^{-1/2} \tilde{\psi}(\mathbf{X}, T, \mathbf{S}) \tag{45}
$$

In Sections 4.2, these formulae are applied to the Niederer map (73).

#### *3.2. The Arnold Map*

The general damped harmonic oscillator with time-dependent driving force *F*(*t*) in 1 + 1 dimensions, (11),

$$
\ddot{\mathbf{x}} + \dot{\lambda}\dot{\mathbf{x}} + \omega^2(t)\mathbf{x} = F(t) \,. \tag{46}
$$

can be solved by an *Arnold transformation* [22,23], which "straightens the trajectories" [21,29–31,40]. To this end, one introduces new coordinates,

$$T = \frac{\mu\_1}{\mu\_2}, \quad X = \frac{\mathbf{x} - \mu\_p}{\mu\_2}, \tag{47}$$

where *u*<sup>1</sup> and *u*<sup>2</sup> are solutions of the associated homogeneous Equation (46) with *F* ≡ 0 and *u<sup>p</sup>* is a particular solution of the full Equation (46). It is worth noting that (47) allows checking, independently, the Junker–Inomata criterion in (10). The initial conditions are chosen as,

$$
\mu\_1(t\_0) = \dot{\mathfrak{u}}\_2(t\_0) = 0, \ \dot{\mathfrak{u}}\_1(t\_0) = \mathfrak{u}\_2(t\_0) = 1, \ \mathfrak{u}\_p(t\_0) = \dot{\mathfrak{u}}\_p(t\_0) = 0. \tag{48}
$$

Then, in the new coordinates, the motion becomes free [22,23],

$$X(T) = aT + b \,, \qquad a, b = \text{const.} \tag{49}$$

Equation (46) can be obtained by projecting a null geodesic of the Bargmann metric:

$$g\_{\mu\nu}d\mathbf{x}^{\mu}d\mathbf{x}^{\nu} = e^{\lambda(t)}d\mathbf{x}^2 + 2dtds - 2e^{\lambda(t)}\left(\frac{1}{2}\omega(t)^2\mathbf{x}^2 - F(t)\mathbf{x}\right)dt^2. \tag{50}$$

Completing (47) by:

$$S = s + e^{\lambda}u\_2^{-1} \left(\frac{1}{2}\dot{u}\_2 x^2 + \dot{u}\_p x\right) + g(t) \quad \text{where} \quad \not{g} = \frac{1}{2}e^{\lambda} \left(\dot{u}\_p^2 - \omega^2 u\_p^2 + 2Fu\_p\right) \tag{51}$$

lifts the Arnold map to Bargmann spaces, (*x*, *t*,*s*) → (*X*, *T*, *S*) (In the Junker–Inomata setting (10), *f* = *u*2*e* <sup>−</sup>*λ*/2 and *g*(*t*) = *u*1/*u*2.),

$$\mathcal{g}\_{\mu\nu}dx^{\mu}dx^{\nu} = e^{\lambda(t)}u\_2^2(t)\left(dX^2 + 2dTdS\right). \tag{52}$$

The oscillator metric (50) is thus carried conformally to the free one, generalizing earlier results [18,19,41]. For the damped harmonic oscillator with *λ*(*t*) = *λ*0*t* and *F*(*t*) ≡ 0, *u<sup>p</sup>* ≡ 0 is a particular solution. When *ω* = *ω*<sup>0</sup> = const., for example,

$$u\_1 = e^{-\lambda\_0 t/2} \frac{\sin \Omega\_0 t}{\Omega\_0}, \ u\_2 = e^{-\lambda\_0 t/2} \left(\cos \Omega\_0 t + \frac{\lambda\_0}{2\Omega\_0} \sin \Omega\_0 t\right), \quad \Omega\_0^2 = \omega\_0^2 - \lambda\_0^2/4 \tag{53}$$

are two independent solutions of the homogeneous equation with initial conditions (48) and provide us with:

$$T\_{\parallel} = \frac{\sin \Omega\_0 t}{\Omega\_0 (\cos \Omega\_0 t + \frac{\lambda\_0}{2\Lambda\_0} \sin \Omega\_0 t)} ' \tag{54}$$

$$X\_{\parallel} = \frac{e^{\lambda\_0 t/2} \,\mathrm{x}}{\cos \Omega\_0 t + \frac{\lambda\_0}{2\Omega\_0} \sin \Omega\_0 t},\tag{55}$$

$$\mathcal{S}\_{\theta} = s - \frac{1}{2} e^{\lambda\_0 t} x^2 \left( \frac{\omega\_0^2}{\Omega\_0} \right) \frac{\sin \Omega\_0 t}{\cos \Omega\_0 t + \frac{\lambda\_0}{2 \Omega\_0} \sin \Omega\_0 t} \,. \tag{56}$$

In the undamped case, *λ*<sup>0</sup> = 0; thus, Ω<sup>0</sup> = *ω*0, and (56) reduces to that of Niederer [16] lifted to the Bargmann space [19,20],

$$T = \frac{\tan \omega\_0 t}{\omega\_0}, \qquad X = \frac{\mathbf{x}}{\cos \omega\_0 t}, \qquad S = s - \frac{1}{2} \mathbf{x}^2 \omega\_0 \tan \omega\_0 t. \tag{57}$$

The Junker–Inomata construction in Section 2 can be viewed as a particular case of the Arnold transformation. We chose *u<sup>p</sup>* ≡ 0 and the two independent solutions:

$$u\_1 = e^{-\lambda/2} \, t^{-1/2} \frac{\sin \bar{\omega} \tau}{\bar{\omega}}, \qquad u\_2 = e^{-\lambda/2} \, t^{-1/2} \cos \bar{\omega} \tau. \tag{58}$$

The initial conditions (48) at *t*<sup>0</sup> = 0 imply *τ*(0) = *ρ*˙(0) = 0, *ρ*(0) = *τ*˙(0) = 1. Then, spelling out (51),

$$S = s - \frac{1}{2}e^{\lambda} \left(\bar{\omega}\dagger\tan\bar{\omega}\,\tau + \frac{1}{2}\dot{\lambda} + \frac{1}{2}\frac{\sharp}{\dagger}\right)\chi^2\tag{59}$$

completes the lift of (22) to Bargmann spaces. In conclusion, the one-dimensional damped harmonic oscillator is described by the conformally flat Bargmann metric,

$$
\delta g\_{\mu\nu}dx^{\mu}dx^{\nu} = \frac{\cos^2\bar{\omega}\,\mathrm{r}}{\dot{\pi}} \Big(dX^2 + 2dTdS\Big).\tag{60}
$$

The metric (60) is manifestly conformally flat; therefore, its geodesics are those of the free metric, *X*(*T*) = *aT* + *b*. Then, using (47) with (58) yields:

$$\mathbf{x}(t) = e^{-\lambda(t)/2} \, t^{-1/2}(t) \left( a \frac{\sin[\bar{\omega}\tau(t)]}{\bar{\omega}} + b \cos[\bar{\omega}\tau(t)] \right). \tag{61}$$

The bracketed quantity here describes a constant-frequency oscillator with "time" *τ*(*t*). The original position, *x*, obtains a time-dependent "conformal" scale factor.

#### **4. The Maslov Correction**

As mentioned before, the semiclassical formula (9) is correct only in the first oscillator half-period, 0 < *t* <sup>00</sup> − *t* <sup>0</sup> < *π*/Ω0. Its extension for all *t* involves the *Maslov correction*. In the constant-frequency case with no friction, for example, assuming that Ω0(*t* <sup>00</sup> − *t* 00)/*π* is not an integer, we have [2,3,6],

$$\left(\mathbf{K}^{\rm ext}(\mathbf{x}'',t''|\mathbf{x}',t')\right) \quad = \left[\frac{\Omega\_0}{2\pi\hbar \left|\sin\Omega\_0(t''-t')\right|}\right]^{\frac{1}{2}} \times e^{-i\frac{\pi}{4}(1+2\ell)}\tag{62}$$

$$\exp\left\{\frac{i\Omega\_0}{2\hbar \sin\Omega\_0(t''-t')} \left[(\mathbf{x}'^2+\mathbf{x}'^2)\cos\Omega\_0(t''-t')-2\mathbf{x}''\mathbf{x}'\right]\right\},$$

where the integer:

$$\ell = \text{Ent}\left[\frac{\Omega\_0(t'' - t')}{\pi}\right] \tag{63}$$

is called the *Maslov index* (Ent[*x*] is the integer part of *x*.). ` counts the completed half-periods and is related also to the *Morse index*, which counts the negative modes of *∂* <sup>2</sup>A/*∂<sup>x</sup>* 0*∂x* 00 [4,5]. Now, we generalize (62) to the *time-dependent* frequency:

**Theorem 3.** *In terms of ω*¯ *and τ introduced in Section 2,*

• *Outside caustics, i.e., for ω*¯(*τ* <sup>00</sup> − *τ* 0 ) 6= *π*`*, the propagator for the harmonic oscillator with the time-dependent frequency and friction is:*

$$K^{\rm ext}(\mathbf{x}^{\prime\prime},t^{\prime\prime}|\mathbf{x}^{\prime},t^{\prime}) = \left[\frac{\omega e^{\frac{i\boldsymbol{\mathcal{V}}\_{\star}+\boldsymbol{\mathcal{V}}}{2}}(t^{\prime\prime}t^{\prime})^{\frac{1}{2}}}{2\pi\hbar[\sin\bar{\omega}(\tau^{\prime\prime}-\tau^{\prime})]}\right]^{1/2} \exp\left\{-\frac{i\pi}{2}\left(\frac{1}{2} + \text{Ent}\left[\frac{\tilde{\omega}(\tau^{\prime\prime}-\tau^{\prime})}{\pi}\right]\right)\right\} \tag{64}$$

$$\times \exp\left\{\frac{i\tilde{\omega}}{2\hbar\sin\bar{\omega}(\tau^{\prime\prime}-\tau^{\prime})}[(\mathbf{x}^{\prime\prime}e^{\boldsymbol{\omega}^{\prime\prime}}t^{\prime\prime}+\mathbf{x}^{\prime}e^{\boldsymbol{\omega}^{\prime}}t^{\prime})\cos[\tilde{\omega}(\tau^{\prime\prime}-\tau^{\prime})]-2\mathbf{x}^{\prime\prime}\mathbf{x}^{\prime}e^{\frac{i\boldsymbol{\mathcal{V}}\_{\star}+\boldsymbol{\mathcal{V}}}{2}}(t^{\prime\prime}t^{\prime})^{\frac{1}{2}}}{2}\right\} \quad; \quad \pm \frac{\omega\_{\rm e}}{2}$$

• *At caustics, i.e., for:*

$$
\bar{\omega}(\pi'' - \pi') = \pi \,\ell, \quad \ell = 0, \pm 1, \ldots \tag{65}
$$

*we have instead [3,6],*

$$K^{\rm ext}\left(\mathbf{x}^{\prime\prime},\mathbf{x}^{\prime},|\mathbf{\tau}^{\prime\prime}-\mathbf{\tau}^{\prime}=\frac{\pi}{\widehat{\boldsymbol{\omega}}}\boldsymbol{\ell}\right)=\left[e^{\frac{\lambda^{\prime\prime}+\lambda^{\prime}}{2}}(t^{\prime\prime}t^{\prime})^{\frac{1}{2}}\right]^{1/2}\tag{66}$$

$$\times \exp\left(-\frac{i\pi\ell}{2}\right)\delta\left(\mathbf{x}^{\prime}\exp(\lambda^{\prime}/2)t^{\prime1/2}-(-1)^{k}\mathbf{x}^{\prime\prime}\exp(\lambda^{\prime\prime}/2)t^{\prime\prime1/2}\right).$$

**Proof.** In terms of the redefined coordinates:

$$
\tau = \tau(t) \quad \text{and} \quad \xi = \ge \exp\left[\frac{\lambda(t)}{2}\right] t^{1/2}(t), \tag{67}
$$

cf. (37), and using the notation ◦ { · }= *d*/*dτ*, the time-dependent oscillator Equation (12) is taken into:

$$\stackrel{\circ}{\mathfrak{F}}^{\circ \circ} + \bar{\omega}^2 \mathfrak{F} = 0, \quad \text{where} \quad \bar{\omega}^2 = \frac{1}{\tau^2} \left( \omega^2(t) - \frac{1}{2} \frac{\ddagger}{\dot{\tau}} + \frac{3}{4} \left( \frac{\ddot{\tau}}{\dot{\tau}} \right)^2 - \frac{\ddot{\lambda}}{2} - \frac{\dot{\lambda}^2}{4} \right). \tag{68}$$

Thus, the problem is reduced to one with a *time-independent* frequency, *ω*¯ in (19) (We record for the sake of later investigations that (turning off *λ*) (68) can be presented as:

$$
\omega^2(t) - \dot{\mathbf{r}}^2 \omega^2 = \frac{1}{2} \mathbf{S}(\tau) \tag{69}
$$

where *S*(*τ*) = ... *τ <sup>τ</sup>*˙ <sup>−</sup> <sup>3</sup> 2 *τ*¨ *τ*˙ 2 is the *Schwarzian derivative* of *τ* [42]).

Let us now recall Formula (19) of Junker and Inomata in [12], which tells us how propagators behave under the coordinate transformation (*ξ*, *τ*) ←→ (*x*, *t*):

$$K\_2(\mathbf{x}^{\prime\prime}, t^{\prime\prime}|\mathbf{x}^{\prime}, t^{\prime}) = \left[ \left( \frac{\partial \xi^{\prime}}{\partial \mathbf{x}^{\prime}} \right) \left( \frac{\partial \xi^{\prime\prime}}{\partial \mathbf{x}^{\prime\prime}} \right) \right]^{\frac{1}{2}} K\_1(\xi^{\prime\prime}, \mathbf{r}^{\prime\prime}|\xi^{\prime}, \mathbf{r}^{\prime}) \,. \tag{70}$$

Here, *K*<sup>2</sup> = *K ext* is the propagator of an oscillator with a time-dependent frequency and friction, *ω*(*t*) and *λ*(*t*), respectively—the one we are trying to find. *K*<sup>1</sup> is in turn the Maslov-extended propagator of an oscillator with no friction and a constant frequency, as in (62). Then, the propagator for the *harmonic oscillator with a time-dependent frequency and friction*, Equation (64), is obtained using (67).

Notice that (64) is *regular* at the points *r<sup>k</sup>* ∈ *J<sup>k</sup>* where sin = ±1. However, at caustics, *τ* <sup>00</sup> − *τ* 0 = (*π*/*ω*¯)`, *K ext* diverges, and we have instead (66).

Henceforth, we limit our investigations to *λ* = 0.

#### *4.1. Properties of the Niederer Map*

More insight is gained from the perspective of the generalized Niederer map (22). We first study their properties in some detail. For simplicity, we chose, in the rest of this section, *x* 0 = *t* 0 = 0 and *x* <sup>00</sup> ≡ *x* and *t* <sup>00</sup> ≡ *t*.

We start with the observation that the Niederer map (22) becomes singular where the cosine vanishes, i.e., where:

$$\cos[\bar{\omega}\tau(r\_k)] = 0, \quad \text{i.e.} \quad \tau(r\_k) = (k + \frac{1}{2})\frac{\pi}{\bar{\omega}}, \quad k = 0, \pm 1, \ldots \tag{71}$$

*r<sup>k</sup>* < *rk*+<sup>1</sup> because *τ*(*t*) is an increasing function by (21). Moreover, each interval:

$$I\_k = \begin{bmatrix} r\_{k\prime} r\_{k+1} \end{bmatrix}, \quad k = 0, \pm 1, \dots \tag{72}$$

is mapped by (22) onto the full range −∞ < *T* < ∞. Therefore, the inverse mapping is *multivalued*, labeled by integers *k*,

$$N\_k: T \to t = \frac{\arctan\_k \tilde{\omega} T}{\tilde{\omega}}, \quad X \to \infty = \frac{X}{\sqrt{1 + \tilde{\omega}^2 T^2}},\tag{73}$$

where arctan*<sup>k</sup>* (·) = arctan0(·) + *kπ* with arctan0(·) the principal determination, i.e., in (−*π*/2, *π*/2).

Then, lim*t*→*rk*<sup>−</sup> tan *t* = ∞ and lim*t*→*rk*<sup>+</sup> tan *t* = −∞ imply that:

$$\lim\_{T \to \infty} N\_k(T) = \, ^\ast r\_{k+1} = \lim\_{T \to -\infty} N\_{k+1}(T) \,. \tag{74}$$

Therefore, the intervals *I<sup>k</sup>* and *Ik*+<sup>1</sup> are joined at *rk*+<sup>1</sup> , and the *I<sup>k</sup>* form a partition of the time axis, − ∞ < *t* < ∞ = ∪*<sup>k</sup> Ik* .

Returning to (64) (which is (62) with Ω<sup>0</sup> ⇒ *ω*¯ , *t* ⇒ *τ*), we then observe that, whereas the propagator is regular at *r<sup>k</sup>* , it diverges at caustics,

$$\sin[\tilde{\omega}\tau(t\_\ell)] = 0 \quad \text{i.e.,} \quad \tau(t\_\ell) = \frac{\pi}{\tilde{\omega}}\,\ell, \quad \ell = 0, \pm 1, \ldots, \tag{75}$$

cf. (65). Thus, *t*` ≤ *t*`+<sup>1</sup> , and:

$$N\_k(-\\\\\infty) = r\_k \\\\\quad N\_k(T=0) = t\_{k+1} \\\quad N\_k(+\\\infty) = r\_{k+1} \\\. \tag{76}$$

Thus, *N<sup>k</sup>* maps the full *T*-line into *I<sup>k</sup>* with *t<sup>k</sup>* an internal point. Conversely, *r<sup>k</sup>* is an *internal point* of *J<sup>k</sup>* . The intervals *J*` = - *t*` , *t*`+<sup>1</sup> cover again the time axis, ∪` *J*` = − ∞ < *t* < ∞ .

By (61) the classical trajectories are regular at *t* = *r<sup>k</sup>* . Moreover, for arbitrary initial velocities,

$$
\sqrt{\dot{\mathbf{r}}(t\_{\ell+1})} \ge (t\_{\ell+1}) = -\sqrt{\dot{\mathbf{r}}(t\_{\ell})} \ge (t\_{\ell}) \tag{77}
$$

implying that after a half-period *ωτ*¯ → *ωτ*¯ + *π*, all classical motions are focused at the same point. The two entangled sets of intervals are shown in Figure 1.

**Figure 1.** The generalized Niederer map (22) maps each interval **I<sup>k</sup>** = (**rk**,**rk**+**<sup>1</sup>** ) onto the entire real line −∞ < *T* < ∞. Its inverse mapping is therefore multivalued, labeled by an integer *k*. The classical motions and the propagator are both regular at the separation points **rk**. All classical trajectories are focused at the caustic points **t**` , where the propagator diverges.

The Niederer map (57) "E-D lifts" to the Bargmann space.

**Theorem 4.** *The E-D lift of the inverse of the Niederer map* (57)*, which we shall denote by N*e *k* : (*X*, *T*, *S*) → (*x*, *t*,*s*) *(t* ∈ *I<sup>k</sup> ), is:*

$$t = \frac{\arctan\_k \bar{\omega} T}{\bar{\omega}}, \quad \mathbf{x} = \frac{\mathbf{X}}{\sqrt{1 + \bar{\omega}^2 T^2}}, \quad s = \mathbf{S} + \frac{\mathbf{X}^2}{2} \frac{\bar{\omega}^2 T}{1 + \bar{\omega}^2 T^2}. \tag{78}$$

**Proof.** These formulae follow at once by inverting (57), at once with the cast *ω*<sup>0</sup> ⇒ *ω*¯ , *t* ⇒ *τ*. Alternatively, it could also be proven as for Theorem 2.

For each integer *k* (78) maps the real line −∞ < *T* < ∞ into the "open strip" [19] - *rk* ,*rk*+<sup>1</sup> <sup>×</sup> <sup>R</sup><sup>2</sup> <sup>≡</sup> *<sup>I</sup><sup>k</sup>* <sup>×</sup> <sup>R</sup><sup>2</sup> with *<sup>r</sup><sup>k</sup>* defined in (71). Their union covers the entire Bargmann manifold of the oscillator.

Now, we pull back the free dynamics by the multivalued inverse (78). We put *ω*¯ = 1 for simplicity. The free motion with initial condition *X*(0) = 0,

$$X(T) = aT, \quad S(T) = S\_0 - \frac{a^2}{2}T,\tag{79}$$

E-D lifts by (78) to:

*x*(*t*) = *a* sin *t s*(*t*) = *S*<sup>0</sup> − *a* 2 4 sin 2*t* , (80)

consistent with *<sup>s</sup>*(*t*) = *<sup>s</sup>*<sup>0</sup> <sup>−</sup> <sup>A</sup>¯ *osc*, as can be checked directly. Note that the *s* coordinate oscillates with a doubled frequency.


#### *4.2. The Propagator by the Niederer Map*

Now, we turn to quantum dynamics. Our starting point is the free propagator (5), which (as mentioned before) is valid only for 0 < *T* <sup>00</sup> − *T* 0 . Its extension to all *T* involves the *sign* of (*T* <sup>00</sup> − *T* 0 ) [19].

Let us explain this subtle point in some detail. First of all, we notice that the usual expression (5) involves a square root, which is double-valued, obliging us to *choose* one of its branches. Which one we choose is irrelevant: it is a mere gauge choice. However, once we do choose one, we must stick to our choice. Take, for example, the one for which √ −*i* = *e* <sup>−</sup>*iπ*/4, then the prefactor in (5) is:

$$\left[\frac{1}{2\pi i \hbar (T'' - T')}\right]^{1/2} = e^{-i\pi/4} \left[\frac{1}{2\pi \hbar \left|T'' - T'| }\right|\right]^{1/2}.$$

Let us now consider what happens when *T* <sup>00</sup> − *T* 0 changes sign. Then, the prefactor becomes multiplied by <sup>√</sup> −1 so it becomes, *for the same choice of the square root*,

$$e^{i\pi/2}e^{-i\pi/4} \left[\frac{1}{2\pi\hbar \left|T'' - T'\right|}\right]^{1/2} = e^{+i\pi/4} \left[\frac{1}{2\pi\hbar \left|T'' - T'\right|}\right]^{1/2}.\tag{81}$$

In conclusion, the formula valid for all *T* is,

$$K\_{free}(X'',T'|X',T') = e^{-i\frac{\pi}{4}\text{sign}(T''-T')} \left[\frac{1}{2\pi\hbar|T'-T'|}\right]^{1/2} \exp\left\{\frac{i}{\hbar}\mathcal{A}\_{free}\right\},\tag{82}$$

where:

$$\bar{\mathcal{A}}\_{free} = \frac{(X'' - X')^2}{2(T'' - T')} \tag{83}$$

is the free action calculated along the classical trajectory. Let us underline that (82) already involves a "Maslov jump" *e* <sup>−</sup>*iπ*/2 , which, for a free particle, happens at *T* = 0. For *T* <sup>00</sup> − *T* <sup>0</sup> = 0, we have *Kf ree* = *δ*(*X* <sup>00</sup> − *X* 0 ).

Accordingly, the wave function Ψ ≡ Ψ*f ree* of a free particle is, by (1),

$$\Psi(\mathbf{X}^{\prime\prime},T^{\prime\prime}) = e^{-i\frac{\pi}{4}\text{sign}(T^{\prime}-T^{\prime})} \left[\frac{1}{2\pi\hbar|T^{\prime\prime}-T^{\prime}|}\right]^{1/2} \int\_{\mathbb{R}} \exp\left\{\frac{i}{\hbar}\vec{\mathcal{A}}\_{free}\right\} \Psi(\mathbf{X}^{\prime},T^{\prime})d\mathbf{X}^{\prime}.\tag{84}$$

Now, we pull back the free dynamics using the multivalued inverse Niederer map. It is sufficient to consider the constant-frequency case *ω*¯ = const. and to denote time by *t*. Let *t* belong to the range of *N<sup>k</sup>* in (73), *t* ∈ *I<sup>k</sup>* = [*r<sup>k</sup>* ,*rk*+<sup>1</sup> ] = *N<sup>k</sup>* {−∞ < *T* < ∞} . Then, applying the general formulae in Section 3.1 yields [19],

$$\begin{split} \tilde{\psi}(\mathbf{x}'',t'',s'') &= \ \cos^{-1/2}[\tilde{\omega}(t''-t')]\Psi(\mathbf{x}'',\mathbf{t''},\mathbf{s}'') = e^{-\frac{i\pi}{\hbar}\text{sign}\left(\frac{\tan\tilde{\omega}(t''-t'')}{\tilde{\omega}}\right)} \times \\ &\quad \cos^{-1/2}[\tilde{\omega}(t''-t')]\exp\left(\frac{i}{\hbar}\mathbf{s}''\right)\exp\left(-\frac{i}{\hbar}(\frac{1}{2}\boldsymbol{\alpha}\mathbf{x}''^{\prime\prime}\tan[\boldsymbol{\omega}(t''-t'')])\right) \\ &= \sqrt{\frac{|\boldsymbol{\omega}|}{2\pi\hbar|\tan[\tilde{\omega}(t''-t')]|}} \int\_{\mathbb{R}}\exp\left\{\frac{i}{\hbar}\frac{\boldsymbol{\mathcal{D}}|\frac{\mathbf{x}''}{\cos[\tilde{\omega}(t''-t')]}-\mathbf{x}'|^{2}}{2\tan[\tilde{\omega}(t''-t')]}\right\}\psi(\mathbf{x}',t')d\mathbf{x}'. \end{split}$$

However, the second exponential in the middle line combines with the integrand in the braces in the last line to yield *the action calculated along the classical oscillator trajectory*,

$$\mathcal{A}\_{\rm osc} = \frac{\varpi}{2 \sin \varpi (t'' - t')} \left( (\mathbf{x''^2} + \mathbf{x'^2}) \cos \varpi (t'' - t') - 2 \mathbf{x''} \mathbf{x'} \right). \tag{85}$$

Thus, using the equivariance, we end up with,

$$\psi\_{\rm osc}(\mathbf{x}'',t'') = \cos^{-1/2}[\hat{\omega}(t''-t')]\exp\left[-\frac{i\pi}{4}\text{sign}\left(\frac{\tan[\hat{\omega}(t''-t')]}{\hat{\omega}}\right)\right] \times \qquad \text{(86)}$$

$$\sqrt{\frac{|\hat{\omega}|}{2\pi\hbar|\tan[\hat{\omega}(t''-t')]|}}\int\_{\mathbb{R}}\exp\left\{\frac{i}{\hbar}\mathcal{A}\_{\rm osc}\right\}\psi\_{\rm osc}(\mathbf{x}',t')d\mathbf{x}\,.$$

Now, we recover the Maslov jump, which comes from the first line here. For simplicity, we consider again *t* 0 = 0, *x* 0 = 0 and denote *t* 00 = *t*, *x* 00 = *x*.

Firstly, we observe that the conformal factor cos *ω*¯ *t* has a constant sign in the domain *I<sup>k</sup>* and changes sign at the end points. In fact,

$$\cos\omega t = \left(-1\right)^{k+1} |\cos\omega t| \quad \Rightarrow \quad \cos^{-1/2}(\omega t) = e^{-i\frac{\pi}{\Delta}(k+1)} |\cos\omega t|^{-1/2}.\tag{87}$$

The cosine enters into the van Vleck factor, while the phase combines with exp<sup>h</sup> <sup>−</sup>*i<sup>π</sup>* 4 sign( tan *ω*¯ *t ω*¯ ) i . Recall now that *tk*+<sup>1</sup> = *N<sup>k</sup> T* = 0 divides *I<sup>k</sup>* into two pieces, *I<sup>k</sup>* = [*r<sup>k</sup>* , *tk*+<sup>1</sup> ] ∪ [*tk*+<sup>1</sup> ,*rk*+<sup>1</sup> ], cf. Figure 1. However, *tk*+<sup>1</sup> is precisely where the tangent changes sign: this term contributes to the phase in [*r<sup>k</sup>* , *tk*+<sup>1</sup> ] −*π*/4 and +*π*/4 in [*tk*+<sup>1</sup> ,*rk*+<sup>1</sup> ]. Combining the two shifts, we end up with the phase:

$$\begin{aligned} -\frac{\pi}{4}(1+2\ell) & \quad \text{for} \quad r\_k < t < t\_{k+1} \\ -\frac{\pi}{4}(1+2(\ell+1)) & \quad \text{for} \quad t\_{k+1} < t < r\_{k+1} \end{aligned} \quad \text{where} \quad \ell = \text{Ent}\left[\frac{\varpi\tau}{\pi}\right] = k+1 \tag{88}$$

which is the Maslov jump at *t*` .

Intuitively, the multivalued *N<sup>k</sup>* "exports" to the oscillator at *t*`+<sup>1</sup> the phase jump of the free propagator at *T* = 0. Crossing from *J*` to *J*`+<sup>1</sup> shifts the index ` by one.

#### **5. Probability Density and Phase of the Propagator: A Pictorial View**

#### *5.1. For a Constant Frequency*

We assume first that the frequency is constant. We split the propagator *K*(*x*, *t*) ≡ *K*(*x*, *t*|0, 0) in (62) as,

$$K(\mathbf{x},t) = \left| K(\mathbf{x},t) \right| P(t), \qquad P(t) = e^{i(\text{phase})}.\tag{89}$$

The *probability density*,

$$|K(\mathbf{x}, t)|^2 = \frac{\Omega\_0}{2\pi\hbar \left| \sin \Omega\_0 t \right|}\tag{90}$$

viewed as a surface above the *x* − *t* plane, diverges at *t* = *t*` = *π*`, ` = 0, ±1, . . . .

Representing the *phase of the propagator* would require four dimensions, though. However, recall that that the dominant contribution to the path integral should come from where the phase is stationary [1], i.e., from the neighborhood of classical paths *x*¯(*t*), distinguished by the vanishing of the first variation, *δ*A*x*¯ = 0. Therefore, we shall study the evolution of the phase along classical paths *x*¯(*t*) for which (61) yields, for ¯*h* = *ω*¯ = 1 and *a* ∈ R, *b* = 0,

$$\mathfrak{x}\_{\mathfrak{a}}(t) = a \sin t \quad \text{and} \quad P\_{\mathfrak{a}}(t) = \exp\left\{-\frac{i\pi}{4} \left[1 - \frac{a^2}{\pi} \sin 2t\right] - \frac{i\pi}{2}\ell\right\},\tag{91}$$

as depicted in Figure 2.

An intuitive understanding comes by noting that when *t* 6= *π*` = *t*` , then different initial velocities *a* yield classical paths *x*¯*a*(*t*) with different end points, and thus contribute to different propagators. However, approaching from the left `-times a half period, *t* → (*π* `)− , all classical paths become focused at the same end point (*x* = 0 for our choice) and for all *a*,

$$P\_{\mathfrak{a}}(t \to \pi\ell -) = e^{-i\frac{\pi}{4}(1+2\ell)} \equiv P\_{\ell}.\tag{92}$$

which is precisely the Maslov phase. Thus, *all classical paths contribute equally*, by *P*` , and to the *same* propagator. Comparing with the right-limit,

$$P\_{\mathfrak{a}}(t \to \pi\ell +) = e^{-i\frac{\pi}{4}(1+2(\ell+1))} = P\_{\ell+1} = e^{-\frac{i\pi}{2}}P\_{\ell}.\tag{93}$$

the Maslov jump is recovered. Choosing instead *y* 6= 0, there will be no classical path from (0, 0) to (*y*, *π*`), and thus no contribution to the path integral.

**Figure 2.** The phase factor **P**(*t*) of the propagator in (89) lies on the unit circle of the complex plane plotted vertically along a classical path *γ*¯(*t*). The orientation is positive if it is clockwise when seen from *t* = +∞. In the time interval *J*` labeled by the Maslov index ` = Ent[*t*/*π*], the factor **P**(*t*) precesses around **P**` = exp[−*i π* 4 (1 + 2`)] with double frequency w.r.t. the classical path, *γ*¯(*t*). Arriving at a caustic, the phase jumps by (−*π*/2) (**red** becoming **purple**) and then continues until the next caustic when it jumps again (and becomes **magenta**), and so on.

To conclude this section, we just mention with that the extended Feynman method [6] with the cast *ω*¯ = constant frequency and *τ* = "fake time" would lead also to (64) and (66) with the integer ` counting the number of negative eigenvalues (Morse index) of the Hessian [2,4,5,24].

#### *5.2. A Time-Dependent Example: The Mathieu Equation*

The combined Junker–Inomata–Arnold method allows us to go beyond the constantfrequency case, as illustrated here for no friction or driving force, *λ* = *F* ≡ 0, but with explicitly time-dependent frequency. For Ω<sup>2</sup> (*t*) = *a* − 2*q* cos 2*t*, for example, (14) becomes the Mathieu equation,

$$
\ddot{\mathbf{x}} + (a - 2q \cos 2t)\mathbf{x} = 0.\tag{94}
$$

This equation can be solved either analytically using Mathieu functions [43], or numerically, providing us for *a* = 2 and *q* = 1 (for which odd Mathieu functions are real) with the dotted curve (in **red**), shown in Figure 3.

Alternatively, we can use the Junker–Inomata–Arnold transformation (47) [22,23,40]. We first achieve *ω*¯ = 1 by a redefinition, *τ* → *τ* 0 = *ωτ*¯ . Inserting Ansatz (15) into (94) yields the pair of coupled Equations (16) and (17). We chose *u<sup>p</sup>* = 0 and two independent solutions *u*1(*t*) and *u*2(*t*), (58), with initial conditions (48) with *t*<sup>0</sup> = 0, i.e., *τ*(0) = *ρ*˙(0) =

0, *ρ*(0) = *τ*˙(0) = 1 , which fix the integration constant, *C* = *ρ* 2 (0)*τ*˙(0) = 1. Then, consistent with the general theory outlined above, the Arnold map (47) lifted to the Bargmann space becomes (22), completed with (59) with *λ* = 0.

**Figure 3.** The analytic solution of the Mathieu equation with *a* = 2, *q* = 1 for *x*(*t*) (**dotted** in **red**) lies on the **black** curve obtained by (15) from combining the numerically obtained *ρ*(*t*) (in **green**) and *τ*(*t*) (in **blue**), which are solutions of the pair (18)–(21). The **black** curve is also obtained by pulling back the free solution (49) by the inverse Niederer map (73).

Equation (17) is solved by following the strategy outlined in Section 2. Carrying out those steps numerically provides us with Figure 3.

From the general formula (24), we deduce, for our choice *x* 00 = *x*, *t* 00 = *t*, *x* 0 = *t* 0 = 0, that the probability density (The wave function is multiplied by the square root of the conformal factor, cf. (39).).

$$|\mathcal{K}(\mathbf{x},t)|^2 = \frac{\sqrt{\dddot{\mathbf{r}}}}{2\pi\hbar|\sin\pi(t)|}\,'\,\tag{95}$$

happens, not depending on the position, and can therefore be plotted as in Figure 4.

The propagator *K* and hence the probability density (95) diverge at **t**` , which are roughly *t*<sup>1</sup> ≈ 1.92, *t*<sup>2</sup> ≈ 4.80, *t*<sup>3</sup> ≈ 7.83 . The classical motions are regular at the caustics, *x*¯(*t*` ) ∝ *ρ*(*t*` ) ≈ 0; see Section 4. The domains **<sup>I</sup><sup>k</sup>** = [*rk*−<sup>1</sup> ,*rk* ] of the inverse Niederer map are shown in Figure 4. Approximately, *r*<sup>1</sup> ≈ 1.52, *r*<sup>2</sup> ≈ 4.49, *r*<sup>3</sup> ≈ 6.75, *r*<sup>4</sup> ≈ 8.44 . The evolution of the phase factor along the classical path is depicted in Figure 5.

**Figure 4.** The probability density |*K*(*x*, *t*)| 2 (95) does not depend on *x* and is regular in each interval **J**` between the adjacent points **t**` (75), where it diverges. The **r<sup>k</sup>** that determines the domains **I***<sup>k</sup>* of the generalized Niederer map (22) lies between the **t**` and conversely.

**Figure 5.** For 0 < *t* < *t*<sup>1</sup> , the Mathieu phase factor **P**(*t*) plotted along a classical path *γ*¯(*t*) = (*x*¯(*t*), *t*) precesses around *e* <sup>−</sup>*iπ*/4. Arriving at the caustic point *<sup>τ</sup>*(*t*1) = *<sup>π</sup>*, its phase jumps by (−*π*/2), then oscillates around *e* <sup>−</sup>3*iπ*/4 until *τ*(*t*2) = 2*π*, then jumps again, and so on.

#### **6. Conclusions**

The Junker–Inomata–Arnold approach yields (in principle) the exact propagator for any quadratic system by switching from a *time-dependent* to a *constant frequency* and redefined time,

$$
\omega(t) \to \bar{\omega} = \text{const.} \quad \text{and} \quad t \to \text{ "false time"} \quad \text{\(\theta\)}.
$$

The propagator (64)–(66) is then derived from the result known for the constant frequency. A straightforward consequence is the Maslov jump for arbitrary time-dependent frequency *ω*(*t*): everything depends only on the product *ω τ* ¯ .

By switching from *t* to *τ*, the Sturm–Liouville-type difficulty is not eliminated, but only transferred to that of finding *τ* = *τ*(*t*) following the procedure outlined in Section 2. We have to first solve EMP Equation (18) for *ρ*(*t*) (which is nonlinear and has time-dependent coefficients) and then integrate *ρ* −2 ; see (21). Although this is as difficult to solve as solving the Sturm–Liouville equation, it provides us with theoretical insights.

When no analytic solution is available, we can resort to numerical calculations.

The Junker–Inomata approach of Section 2 is interpreted as a Bargmann-conformal transformation between time-dependent and constant frequency metrics; see Equation (39).

Alternatively, the damped oscillator can be converted to a free system by the generalized Niederer map (22), whose Eisenhart–Duval lift (47)–(51) carries the conformally flat oscillator metric (60) to the flat Minkowski space.

Two sets of points play a distinguished role in our investigations: the *r<sup>k</sup>* in (71) and the *t*` in (75). The *r<sup>k</sup>* divides the time axis into domains *I<sup>k</sup>* of the (generalized) Niederer map (22). Both classical motions and quantum propagators are *regular* at *r<sup>k</sup>* , where these intervals are joined. The *t*` are in turn the caustic points where all *classical trajectories are focused*, and the *quantum* propagator becomes *singular*.

While the "Maslov phase jump" at caustics is well established when the frequency is constant, *ω* = *ω*<sup>0</sup> = const., its extension to the time-dependent case *ω* = *ω*(*t*) is more subtle. In fact, the proofs we are aware of [25–28] use sophisticated mathematics, or a lengthy direct calculation of the propagator [44]. A bonus from the Junker–Inomata transcription (10) we followed here is to provide us with a straightforward extension valid to an arbitrary *ω*(*t*). Caustics arise when (65) holds, and then, the phase jump is given by (88).

The subtle point mentioned above comes from the standard (but somewhat sloppy) expression (5), which requires choosing a branch of the double-valued square root function. Once this is done, the sign change of *T* <sup>00</sup> − *T* 0 induces a phase jump *π*/2. Our "innocentlooking" factor *is* in fact the Maslov jump for a free particle at *T* = 0 (obscured when one considers the propagator for *T* > 0 only). Moreover, it then becomes the key tool for the oscillator: intuitively, the multivalued inverse Niederer map repeats, again and again, the same jump. The details are discussed in Section 4.

The transformation (10) is related to the *nonrelativistic "Schrödinger" conformal symmetries* of a free nonrelativistic particle [45–47], later extended to the oscillator [16] and an inverse-square potential [48]. These results can in fact be derived using a time-dependent conformal transformation of the type (10) [19,42].

The above results are readily generalized to higher dimensions. For example, the oscillator frequency can be time-dependent, uniform electric and magnetic fields, and a curl-free "Aharonov–Bohm" potential (a vortex line [49]) can also be added [41]. Further generalization involves a Dirac monopole [50].

Alternative ways to relate free and harmonically trapped motions are studied, e.g., in [51–54]. Motions with the Mathieu profile were considered also in [55].

**Author Contributions:** All authors have contributed equally and substantially to this paper, including its conception, development, preparation, writing and editing. All authors have read and agreed to publish the manuscript in its present form.

**Funding:** This research was funded by the National Natural Science Foundation of China (grant number 11975320).

**Institutional Review Board Statement:** Exclude this statement since the study did not involve humans or animals.

**Informed Consent Statement:** Exclude this statement since the study did not involve humans.

**Data Availability Statement:** Exclude this statement since the study did not report any data.

**Acknowledgments:** This work was partially supported by the National Natural Science Foundation of China (Grant No. 11975320).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **What Is the Size and Shape of a Wave Packet?**

**Larry S. Schulman**

Physics Department, Clarkson University, Potsdam, NY 13699-5820, USA; lschulma@clarkson.edu

**Abstract:** Under pure quantum evolution, for a wave packet that diffuses (like a Gaussian), scattering can cause localization. Other forms of the wave function, spreading more rapidly than a Gaussian, are unlikely to localize.

**Keywords:** spreading wave function; scattering; Localization

What is the size and shape of a wave packet? I am talking about a wave packet of a particle (atom or molecule) in a gas. Is it a plane wave that fills the container? Or is it a microscopic (perhaps ≤ 100 nm) object? I am not talking about a situation where there is a potential holding the particle in some region, like a hydrogen atom. The only things around are other particles, the same or different. (Please see Appendix A). Strangely, this subject has not attracted a lot of attention.

Our conclusion considers a number of possibilities. Should the eventual wave functions be Gaussian or Gaussian-like (to be later defined) then, yes, there is localization. If not, probably no. However, we give an argument that the wave function most likely does become Gaussian. (Generally speaking, people use Gaussians in descriptions, but this is not always warranted.)

This paper has three parts: arguments for a Gaussian, localization in that case and the case(s) that it is not a Gaussian. We emphasize that the first part (eventually) treats non-Gaussians that become Gaussian after a number of collisions.

*Arguments for a Gaussian.* The wave function for a Gaussian would be

$$\psi(\mathbf{r}) = \frac{1}{(\pi \sigma^2)^{3/4}} \exp\left(-\frac{(\mathbf{r} - \mathbf{r}\_0)^2}{2\sigma^2} + \frac{i}{\hbar}p\_0(\mathbf{r} + \mathbf{r}\_0)\right). \tag{1}$$

In one dimension, a Gaussian in momentum space is

$$\phi(p) = \left(\frac{\sigma^2}{\pi\hbar^2}\right)^{1/4} \exp\left(-\frac{\sigma^2(p-p\_0)^2}{2\hbar^2} - \frac{i}{\hbar}p\mathbf{x}\_0\right). \tag{2}$$

(The following paragraph is the treatment in [1]; for more details, see that reference.) We suppose there is a particle of a different mass and that they scatter. The conservation laws (once they are far enough apart that they do not interact) are

$$P\_{\text{final}} = P\_{\text{initial}} \text{ and } \ p\_{\text{final}} = -p\_{\text{initial}} \text{ } \tag{3}$$

where *P* and *p* are collective coordinates: *P* = *p*<sup>1</sup> + *p*<sup>2</sup> and *p* = (*m*<sup>2</sup> *p*<sup>1</sup> − *m*<sup>1</sup> *p*2)/*M*. In these relations, *p<sup>j</sup>* and *m<sup>j</sup>* (*j* = 1, 2) are the momenta and masses of each of the particles, 1 *<sup>µ</sup>* <sup>=</sup> <sup>1</sup> *m*<sup>1</sup> + <sup>1</sup> *m*<sup>2</sup> and *M* = *m*<sup>1</sup> + *m*2. Furthermore, we let *µ<sup>i</sup>* = *mi <sup>M</sup>* (*i* = 1, 2). It turns out that, for calculation of the spread, all that matters is the real part of the (2× negative) exponent. This changes from - *σ* 2 1 *p* 2 <sup>1</sup> + *σ* 2 2 *p* 2 2 = - *σ* 2 1 (*µ*1*P* + *p*) + *σ* 2 2 (*µ*2*P* − *p*) , following Equation (3), to

$$\begin{array}{rcl} Q(p\_1, p\_2) & \equiv & p\_1^2 [\sigma\_1^2 (\mu\_1 - \mu\_2)^2 + 4\mu\_2^2 \sigma\_2^2] + p\_2^2 [\sigma\_2^2 (\mu\_2 - \mu\_1)^2 + 4\mu\_1^2 \sigma\_1^2] \\ & & + 4(\mu\_1 - \mu\_2) p\_1 p\_2 (\mu\_1 \sigma\_1^2 - \mu\_2 \sigma\_2^2) \end{array} \tag{4}$$


**Citation:** Schulman, L.S. What Is the Size and Shape of a Wave Packet?. *Symmetry* **2021**, *13*, 527. https:// doi.org/10.3390/sym13040527

Academic Editor: Georg Junker

Received: 17 February 2021 Accepted: 16 March 2021 Published: 24 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

As shown in [1], provided *µ*<sup>1</sup> and *µ*<sup>2</sup> are significantly different from one another, within a few scattering events the "cross" term in *p*<sup>1</sup> *p*<sup>2</sup> vanishes. This means that the wave functions have ratio of spread such that *mσ* 2 is constant and there is no cross term—nothing to decohere (and the von Neumann entropy is maximum). In three dimensions the same thing happens, but is more difficult to show [2–4]. That, however, is not our main point.

What happens in one dimension if the initial functions are not a Gaussian? We suppose the wave function has the form exp(− ∑ ∞ 2 *anx n* ). This form can fit various other functional forms, e.g., 1/(1 + *x* 2 ) 2 to within 0.06 using 4 coefficients (i.e., elements of the set {*an*}). (Please see Appendix B) . Following the earlier method (based on Equation (3)), we find that 4th order terms obey

$$a\_4^{(1)'} = \begin{array}{c} a\_4^{(1)}(\mu\_1 - \mu\_2)^4 + a\_4^{(2)}2^4\mu\_2^4\\ a\_4^{(2)'} = \begin{array}{c} a\_4^{(2)}(\mu\_1 - \mu\_2)^4 + a\_4^{(1)}2^4\mu\_1^4 \end{array} \tag{5}$$

with *a* (*j*) 4 (*j* = 1, 2) the coefficients of *x* 4 for the respective wave functions. In general, if the deviation from a Gaussian begins with a term *p* 2*n k* there is a matrix that takes one from the values of the coefficients multiplying these terms from before scattering to after scattering. That matrix is

$$
\begin{pmatrix}
\left(\mu\_1-\mu\_2\right)^{2n} & \left(2\mu\_2\right)^{2n} \\
& \left(2\mu\_1\right)^{2n} & \left(\mu\_1-\mu\_2\right)^{2n}
\end{pmatrix}.
\tag{6}
$$

The eigenvalues of this matrix are (for all *n* > 1) below 1 for *m*<sup>1</sup> 6= *m*<sup>2</sup> and *m<sup>j</sup>* 6= 0 (*j* = 1, 2).

What happens if there is no other-mass particle? I do not know. I would not have thought it should make a difference, but I do not have a proof. In practice, there is almost always some impurity, but it may scatter rarely.

Thus, there is an indication that in one dimension the wave function approaches a Gaussian. In higher dimension—in particular 3—I do not have definitive results. It is still true [1] that for Gaussians the spread approaches a maximum of von Neumann entropy and (if there are two types of particles, #1 and #2 then) *m*1*σ* 2 <sup>1</sup> = *m*2*σ* 2 2 .

For three dimensions, there are many ways that *n* th power terms can occur; for #1 one can have anything of the form *p*<sup>1</sup> *n*1 *<sup>x</sup> p*<sup>1</sup> *n*2 *<sup>y</sup> p*<sup>1</sup> *n*3 *<sup>z</sup>* with *n*<sup>1</sup> + *n*<sup>2</sup> + *n*<sup>3</sup> = *n*, and similarly for #2, leading to (*n* + 2)(*n* + 1) coefficients for the two of them. (Please see Appendix C). Moreover, as discussed in [1], the post-scattering values of *p*<sup>1</sup> and *p*<sup>2</sup> involve a rotation, *R* ∈ SO(3), not just a flip. Thus, *p* 0 <sup>1</sup> = (*µ*<sup>1</sup> *I* + *µ*2*R*)*p*<sup>1</sup> + *µ*1(*I* − *R*)*p*<sup>2</sup> (and 1 ↔ 2 for *p* 0 2 ).

I have examined cubic and quartic components of the logarithm of the wave function. All the indicated operations have been carried out, the cross terms involving momenta of #1 and of #2 have been dropped, and a rather complex recursion for the plethora of coefficients evaluated numerically. Sample results are shown in Figures 1 and 2. The "time" represents the number of scattering events, and the ordinate is the sum total of the absolute values of all cubic or quartic coefficients. Remarkably, they all tend to zero.

Evaluations beyond quartic represent a further problem in symbolic manipulation, but based on the higher-power evidence of one dimension together with the cubic and quartic evidence in three dimensions, it is reasonable to make the claim that all higher power coefficients tend to zero under decoherence.

Is this a proof that eventually everything decoheres to a Gaussian? Absolutely not, but it is an indication.

*Localization with Gaussian-like behavior.* Scattering can localize. This may be surprising, since some may hold that a wave function can only spread. It turns out (as shown in [2]) that scattering can act like a measurement, that scattering alone can localize. Here we extend that result.

I do not deal with the effects of temperature (cf. [5]), nor with off-diagonal elements of the density matrix [6,7]. I am concerned with pure quantum behavior. Nor do the conclusions depend on interpretation, i.e., they are independent of whether one subscribes to the Copenhagen interpretation (in its many variants), Many Worlds, or some other theory.

**Figure 1.** Reduction of non-Gaussian terms with unequal mass scattering in three dimensions.Because that the problem has an intrinsic nonlinearity, the initial coefficients were varied. Usually they were taken as random between 0 and 1, but on some occasions they were taken to be 20 times that. In all cases there was convergence to zero. The programs to establish this were combinations of symbolic manipulation and numerical evaluation.

**Figure 2.** As in Figure 1, except that this looks at quartic terms. As in the other example, all terms shrink to zero.

I will briefly review the results of [2] and then turn to the extension. (There is a slight change in notation: instead of *σ* 2 , we use 2∆ 2 for convenience in matching results.) The principal consequence of [2] is that, assuming the wave function is a Gaussian, particles do not spread indefinitely. The proof in [2] is a self-consistency argument. We assume two normalized Gaussians in 3 dimensions of the form

$$\psi(\mathbf{r}\_1, \mathbf{r}\_2, 0) = \frac{1}{(2\pi\Lambda^2)^{3/2}} \exp\left(-\frac{(\mathbf{r}\_1 + \mathbf{r}\_0)^2}{4\Delta^2} + \frac{i}{\hbar}p\_1(\mathbf{r}\_1 + \mathbf{r}\_0) - \frac{(\mathbf{r}\_2 - \mathbf{r}\_0)^2}{4\Delta^2} + \frac{i}{\hbar}p\_2(\mathbf{r}\_2 - \mathbf{r}\_0)\right). \tag{7}$$

with parameters *r*0, *p*1, *p*<sup>2</sup> and ∆. We assume *p*<sup>1</sup> ≈ −*p*2, so that at time *m*|*r*/*p*1| they scatter. Assuming no interaction, at time *t* this becomes

$$\psi(\mathbf{r}\_1, \mathbf{r}\_2, t) \quad = \left(\frac{\Delta^4 + (\frac{\hbar t}{2m})^2}{2\pi\Delta^2}\right)^{-3/2} \exp\left(-\frac{(\mathbf{r}\_1 + \mathbf{r}\_0 - \frac{\mathbf{p}\_1 t}{m})^2}{4\left(\Delta^2 + i\frac{\hbar t}{2m}\right)} + \frac{i}{\hbar}p\_1(\mathbf{r}\_1 + \mathbf{r}\_0) - \frac{i}{\hbar}\frac{p\_1^2 t}{2m}\right)$$

$$\times \exp\left(-\frac{(\mathbf{r}\_2 - \mathbf{r}\_0 - \frac{\mathbf{p}\_2 t}{m})^2}{4\left(\Delta^2 + i\frac{\hbar t}{2m}\right)} + \frac{i}{\hbar}p\_2(\mathbf{r}\_2 - \mathbf{r}\_0) - \frac{i}{\hbar}\frac{p\_2^2 t}{2m}\right). \tag{8}$$

using center of mass coordinates, the exponent is

$$-\frac{(\mathbf{R} + \frac{\mathbf{P}t}{2m})^2}{2\left(\Delta^2 + i\frac{\hbar t}{2m}\right)} - \frac{(r + 2r\_0 + \frac{pt}{m/2})^2}{2\left(4\Delta^2 + i\frac{\hbar t}{m/2}\right)} - \frac{i}{\hbar}(\mathbf{R}\mathbf{P} + (r - 2r\_0)\mathbf{p}) - \frac{i}{\hbar}\left(\frac{\mathbf{P}^2}{4m} + \frac{\mathbf{p}^2}{m}\right)t,\tag{9}$$

where *R* = <sup>1</sup> 2 (*r*<sup>1</sup> + *r*2), *r* = *r*<sup>1</sup> − *r*2, and correspondingly for *P* and *p*. Now set *t* ≈ *m*|*r*/*p*1| (or just afterward). They scatter, but nothing happens to *P*. In the center of mass, we use the Born approximation. Up to normalization, the new wave function is *δt i h*¯ *V*(*r*)Ψ(*R*,*r*, *t*), where we also assumed the interaction was brief, taking place during a short time *δt*. With a further condition that *<sup>V</sup>*(*r*) = *<sup>V</sup>*<sup>0</sup> exp − *r* 2 4*a* 2 , for convenience in integrating, we obtain

$$|\Psi|^2 = \exp\left(-\frac{r^2}{2a^2} - \frac{(\mathbf{R} + \frac{\mathbf{P}t}{2m})^2}{\left|\Delta^2 + i\frac{\hbar t}{2m}\right|} - \frac{(r + 2r\_0 + \frac{pt}{m/2})^2}{\left|4\Delta^2 + i\frac{\hbar t}{m/2}\right|}\right). \tag{10}$$

(Normalization cancels and can be ignored.) Using this wave function, one can calculate the spread in *r* (∆*r*), in *R* (∆*R*) and from them ∆*r*<sup>1</sup> and ∆*r*2. The latter quantities are then set equal to the original ∆*r*<sup>1</sup> and ∆*r*<sup>2</sup> (both equal to ∆). Using as the time between collisions scattering length velocity (and mass values) we find that indeed the original <sup>∆</sup> can be set equal to

the final values of spread. To make equations simpler, we define

$$
\theta \equiv \frac{\hbar t}{m} = \frac{\hbar \ell}{m v} \,\prime \tag{11}
$$

where ` is the scattering length and *v* a typical velocity. For any given gas, the range of *θ* is fixed.

The essential steps in the foregoing derivation are the estimation of ∆*r*, *R* and ∆*R*. It is these that can be generalized.

The first step is to formalize *V*. Instead of an exponential of range *a*, we take a potential that is cut off at distance *a*. This restricts the location of *r* to be within a distance *a* of *R*. In other words, ∆*r* ≤ √ *r* · *r* = 3*a* 2 . This is an additional assumption and may require a larger "*a*" than was previously posited.

The expectation of *<sup>R</sup>* is simply <sup>h</sup>*R*<sup>i</sup> <sup>=</sup> *<sup>P</sup><sup>t</sup>* 2*m* , since it is unaffected by the interaction/scattering.

The *spread* in *R* is another matter and is the principal source of uncertainty in our calculation. For a Gaussian-like initial wave function, we can give estimates. From our previous work [1], we have

$$
\langle (\Delta \mathbf{R})^2 \rangle = \langle \mathbf{R}^2 \rangle - \langle \mathbf{R} \rangle^2 = 3 \frac{4\Delta^4 + \theta^2}{8\Delta^2} \,. \tag{12}
$$

Now we would like to weaken the assumptions. It turns out that the most important property (for our purposes) of the Gaussian is its diffusive behavior. This assumption about the spread in the center of mass coordinate is weakened in a specific way: in place of the 8 that appears in the denominator, we allow smaller values, *γ*. (In this sense, the wave function is Gaussian-like.) Specifically, 4 < *γ* ≤ 8 and

$$
\langle (\Delta \mathbf{R})^2 \rangle = 3 \frac{4\Delta^4 + \theta^2}{\gamma \Delta^2} \,. \tag{13}
$$

We now use h∆*r* 2 1 i = h∆*r* 2 2 i = h∆*R* 2 <sup>i</sup> <sup>+</sup> <sup>1</sup> 4 h∆*r* 2 i to arrive at the denominators for the new wave functions of *r*<sup>1</sup> and *r*2:

$$
\langle (\Delta \mathbf{r}\_{\text{new}})^2 \rangle = \langle (\Delta \mathbf{r}\_k)^2 \rangle = 3a^2 + 3 \frac{4\Delta^4 + \theta^2}{\gamma \Delta^2}, \qquad k = 1.2. \tag{14}
$$

Setting the old ∆ equal to the new one, we arrive at a self-consistency criterion for the spread

$$3\Delta^2 = 3a^2 + 3\frac{4\Delta^4 + \theta^2}{\gamma \Delta^2} \,. \tag{15}$$

This is a quadratic equation whose solution is

$$
\Delta^2 = \frac{a^2 \gamma}{4(\gamma - 4)} \pm \sqrt{\frac{\theta^2}{\gamma - 4} + \left(\frac{a^2 \gamma}{4(\gamma - 4)}\right)^2}.\tag{16}
$$

Taking ` <sup>=</sup> <sup>70</sup> nm, *<sup>v</sup>* <sup>=</sup> <sup>500</sup> m/s, *<sup>a</sup>* <sup>=</sup> <sup>1</sup> nm and *<sup>m</sup>* <sup>=</sup> <sup>29</sup> gm/6 <sup>×</sup> <sup>10</sup><sup>23</sup> <sup>≈</sup> <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>23</sup> gm and *γ* = 4.1 gives a value of 4.5 nm, much less than the mean free path. (Please see Appendix D). These are typical parameters for the air and the wave function is localized. Note though that (for air) there is overlap: (number density)1/3 <sup>≈</sup> (0.025 <sup>×</sup> <sup>10</sup>−27) 1/3 <sup>≈</sup> <sup>3</sup> Å. Therefore, although the wave functions occupy common volumes, they mostly do not interact.

*Not a Gaussian.* The essential feature of non-Gaussian wave functions is that the spread in *R* grows. I have not examined the "boundary," that is, the form of, or parameters in, wave functions that eventually become Gaussian and those that do not. I will though examine various wave functions that are non-Gaussian. Consider, for example, an exponential

$$
\psi(r,0) = \frac{\lambda^3}{8\pi} \exp(-\lambda r) \,. \tag{17}
$$

We assume that both scattering particles have this form, with the same value of *λ*. The spread in the relative coordinate ∆*r* is still bounded by *a* (or 3*a*) but the center of mass coordinate grows. The spread for each, before collision, is 1/*λ*. The center of mass coordinate for equal mass particles is *R* = (*r*<sup>1</sup> + *r*2)/2, so that the spread for of the center of mass coordinate is also 1/*λ*. As a result, the center of mass can also be taken of the form Equation (17). Now apply the free propagator numerically and fit the result to an exponential. Except for particular values of *λ* the value of the spread has *increased*. This means there can be no self-consistent solution (as there was for a Gaussian).

The same happens for the wave function taken as a power.

*Conclusions.* The point of this is not that no wave function spreads. Rather, it places bounds on spreading for certain wave functions and makes it plausible that scattering is sometimes like a measurement, pinning a particle to a small, localized region.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

In particular, we do not deal with chains of oscillators or single oscillators, which according to [8,9] (and other literature) become, by decoherence, coherent states, i.e., Gaussian.

#### **Appendix B**

The function 1/(1 + *x* 2 ) 2 is not symmetric about *x* = 0; hence, it requires odd terms in the expansion exp(− ∑ ∞ 2 *anx n* ). This is accomplished with coefficients *a*<sup>3</sup> and *a*<sup>5</sup> that are 10−<sup>8</sup> or less. The fourth order term is about 1/100 of the quadratic term (which is about 1.46).

#### **Appendix C**

This is the number of possibilities for both particles and is calculated by imagining the interval [0, *n*] as having partitions at 2 (integer) locations (and multiplying by 2). Just to be clear, the 3-dimensional tendency to have a Gaussian wave function is new material.

#### **Appendix D**

The minus sign in Equation (16) is spurious and gives an imaginary value for <sup>√</sup> ∆2 .

#### **References**


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