# The mean square displacement of a ballistic quantum particle

Oussama Bindech, Fabien Gatti, Souvik Mandal, Roberto Marquardt, L. Shi, Jean Christophe Tremblay

► To cite this version:

Oussama Bindech, Fabien Gatti, Souvik Mandal, Roberto Marquardt, L. Shi, et al.. The mean square displacement of a ballistic quantum particle. Molecular Physics, 2024, 122 (15-16), 10.1080/00268976.2024.2322023. hal-05368486

HAL Id: hal-05368486

https://hal.science/hal-05368486v1

Submitted on 17 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# The mean square displacement of a ballistic quantum particle

O. Bindech<sup>a</sup>, F. Gatti<sup>b</sup>, S. Mandal<sup>c</sup>, R. Marquardt<sup>a</sup>, L. Shi<sup>b</sup>, and J. C. Tremblay<sup>c</sup>

<sup>a</sup> Laboratoire de Chimie Quantique, Institut de Chimie, UMR 7177 CNRS/Unistra Université de Strasbourg, 4, Rue Blaise Pascal, CS 90032, 67081 Strasbourg Cedex, France;

<sup>b</sup> Institut de Sciences Moléculaires d'Orsay, UMR 8214 CNRS/Université Paris-Saclay, Bât 520, Rue André Rivière, 91405 Orsay Cedex, France;

<sup>c</sup> Laboratoire de Physique et Chimie Théoriques, UMR 7019 CNRS/Université de Lorraine, 1 Blvd. Arago, 57070 Metz, France

## ARTICLE HISTORY

Compiled December 12, 2023

## ABSTRACT

The quantum mechanical definition of the mean square displacement proposed in a previous work (R. Marquardt, Mol. Phys. 119, e1971315 (2021)) is extended to the continuum of a free particle in the interval topology. Integrals are evaluated in the formalism of generalized functions (distributions). A comparison with the definition based on the quantum correlation function is carried out. It is found that the latter diverges, when evaluated in a ring topology with infinitely large rings.

## KEYWORDS

mean square displacement, ballistic motion, quantum dynamics, thermal wave packets, Fourier transformation

## 1. Introduction

Quantum dynamics and spectroscopy [1–4] have been constant objects of Attila G. Cszászár’s research, in particular in relation to thermodynamical molecular properties [5–7]. The present work is about another thermodynamically relevant quantum dynamical quantity: the definition of the mean square displacement (MSD) of a quantum particle at thermal equilibrium.

Consider the following definition:

$$ (\text{QCF}) \delta_x^2(t) = \text{Tr} (\hat{\rho}^{(T)} \hat{d}(t)^2) \qquad (1) $$

where $\hat{\rho}^{(T)}$ is the thermal density operator and

$$ \hat{d}(t) \equiv \hat{x}(t) - \hat{x}(0) \qquad (2) $$

is the time differential displacement operator. Here $\hat{x}(t)$ is the position operator of the particle at time $t$ in the Heisenberg representation, $\hat{x}(t) = e^{i\hat{H}t/\hbar}\hat{x}(0)e^{-i\hat{H}t/\hbar}$, with
$\hat{H}$ the particle's Hamiltonian, $\hbar = h/(2\pi)$ and $h$ the Planck constant. “Tr” is the trace of a matrix representing the operator product.

The definition given in Eq. (1) was used by Ford et. al in a paper on quantum measurement and decoherence [8] (the material is based on their work in ref. 9), then in a paper by Wojcik and Dorfman [10], and in the book of Gardiner and Zoller [11, page 63 therein]. The acronym “QCF” indicates that this definition is related to the quantum correlation function of the position operator, as will be further explained below.

Earlier work of Ford, Kac and Mazur use the expression $\langle : (\hat{x}(t) - \langle \hat{x}(t) \rangle_{\text{bath}})^2 : \rangle$, where $\langle \cdot \rangle = \text{Tr}(\hat{\rho}^{(T)} \cdot)$, the colon sign indicates normal order, and $\langle \cdot \rangle_{\text{bath}}$ is the average over the initial states of the bath [12]. Normal order is the current usual treatment, i.e. one evaluates explicitly the anti-commutator of $\hat{x}(t)$ and $\hat{x}(0)$.

A different definition of the MSD is

$$ (\text{TWP}) \delta_x^2(t) = \langle (x_\theta(t) - x_\theta(0))^2 \rangle_\theta \quad (3) $$

Eq. (3) was proposed in ref. 13. Here, $x_\theta(t) = \langle \psi_\theta^{(T)}(t) | \hat{x} | \psi_\theta^{(T)}(t) \rangle$, $| \psi_\theta^{(T)}(t) \rangle$ is the time dependent state representing a typical member of a thermal statistical ensemble, a so called thermal wave packet [14], and $\langle \cdot \rangle_\theta$ is the average in the sense of the arithmetic mean over these states. Eq. (3) was also used in a different form in the context of intra-chain exciton diffusion in ref. 15.

In the present work we compare these two definitions at the example of a freely moving particle of mass $m$ at thermal equilibrium in the ballistic regime, i.e. during a time interval in which it moves by inertia between two successive collisions. In this case, the classical expression of the MSD is

$$ (\text{cl}) \delta_x^2(t) = v_{\text{th}}^2 t^2 \quad (4) $$

where $v_{\text{th}} = \sqrt{k_B T/m}$ is the average thermal velocity in one dimension, $T$ is the temperature and $k_B$ is the Boltzmann constant.

The MSD of the free particle is evaluated analytically, for both definitions, in the true continuum. The analytical approach needs some care, however, as some of the integrals can be evaluated only in the context of generalized functions (distributions).

Furthermore, the MSD is also evaluated for a thermalized particle moving freely on a ring, i.e. under periodic boundary conditions. In this case, a simple analytical formula was derived for $(\text{TWP})\delta_x^2(t)$ in ref. 13 in the case of an infinitely large ring. Periodic boundary conditions are often used in numerical evaluations of particle dynamics in periodic environments, such as in crystal lattices or surfaces. The classical expression of the MSD, Eq. (4), holds also in the ring topology.

In this paper the MSD of a particle moving freely will thus be evaluated by four different approaches: the two aforementioned definitions will be evaluated in two distinct situations which correspond to mathematically distinct topological spaces. The true continuum situation can be referred to as the interval topology, periodical boundary conditions correspond to a ring or circle topology. Both situations were considered in ref. 16 to calculate $(\text{TWP})\delta_x^2$ in the Hilbert space of square integrable functions in $\mathbb{R}$ in terms of thermal Gaussian wave packets [17]. In the present work, the analysis is extended to the true continuum of functions that are not square integrable.

The paper is organized as follows: in section 2 the basic theory is developed for general systems; in section 3 the quantities needed for the two different evaluations of
the MSD are evaluated for the specific case of the freely moving particle in the two topologies; section 4 is essentially a technical report on the evaluation of integrals and contains a derivation of the distribution yielding the Fourier transform $F[f(x)](y) = \int_{-\infty}^{\infty} dx f(x) e^{ixy}$ of the line function $f(x) = x$; section 5 presents the four results for the MSD in terms of simple mathematical formulae or figures with numerical evaluations. Section 6 concludes this work.

## 2. General expressions for the MSD

### 2.1. Energy spectrum, representations of the thermal density operator and partition functions

In the case the spectrum of the system's Hamiltonian $\hat{H}$ is discrete, such as under periodic boundary conditions considered in ref. 13, the stationary states $|\phi_n\rangle$ satisfy

$$ \hat{H} |\phi_n\rangle = E_n |\phi_n\rangle \qquad (5) $$

and are countable, with $n \in \mathbb{Z}$. Discrete states can be normalized with the standard norm of Hilbert spaces, i.e. the inner Hilbert space product yields

$$ \langle \phi_{n'} | \phi_n \rangle = \delta_{n'n} \qquad (6) $$

where $\delta_{n'n}$ is the Kronecker $\delta$-symbol.

The partition function is given by

$$ Q = \mathrm{Tr} \left( e^{-\beta \hat{H}} \right) = \sum_n e^{-\beta E_n} \qquad (7) $$

where $\beta \equiv 1/(k_B T)$ and the matrix representing the thermal density operator in the basis of stationary states, the thermal density matrix, is diagonal:

$$ \langle \phi_{n'} | \hat{\rho}^{(T)} | \phi_n \rangle = \rho_{n'n}^{(T)} = \delta_{n'n} \frac{e^{-\beta E_n}}{Q} \qquad (8) $$

In the case the spectrum is a continuum, the stationary states $|\phi_q\rangle$ satisfying

$$ \hat{H} |\phi_q\rangle = E(q) |\phi_q\rangle \qquad (9) $$

and the corresponding energies $E(q)$ are continuous functions of the continuous index $q$. Contrary to the discrete case, the inner product yields

$$ \langle \phi_{q'} | \phi_q \rangle = \delta(q' - q) \qquad (10) $$

where $\delta(x)$ is the Dirac $\delta$-function [18, vol. I, chap. VII].

The canonical partition function is given as

$$ Q = \mathrm{Tr} \left( e^{-\beta \hat{H}} \right) = \int_{\mathcal{D}(q)} dq e^{-\beta E(q)} \langle \phi_q | \phi_q \rangle \qquad (11) $$
where $\mathcal{D}(q)$ is the set containing all possible values of $q$. It is dimensionless and all levels are non-degenerate. The “norm” $\langle \phi_q | \phi_q \rangle$ deserves special attention. From Eq. (10) one concludes $\langle \phi_q | \phi_q \rangle = \delta(0) = \infty$, the partition function is factually infinitely large. Technically, problems arising from infinite norms may be overcome in specific evaluations of the partition function by renormalization, as will be shown below.

## 2.2. MSD on the basis of the positional quantum correlation function

The definition in Eq. (1) is related to the quantum correlation function of the position operator [19]

$$C_x(t) = \mathrm{Tr} \left( \hat{\rho}^{(T)} [\hat{x}(t) \hat{x}(0) + \hat{x}(0) \hat{x}(t)] \right) \quad (12)$$

Because the thermal density operator commutes with the Hamiltonian, one has, in general

$$\mathrm{Tr} \left( \hat{\rho}^{(T)} \hat{x}^2(t) \right) = \mathrm{Tr} \left( \hat{\rho}^{(T)} \hat{x}^2(0) \right) \quad (13)$$

hence

$$(\mathrm{QCF}) \delta_x^2(t) = 2 \mathrm{Tr} \left( \hat{\rho}^{(T)} \hat{x}^2(0) \right) - C_x(t) = C_x(0) - C_x(t) \quad (14)$$

It is straightforward to show, upon use of the resolution of identity, that the definition in Eq. (1) yields, in the case of a discrete spectrum, the following general expression:

$$(\mathrm{QCF}) \delta_x^2(t) = \frac{4}{Q} \sum_n \sum_j e^{-\beta E_n} |x_{nj}|^2 \sin^2 \left[ \frac{(E_n - E_j) t}{2\hbar} \right] \quad (15)$$

Here

$$x_{nj} = \langle \phi_n | \hat{x} | \phi_j \rangle \quad (16)$$

is the matrix element of the position operator in the basis of the stationary states. By analogy, the expression in the case of a continuous spectrum could be given as

$$(\mathrm{QCF}) \delta_x^2(t) = \frac{4}{Q} \int_{-\infty}^{\infty} dq \, e^{-\beta E(q)} \int_{-\infty}^{\infty} dq' |x(q', q)|^2 \sin^2 \left[ \frac{(E(q) - E(q')) t}{2\hbar} \right] \quad (17)$$

where

$$x(q', q) = \langle \phi_{q'} | \hat{x} | \phi_q \rangle \quad (18)$$

are the matrix elements of the position operator in the continuum.

The position operator is unbound and the integrals in Eq. (18) diverge in the continuum. Eq. (17) must therefore be viewed as a formal equation with no practical value
at first sight. The evaluation of the matrix elements Eq. (18) needs special treatment. As will be shown below, this evaluation can be carried out in specific situations within the same renormalization procedure adopted for the partition function, Eq. (11). Indeed, dimensional analysis reveals that the position matrix elements in Eq. (18) have the dimension of an area, rather than a length, so that some renormalization will be needed in any case when final results are obtained.

## 2.3. MSD on the basis of thermal wave packets

Following Tolman [14], a time dependent thermal state (thermal wave packet) is given in the case of a discrete spectrum as

$$|\psi_{\theta}^{(T)}(t)\rangle = \sum_{n} \frac{e^{-\beta E_{n}/2 - iE_{n}t/\hbar + i\theta_{n}}}{\sqrt{Q}} |\phi_{n}\rangle \quad (19)$$

The quantities $\theta_n$ appearing in Eq. (19) are random angles which satisfy the following correlation functions:

$$\langle e^{i\theta_n} \rangle_{\theta} = 0 \quad (20)$$

$$\langle e^{i\theta_n} e^{-i\theta_{n'}} \rangle_{\theta} = \delta_{nn'} \quad (21)$$

with the Kronecker $\delta$-symbol $\delta_{nn'}$.

With this definition of the thermal wave packet, one derives the general expression for the MSD following Eq. (3) (see ref. 13)

$$^{(TWP)}\delta_x^2(t) = \frac{4}{Q^2} \sum_n \sum_j e^{-\beta(E_n + E_j)} |x_{nj}|^2 \sin^2\left[\frac{(E_n - E_j)t}{2\hbar}\right] \quad (22)$$

where $x_{nj}$ is defined as in given in Eq. (16).

It is interesting to note that the resulting expression for the MSD following the definition in terms of thermal wave packets, Eq. (22), differs from that obtained from the definition based on the quantum correlation function of the position operator, Eq. (15), by the presence of additional factors $e^{-\beta E_j / Q}$ in the individual summation terms.

In the case of a continuous spectrum, a natural expansion of Eq. (19), the thermal wave packet may be expressed as

$$|\psi_{\theta}^{(T)}(t)\rangle \propto \int_{D(q)} dq \frac{e^{-\beta E(q)/2 - iE(q)t/\hbar + i\theta(q)}}{\sqrt{Q}} |\phi_q\rangle \quad (23)$$

The unspecified proportionality constant may be chosen in specific evaluations of the thermal wave packet upon renormalization, which also affects the partition function occurring in the denominator. Renormalization will also render $\psi_{\theta}^{(T)}$ and $\phi_q$ of the same dimension. Because of the inner product Eq. (10), eigenstates $\phi_q$ have the dimension of the square root of the density in $D(q)$.
The quantities $\theta(q)$ are random angles and satisfy the correlation functions

$$ \langle e^{i\theta(q)} \rangle_{\theta} = 0 \qquad (24) $$

$$ \langle e^{i\theta(q)} e^{-i\theta(q')} \rangle_{\theta} = \begin{cases} 0, & q \neq q' \\ 1, & q = q' \end{cases} \qquad (25) $$

These relations will be used for the evaluation of the MSD in the continuum.

Eq. (17) proposes a formal expression for $^{(QCF)}\delta_x^2(t)$ in the continuum. Correspondingly, we may give here a formal expression for $^{(TWP)}\delta_x^2(t)$ in the continuum, by analogy:

$$ {}^{(TWP)}\delta_x^2(t) = \frac{4}{Q^2} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq e^{-\beta(E(q') + E(q))} |x(q', q)|^2 \sin^2 \left[ \frac{(E(q) - E(q'))t}{2\hbar} \right] \qquad (26) $$

where $x(q', q)$ are defined in Eq. (18).

### 3. Freely moving particle

The basic theory presented in the previous section is general. We now address the specific case of a particle of mass $m$ moving freely in space, i.e. the Hamiltonian

$$ \hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \qquad (27) $$

#### 3.1. Continuum case

Without further constraints, the projections of the stationary states of the Hamiltonian in Eq. (27) onto the position states yield the spectral wave functions

$$ \phi_q(x) = \langle x | \phi_q \rangle = \frac{e^{i q x}}{\sqrt{2\pi}} \qquad (28) $$

and the energies

$$ E(q) = \frac{\hbar^2 q^2}{2m} \qquad (29) $$

with $D(q) = \mathbb{R}$, such that

$$ \beta E(q) = \frac{\hbar^2 \beta}{2m} q^2 = \frac{\lambda_{\text{th}}^2 q^2}{4\pi} \qquad (30) $$

$$ \frac{E(q) t}{\hbar} = \frac{\hbar^2 \beta q^2 t}{2m \hbar \beta} = \frac{\lambda_{\text{th}}^2 q^2 t}{4\pi \tau_{\text{th}}} \qquad (31) $$
In these equations, $\lambda_{th} = h/\sqrt{2\pi m k_B T}$ is the thermal de Broglie wavelength [20, 21], and $\tau_{th} = \hbar/(k_B T)$ is the thermal time (see section 1). In ref. 13 the quantity $\lambda_T = \lambda_{th}/2\pi$ was used to denote the thermal de Broglie wavelength, and the symbol $t_T$ was used to denote the thermal time. In the present work, we shall consider $\lambda_{th}$ and $\tau_{th}$ as units for lengths and times, respectively.

For the free particle, it is interesting to consider the quantity $D$, defined by

$$D \equiv \frac{\hbar}{2m} \qquad (32)$$

which has the dimension of a diffusion coefficient [13]. Useful relations involving $D$, $v_{th}$, $\lambda_{th}$ and $\tau_{th}$ are:

$$D = \frac{\lambda_{th}^2}{4\pi \tau_{th}} = \frac{1}{\sqrt{8\pi}} v_{th} \lambda_{th} \qquad (33)$$

$$v_{th}^2 = \frac{\lambda_{th}^2}{2\pi \tau_{th}^2} = \frac{8\pi D^2}{\lambda_{th}^2} \qquad (34)$$

In terms of $D$, time dependent stationary states can be written as

$$\phi_q(t, x) = \frac{1}{\sqrt{2\pi}} e^{i q \cdot x} (x - D q \cdot t) \qquad (35)$$

For simplicity of the notation, we shall normally set $|\phi_q\rangle \equiv |q\rangle$ in the remainder of this work.

## 3.2. Periodic boundary conditions

A particle moving freely on a ring of length $L$, is described by the Hamiltonian defined in Eq. (27) with the conditions that wave functions are periodic in $L$, leading to a discretization of the spectrum. The spectral wave functions are now the normalized plane waves (PL)

$$\phi_n(x) = \langle x | \phi_n \rangle = \frac{e^{i q_n x}}{\sqrt{L}} \qquad (36)$$

and the energies

$$E_n = \frac{\hbar^2 q_n^2}{2m} \qquad (37)$$

with $q_n = 2\pi n/L$, $n \in \mathbb{Z}$ and $D(q_n) = \mathbb{R}$, such that

$$\beta E_n = \frac{\lambda_{th}^2 q_n^2}{4\pi} = \pi n^2 \left(\frac{\lambda_{th}}{L}\right)^2 \qquad (38)$$

$$\frac{E_n t}{\hbar} = \frac{\lambda_{th}^2 q_n^2}{4\pi} \frac{t}{\tau_{th}} = \pi n^2 \left(\frac{\lambda_{th}}{L}\right)^2 \frac{t}{\lambda_{th}} \qquad (39)$$
Here, too, the time dependent stationary states can be expressed in terms of the constant D as

$$ \phi_n(t, x) = \frac{1}{\sqrt{L}} e^{i q_n} (x - D q_n t) \quad (40) $$

For simplicity of the notation, we shall set $|\phi_n\rangle \equiv |n\rangle$ for the spectral states in the remainder of this work.

Under periodic boundary conditions, the vector space spanned by the spectral states of Eq. (36) becomes the Hilbert space

$$ \mathcal{H}_{\text{PL}}(L) = \{\psi \in \mathbb{C}^{\mathbb{R}} \mid \exists \int_{0}^{L} |\psi(x)|^2 dx, \psi(x) = \psi(x+L)\} \quad (41) $$

The matrix element of the position operator, Eq. (16), are complex numbers that can be evaluated analytically, see appendix A.

In the absence of these conditions, the spectrum is continuous, the space spanned by the spectral states is not closed and hence not a Hilbert space, and special care has to be taken to evaluate the matrix elements of the position operator. We shall show in the following, that these quantities are distributions in q-space. To that end, we shall first consider the role of the renormalization factor mentioned above in connection with the partition function given by Eq. (11).

Before doing that, it is worth to mention that the Hilbert space of square integrable functions composes a subset of the continuum. That space can in turn be approximated by a sequence of Hilbert spaces with a discrete spectrum that we recall next.

### 3.3. Quasi-continuum states

Let

$$ \mathcal{H}_{\text{PB}}(L) = \{\psi \in \mathbb{C}^{\mathbb{R}} \mid \exists \int_{0}^{L} |\psi(x)|^2 dx, \psi(x \le 0) = \psi(L \le x) = 0\} \quad (42) $$

be the set of square integrable functions in an interval of length L, which vanish identically outside that interval. This space contains the spectral wave functions of the particle-in-a-box Hamiltonian, Eq. (43):

$$ \hat{H}_L = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V_L(x) \quad (43) $$

where $V_L(x) = 0$ for $0 < x < L$ and $\infty$ outside this interval. In that interval, $\hat{H}_L$ is identical with $\hat{H}$, Eq. (27), and one has $\hat{H}_L \to \hat{H}$ throughout $\mathbb{R}$ in the limit $L \to \infty$. The stationary particle-in-a-box waves (PB) are given by

$$ \phi_n(x) = \begin{cases} \sqrt{\frac{2}{L}} \sin\left(n\pi \frac{x}{L}\right) & 0 \le x \le L \\ 0 & x \le 0 \text{ or } L \le x \end{cases} \quad (44) $$
where $n = 1, 2, 3, \dots$, and $q_n = n\pi/L$. If the interval $(-L/2, L/2)$ is chosen instead of $(0, L)$, the spectral wave functions are the sine functions in Eq. (44) for even values of $n$, and cosine functions of the same argument for odd values of $n$.

The spectrum is formally identical to that of periodic boundary conditions, i.e. $E_n = \hbar^2 q_n^2/(2m)$. However, the expressions for $\beta E_n$ and $E_n t/\hbar$ in terms of $n$ differ from those of Eqs. (38) and (39) by a factor 1/4.

For $L \to \infty$, $E_n$ become densely distributed and $\mathcal{H}_{\text{PB}}(L \to \text{inf})$ spans the Hilbert space of square integrable functions in $\mathbb{R}$. We therefore may call this space the quasi-continuum. Matrix elements of the position operator in the PB-basis can be evaluated analytically (see appendix B).

### 3.4. Renormalization factor

In addition to the inner product Eq. (10), eigenstates of the free particle in the continuum form a complete set, i.e. they satisfy the resolution of identity

$$ \hat{I} = \int_{-\infty}^{\infty} dq |q\rangle\langle q| \qquad (45) $$

The set of states of the position operator,

$$ \hat{x}|x\rangle = x|x\rangle \qquad (46) $$

is also orthogonal in the sense that

$$ \langle x'|x\rangle = \delta(x' - x) \qquad (47) $$

and also satisfies the resolution of identity

$$ \hat{I} = \int_{-\infty}^{\infty} dx |x\rangle\langle x| \qquad (48) $$

The latter expression can be used to derive the renormalization factor:

$$ \begin{aligned} \langle q|q\rangle &= \langle q| \int_{-\infty}^{\infty} dx|x\rangle\langle x||q\rangle = \int_{-\infty}^{\infty} dx \langle q|x\rangle\langle x|q\rangle \\ &= \int_{-\infty}^{\infty} dx |\phi_q(x)|^2 = \frac{1}{2\pi} \int_{-\infty}^{\infty} dx \equiv \frac{L_{\infty}}{2\pi} \end{aligned} \qquad (49) $$

The formal length $L_{\infty}(=\int_{-\infty}^{\infty} dx = \infty)$ is the infinitely large measure of the particle's position space $x \in \mathbb{R}$. It is useful in relating the Dirac $\delta$-function with the Kronecker $\delta$-symbol. Consider the discrete set of wave numbers $q_n = 2\pi n/L \in \mathbb{R}, \forall L \in \mathbb{R}^+, n \in \mathbb{Z}$. For $L \to \infty$ this set becomes dense in $\mathbb{R}$. Furthermore, $1 = \sum_n \delta_{n'n}$ can be replaced by the Riemann sum $1 = L/(2\pi) \sum_n \delta_{n'n} \Delta q_n \approx \int_{-\infty}^{\infty} dq \delta(q' - q)$, with $\delta(q_{n'} - q_n) \approx L/(2\pi) \delta_{n'n}$, where the approximation is the better, the larger $L$. It
is therefore meaningful to formally set $\delta(q_{n'} - q_n) = L_\infty/(2\pi) \delta_{n'n}$, i.e. discrete in $n$, but continuous in $q_n$.

This formal length will be used in the remainder of this work to renormalize the partition function, the thermal density matrix and the thermal wave packet in the continuum. It will also prove useful in the calculation of matrix elements of the position operator in the continuum.

### 3.5. Thermal states of the free particle

We start with the free particle in the continuum. Evaluation of Eq. (11) yields the formal result

$$ 
\begin{aligned}
Q &= \int_{-\infty}^{\infty} dq \, e^{-\beta E(q)} \frac{L_{\infty}}{2\pi} \\
&= \frac{L_{\infty}}{2\pi} \underbrace{\int_{-\infty}^{\infty} dq \, e^{-\lambda_{\text{th}}^2 q^2 / (4\pi)}}_{2\pi/\lambda_{\text{th}}} \\
&= \frac{L_{\infty}}{\lambda_{\text{th}}}
\end{aligned} 
\quad (50) $$

The thermal density matrix is diagonal and time independent; formally:

$$ 
\begin{aligned}
\rho^{(T)}(q', q) &= \frac{1}{Q} \langle q'|e^{-\beta \hat{H}}|q \rangle \\
&= \frac{\lambda_{\text{th}}}{L_{\infty}} \delta(q' - q) e^{-\frac{\lambda_{\text{th}}^2 q^2}{4\pi}}
\end{aligned} 
\quad (51) $$

The thermal wave packet in the continuum is derived from Eq. (23) upon insertion of the specific values for the energies and partition function for the free particle:

$$ |\psi_{\theta}^{(T)}(t)\rangle = C \sqrt{\frac{\lambda_{\text{th}}}{L_{\infty}}} \int_{-\infty}^{\infty} dq \, e^{-\frac{\lambda_{\text{th}}^2 q^2}{4\pi}} \left( \frac{1}{2} + i \frac{t}{\tau_{\text{th}}} \right) + i \theta(q) |\phi_q\rangle \quad (52) $$

This equation is formal, as it contains the formal parameter $L_\infty$. And the factor $C$ is now introduced as an appropriate proportionality constant. It can indeed be determined by evaluation of the thermal density matrix, as shown in the following.
The density matrix underlying the thermal wave packet can then be expressed as

$$
\begin{align}
\rho_{\theta}^{(T)}(t; q', q) &= \langle q' | \psi_{\theta}^{(T)}(t) \rangle \langle \psi_{\theta}^{(T)}(t) | q \rangle \nonumber \\
&= |C|^2 \frac{\lambda_{\text{th}}}{L_{\infty}} \int_{-\infty}^{\infty} dq'' \int_{-\infty}^{\infty} dq''' e^{-\frac{\lambda_{\text{th}}^2}{8\pi}} (q''^2 + q'''^2) \nonumber \\
&\quad \times e^{-i \frac{\lambda_{\text{th}}^2}{4\pi} \frac{t}{\tau_{\text{th}}} (q'''^2 - q''^2) + i (\theta(q'') - \theta(q'''))} \times \delta(q''' - q') \delta(q'' - q) \nonumber \\
&= |C|^2 \frac{\lambda_{\text{th}}}{L_{\infty}} e^{-\frac{\lambda_{\text{th}}^2}{8\pi} (q'^2 + q'^2)} - i \frac{\lambda_{\text{th}}^2}{4\pi} \frac{t}{\tau_{\text{th}}} (q'^2 - q'^2) + i (\theta(q') - \theta(q)) \tag{53}
\end{align}
$$

The ensemble averaged density matrix is diagonal and time independent:

$$
\begin{align}
\langle \rho_{\theta}^{(T)}(t; q', q) \rangle_{\theta} &= |C|^2 \frac{\lambda_{\text{th}}}{L_{\infty}} e^{-\frac{\lambda_{\text{th}}^2}{8\pi} (q'^2 + q''^2)} - i \frac{\lambda_{\text{th}}^2}{4\pi} \frac{t}{\tau_{\text{th}}} (q'^2 - q''^2) \left\langle e^{i(\theta(q') - \theta(q))} \right\rangle_{\theta} \nonumber \\
&= \begin{cases} |C|^2 \frac{\lambda_{\text{th}}}{L_{\infty}} e^{-\frac{\lambda_{\text{th}}^2}{4\pi} q'^2} & , q = q' \\ 0 & , q \neq q' \end{cases} \tag{54}
\end{align}
$$

where the last equation results from the correlations between random phases given
in Eq. (25).

Eq. (54) is to be identified with the thermal density matrix from Eq. (51). We
consider again the discrete but dense set of wave numbers $q_n = 2\pi n/L \in \mathbb{R}, \mathbb{R}^+ \ni L \to L_\infty, n \in \mathbb{Z}$, such that $\delta(q_{n'} - q_n) = L_\infty/2\pi \delta_{n'n}$; then, Eq. (54) can also be
written as

$$
\langle \rho_{\theta}^{(T)}(t; q_{n'}, q_n) \rangle_{\theta} = |C|^2 \frac{\lambda_{\text{th}}}{L_{\infty}} \delta_{n'n} e^{-\frac{\lambda_{\text{th}}^2 q_n^2}{4\pi}} \quad (55)
$$

and the identity $\langle \rho_{\theta}^{(T)}(t; q_{n'}, q_n) \rangle_{\theta} \equiv \rho^{(T)}(q_{n'}, q_n)$ is verified by defining the propor-
tionality constant $C$, modulo a phase factor, as

$$
C = \sqrt{\frac{L_{\infty}}{2\pi}} \qquad (56)
$$

The wave function of the thermal wave packet

$$
\langle x | \psi_{\theta}^{(T)}(t) \rangle = \psi_{\theta}^{(T)}(t, x) = \frac{\sqrt{\lambda_{\text{th}}}}{2\pi} \int_{-\infty}^{\infty} dq \, e^{-\frac{\lambda_{\text{th}}^2}{4\pi} q^2 \left(\frac{1}{2} + i \frac{t}{\tau_{\text{th}}}\right)} + i q x + i \theta(q) \quad (57)
$$

has hence the appropriate dimension of the square root of a reciprocal length.
In the case periodic boundary conditions apply, the partition function in Eq. (7) becomes

$$Q = \sum_n e^{-\lambda_{\text{th}}^2 q_n^2 / (4\pi)} \qquad (58)$$

the density matrix in the basis of stationary states is diagonal and time independent,

$$\langle \phi_{n'} | \hat{\rho}^{(T)} | \phi_n \rangle = \delta_{n'n} \frac{\lambda_{\text{th}}}{L} e^{-\lambda_{\text{th}}^2 q_n^2 / (4\pi)} \qquad (59)$$

and the thermal wave packet is

$$|\psi_{\theta}^{(T)}(t)\rangle = \sum_{n} \frac{e^{-\lambda_{\text{th}}^2 q_n^2 / (8\pi) - iD q_n^2 t + i\theta_n}}{\sqrt{Q}} |\phi_n\rangle \qquad (60)$$

The sums extend over the entire set of integer numbers, $n \in \mathbb{Z}$. These equations apply formally also for the quasi-continuum of particle-in-the-box states, where the sums extend over the set of positive integers $n = 1, 2, \dots, \infty$.

Because $q_n = 2\pi n/L$, in the limit of large $L$ one can approximate $q_n$ by the continuous wave number $q$, the sums by Riemann sums and the corresponding integrals:

$$\lim_{L \to \infty} Q = \frac{L_{\infty}}{\lambda_{\text{th}}} \qquad (61)$$

$$\lim_{L \to \infty} \langle \phi_{n'} | \hat{\rho}^{(T)} | \phi_n \rangle = \langle \phi_{q'} | \hat{\rho}^{(T)} | \phi_q \rangle \, dq = \delta(q' - q) \, dq \frac{\lambda_{\text{th}}}{L_{\infty}} e^{-\lambda_{\text{th}}^2 q^2 / (4\pi)} \qquad (62)$$

$$\lim_{L \to \infty} |\psi_{\theta}^{(T)}(t)\rangle = \sqrt{\frac{\lambda_{\text{th}}}{2\pi}} \int_{-\infty}^{\infty} dq \, e^{-\frac{\lambda_{\text{th}}^2}{4\pi} q^2 \left(\frac{1}{2} + i \frac{t}{\tau_{\text{th}}}\right)} + i \theta(q) |q\rangle \qquad (63)$$

It would be wrong, however, to anticipate that the results for the MSD in the continuum can be obtained from those under periodic boundary conditions in the limits of infinitely large rings. As will be shown in section 5, the results are quite different, indeed.

## 4. Matrix elements in the continuum

To evaluate the MSD in the continuum of a free particle, one needs to calculate the matrix elements

$$x(q', q) = \langle q' | \hat{x} | q \rangle = \frac{1}{2\pi} \int_{-\infty}^{\infty} dx \, x e^{i(q - q')x} \qquad (64)$$

These are Fourier transformations of the line function $f(x) = x$. In standard algebra, this transformation does not exist. However, in the formalism of generalized functions [18], it can be viewed as a distribution, in the same spirit as the inner product of two stationary states of the free particle yield the Dirac $\delta$-function.
One possibility is to write

$$
\begin{align}
x(q', q) &= \lim_{L \to \infty} \frac{1}{2\pi} \int_{-L/2}^{L/2} dx \, x e^{i(q - q')x} \nonumber \\
&= \lim_{L \to \infty} \frac{1}{2\pi i} \left( L \frac{\cos((q - q')L/2)}{q - q'} - \frac{2 \sin((q - q')L/2)}{(q - q')^2} \right) \tag{65}
\end{align}
$$

whenever this limes can be appropriately evaluated.

A second possibility is to evaluate the matrix elements in a series of appropriate Gaussian wave functions, which individually converge to the stationary states of the free particle. The resulting matrix elements will then be the limes of the matrix elements of that series, whenever that limes exists and makes sense.

4.1. Zel'dovich states

For $L \in \mathbb{R}^+$ consider a family of functions defined for $x \in \mathbb{R}$ of the type

$$
\zeta_{L,q}(x) = e^{-\frac{x^2}{4L^2}} \frac{e^{iqx}}{\sqrt{2\pi}} \quad (66)
$$

Such states are dubbed Zel'dovich states here, in accordance to the theory developed
on pages 130 ff. of ref. 22. Because

$$
\langle \zeta_{L,q'} | \zeta_{L,q} \rangle = L \frac{e^{-L^2 (q' - q)^2 / 2}}{\sqrt{2\pi}} \quad (67)
$$

$$
\int_{-\infty}^{\infty} dq \, \langle \zeta_{L,q'} | \zeta_{L,q} \rangle = \int_{-\infty}^{\infty} dq' \, \langle \zeta_{L,q'} | \zeta_{L,q} \rangle = 1 \quad (68)
$$

from which we conclude $\lim_{L \to \infty} \langle \zeta_{L,q'} | \zeta_{L,q} \rangle = \delta(q' - q)$ and hence prove that

$$
\lim_{L \to \infty} |\zeta_{L,q}\rangle = |\phi_q\rangle (\equiv |q\rangle) \qquad (69)
$$

4.2. Time evolution of the Zel'dovich states for the free particle

Under the action of the Hamiltonian for the free particle in the continuum, Eq. (27),
the time evolution of the Zel'dovich states can be given as

$$
\zeta_{L,q}(x,t) = \int_{-\infty}^{\infty} dq' \langle \phi_{q'} | \zeta_{L,q} \rangle e^{-iDq'^2 t} \phi_{q'}(x) \quad (70)
$$

with

$$
\langle \phi_{q'} | \zeta_{L,q} \rangle = \frac{L}{\sqrt{\pi}} e^{-L^2 (q' - q)^2} \quad (71)
$$
The quantity $D = \hbar/(2m)$ was introduced in Eq. (32); it has the dimension of a diffusion coefficient and potentially bears important quantitative information about the diffusion process.

The integral yields

$$ \begin{aligned} \zeta_{L,q}(x,t) &= e^{-\frac{x^2 + i 4 q L^2 (Dqt - x)}{4 (L^2 + i Dt)}} \frac{L}{\sqrt{2 \pi (L^2 + i Dt)}} \\ &= \frac{e^{-\frac{(x - 2Dqt)^2}{4 L^2 w^2(t)}} + i \left( \frac{(qx - Dq^2t) - Dtx^2 / (4L^2)}{w^2(t)} + \arctan(Dt / L^2) \right)}{\sqrt{2 \pi w(t)}} \end{aligned} \quad (72) $$

with

$$ w(t) = \sqrt{1 + \frac{(Dt)^2}{L^4}} \quad (73) $$

One concludes, furthermore,

$$ \lim_{L \to \infty} \zeta_{L,q}(x,t) = \phi_q(x,t) \left( \equiv \frac{e^{i q (x - Dqt)}}{\sqrt{2\pi}} \right) \quad (74) $$

### 4.3. Position matrix element in a Zel'dovich state

Integrals such as $\langle \zeta_{L,q'}(t)|\hat{x}|\zeta_{L,q}(t) \rangle$ exist for all times and finite values of $L$; and one can straightforwardly show that

$$ \langle \zeta_{L,q'}(t)|\hat{x}|\zeta_{L,q}(t) \rangle = \frac{L}{\sqrt{2\pi}} e^{-\frac{L^2 (q' - q)^2}{2}} (D(q' + q)t - i L^2 (q' - q)) \quad (75) $$

The second term can be written as

$$ -i \frac{L}{\sqrt{2\pi}} e^{-\frac{L^2 (q' - q)^2}{2}} L^2 (q' - q) = i \frac{\partial}{\partial(q' - q)} \frac{L}{\sqrt{2\pi}} e^{-\frac{L^2 (q' - q)^2}{2}} \quad (76) $$

and one might therefore identify (see Eq. (67) ff.)

$$ \lim_{L \to \infty} \langle \zeta_{L,q'}(t) | \hat{x} | \zeta_{L,q}(t) \rangle = \delta(q' - q) \underbrace{2 D q t}_{\hbar q t / m} + i \frac{\partial \delta(q' - q)}{\partial(q' - q)} \quad (77) $$

Let the time differential matrix element be defined as

$$ \Delta x_{L;q'q}(t) \equiv \langle \zeta_{L,q'}(t) | \hat{x} | \zeta_{L,q}(t) \rangle - \langle \zeta_{L,q'}(0) | \hat{x} | \zeta_{L,q}(0) \rangle \quad (78) $$
One has

$$ \Delta x_{L;q'q}(t) = \frac{L}{\sqrt{2\pi}} e^{-\frac{L^2(q'-q)^2}{2}} (q'+q) Dt \quad (79) $$

and

$$ \lim_{L \to \infty} \Delta x_{L; q'q}(t) = \delta(q' - q) 2 D q t \left( = \delta(q' - q) \frac{\hbar q}{m} t \right) \quad (80) $$

### 4.4. Matrix elements of the position and displacement operators in the basis of energy eigenstates of the free particle

From Eq. (77) one can hence conclude that

$$ \langle \phi_{q'}(t) | \hat{x} | \phi_q(t) \rangle = \delta(q' - q) \frac{\hbar q}{m} t + i \frac{\partial \delta(q' - q)}{\partial(q' - q)} \quad (81) $$

Here, use is made of the Schrödinger representation. The same result holds in the Heisenberg representation, because

$$ \langle \phi_{q'}(t) | \hat{x} | \phi_q(t) \rangle = \langle \phi_{q'} | \hat{x}(t) | \phi_q \rangle \quad (\equiv \langle q' | \hat{x}(t) | q \rangle) \quad (82) $$

The matrix elements of the displacement operator $\hat{d}(t)$ (Eq. (2)) in the basis of eigenstates are then

$$ d_{q'q}(t) = \langle q' | \hat{d}(t) | q \rangle = \delta(q' - q) 2Dqt \quad (83) $$

From Eq. (81) it follows for $t = 0$ that the matrix element of the position operator, Eq. (65), is the distribution

$$ x(q', q) = i \frac{\partial \delta(\tilde{q})}{\partial \tilde{q}} \Big|_{\tilde{q}=q'-q} \quad (84) $$

### 4.5. Matrix element of the squared displacement operator $\hat{d}(t)^2$

From Eq. (83) one concludes that, for continuum states $|q\rangle \neq |q'\rangle$

$$ \begin{aligned} \langle q'|\hat{d}(t)^2|q\rangle &= \int_{-\infty}^{\infty} dq'' \langle q'|\hat{d}(t)|q''\rangle \langle q''|\hat{d}(t)|q\rangle \\ &= \int_{-\infty}^{\infty} dq'' d_{q'q''}(t) d_{q''q}(t) \\ &= (2Dt)^2 \int_{-\infty}^{\infty} dq'' \delta(q' - q'') \delta(q'' - q) q''q \\ &= \delta(q' - q) q'q (2Dt)^2 \end{aligned} \quad (85) $$
Here the resolution of identity in $q$-space was used.

To calculate the diagonal matrix elements, the expression

$$ \begin{aligned} \langle q | \hat{d}(t)^2 | q \rangle &= \int_{-\infty}^{\infty} dq' \langle q | \hat{d}(t) | q' \rangle \langle q' | \hat{d}(t) | q \rangle \\ &= \int_{-\infty}^{\infty} dq' (d_q q'(t))^2 = (2Dt)^2 \int_{-\infty}^{\infty} dq' \delta^2(q' - q) \end{aligned} \quad (86) $$

is not solvable, as we do not know how to evaluate $\delta^2(q)$ [18].

To circumvent this problem, one may consider the matrix element between an eigen-state $|x\rangle$ of the position operator, and a stationary state $|q\rangle$, again by using the resolution of identity on $q$-space, and write

$$ \begin{aligned} \langle x | \hat{d}(t) | q \rangle &= \int_{-\infty}^{\infty} dq' \langle x | q' \rangle \langle q' | \hat{d}(t) | q \rangle \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dq' e^{i x q'} d_{q'q}(t) = \frac{2Dt}{\sqrt{2\pi}} e^{i x q} q \end{aligned} \quad (87) $$

One then obtains

$$ \begin{aligned} \langle q | \hat{d}(t)^2 | q \rangle &= \int_{-\infty}^{\infty} dx \langle q | \hat{d}(t) | x \rangle \langle x | \hat{d}(t) | q \rangle \\ &= \frac{1}{2\pi} (2Dqt)^2 \int_{-\infty}^{\infty} dx = \frac{L_{\infty}}{2\pi} (2Dqt)^2 \end{aligned} \quad (88) $$

In the latter equation, use was made of the resolution of identity in $x$-space and of the formal renormalization length $L_{\infty}$ defined in Eq. (49). This provides a robust way of evaluating the matrix element of the displacement operator in the continuum. It reveals a quadratic dependency on time, while allowing the matrix elements to be renormalized.

A further interesting matrix element to consider is

$$ \begin{aligned} \langle x | \hat{d}(t)^2 | q \rangle &= \int_{-\infty}^{\infty} dq' \langle x | \hat{d}(t) | q' \rangle \langle q' | \hat{d}(t) | q \rangle \\ &= \int_{-\infty}^{\infty} dq' \frac{e^{i x q'} 2Dt}{\sqrt{2\pi}} \delta(q' - q) 2Dt \\ &= \frac{1}{\sqrt{2\pi}} e^{-i x q} (2Dt)^2 \end{aligned} \quad (89) $$

where use was made of the resolution of identity in $q$-space again.
## 5. Calculation of the MSD of the freely moving particle

We now evaluate the formulae for $(QCF)\delta_x^2$ and $(TWP)\delta_x^2$ for a free particle moving in the continuum and under periodic boundary conditions. In the first place, the specific expressions for the expectation value of the position operator in a thermal wave packet state are discussed, both in the interval and in the ring topology.

### 5.1. Expectation value of the position operator in a thermal wave packet

Before evaluating Eq. (3), one needs to determine the expectation value of the position operator in a thermal wave packet state. For the particle moving on a ring, this evaluation was carried out in ref. 13:

$$x_{\theta}(t) = i \frac{\lambda_{\text{th}}}{2\pi} \sum_{n=-\infty}^{\infty} \sum_{j=-\infty}^{\infty} e^{-\pi \frac{\lambda_{\text{th}}^2}{2L^2} (n^2 + j^2)} \frac{e^{i(\theta_j - \theta_n + \pi(n-j+1))}}{(n-j)} e^{i(n^2 - j^2)} \frac{4\pi^2 Dt}{L^2} \quad (90)$$

This is a non-trivial, random function of time; the primes indicate that the terms with $n = j$ are excluded from the sums. Figure 1 shows such functions for two different stochastic realizations and two different values of the ring length $L$.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** $x_{\theta}(t) - x_{\theta}(0)$ for a freely moving particle on a ring. Dashed lines are for $L \approx 25\lambda_{\text{th}}$, continuous lines for $L \approx 100\lambda_{\text{th}}$; color codes are two different sets of random phases $\theta_n$.</figcaption></figure>

From these simulations there is no numerical indication that the quantity in Eq. (90) should vanish for infinitely large rings.
For a particle moving in the continuum, one first has

$$ 
\begin{aligned}
x_{\theta}(t) &= \langle \psi_{\theta}^{(T)}(t) | \hat{x} | \psi_{\theta}^{(T)}(t) \rangle \\
&= \frac{\lambda_{\text{th}}}{2\pi} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2}{4\pi} (q'^2 + q^2)} e^{i(\theta(q) - \theta(q'))} \langle \phi_{q'}(t) | \hat{x} | \phi_q(t) \rangle
\end{aligned} 
\quad (91) $$

It is more interesting to evaluate directly the displacement

$$ 
\begin{aligned}
x_{\theta}(t) - x_{\theta}(0) &= \langle \psi_{\theta}^{(T)}(t) | \hat{x} | \psi_{\theta}^{(T)}(t) \rangle - \langle \psi_{\theta}^{(T)}(0) | \hat{x} | \psi_{\theta}^{(T)}(0) \rangle \\
&= \frac{\lambda_{\text{th}}}{2\pi} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2}{4\pi} (q'^2 + q^2)} e^{i(\theta(q) - \theta(q'))} d_{q'q}(t)
\end{aligned} 
\quad (92) $$

Insertion of Eq. (83) yields immediately

$$ 
\begin{aligned}
x_{\theta}(t) - x_{\theta}(0) &= \frac{\lambda_{\text{th}}}{2\pi} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2}{4\pi} (q'^2 + q^2)} e^{i(\theta(q) - \theta(q'))} \delta(q' - q) 2Dq \\
&= \frac{2D\lambda_{\text{th}}t}{2\pi} \underbrace{\int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2}{2\pi} q^2}}_{\equiv 0} q \quad (93)
\end{aligned} $$

The differential expectation value of the position operator in a thermal state in the continuum vanishes identically and independently of the random phases: $x_{\theta}(t) = x_{\theta}(0), \forall \theta$.

This result is in stark contrast to the particle moving on a ring, Eq. (90). Figure 1 indicates that the displacement $x_{\theta}(t) - x_{\theta}(0)$ for the particle moving on a ring does not seem to decrease with increasing ring length. Eq. (93) therefore does not capture the limiting situation for infinitely large ring lengths $L \to \infty$.

## 5.2. (TWP) $\delta_x^2$ for a free particle in the continuum

It is instructive to first evaluate Eq. (22) in the discrete case of the quasi-continuum of particle-in-the-box states. Using the matrix elements derived in appendix B one obtains

$$ 
\begin{aligned}
(\text{TWP}) \delta_x^2(t) &= \frac{64 L^2}{\pi^4 Q^2} \sum_{n=1}^{\infty} \sum_{j=1}^{\infty} e^{-\frac{\pi}{4} \left(\frac{\lambda_{\text{th}}}{L}\right)^2 (n^2 + j^2)} \times \\
&\quad \sin^2 \left[ (n^2 - j^2) \frac{\pi}{8} \left(\frac{\lambda_{\text{th}}}{L}\right)^2 \frac{t}{\tau_{\text{th}}} \right] \left( \frac{n j \left((-1)^{n+j} - 1\right)}{(n^2 - j^2)^2} \right)^2
\end{aligned} 
\quad (94) $$
where the prime indicates $j \neq n$ and $Q$ is given in Eq. (58); note that for $L \gg \lambda_{th}$, $Q \approx L/\lambda_{th}$.

Eq. (94) can be explored numerically. Figure 2 shows the results for increasing values of the box length $L$.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** $(^{(TWP)}\delta_x^2(t)$ for a freely moving particle in a box. Black lines are numerical results from Eq. (94), for different values of $L/\lambda_{th}$. The blue line is the classical result $v_{th}^2 t^2$ from Eq. (4) for a particle moving in a continuum.</figcaption></figure>

The figure suggests that $(^{(TWP)}\delta_x^2(t) \to 0$ identically, when the box length becomes infinitely large, i.e. when the quasi-continuum is approached. While the numerical result excludes non-square integrable wave functions, Eq. (93) allows one to immediately draw the same conclusion for true continuum states.

$$ (^{(TWP)}\delta_x^2(t) \equiv 0 \qquad (95) $$

for the free particle in the continuum.

This result could have been anticipated from ref. 16. The result of that analysis, which was carried out using thermal Gaussian wave packets [17], can indeed also be drawn in the basis of Zel'dovich states (appendix C).
### 5.3. $(\text{QCF})\delta_x^2$ for a free particle in the continuum

Here, too, it is first interesting to evaluate Eq. (15) in the quasi-continuum of particle-in-the-box states. Similarly to Eq. (94) one obtains

$$ (\text{QCF})\delta_x^2(t) = \frac{64 L^2}{\pi^4 Q} \sum_{n=1}^{\infty} e^{-\frac{\pi}{4} \left(\frac{\lambda_{\text{th}}}{L}\right)^2} n^2 \times \\ \sum_{j=1}^{\infty} \sin^2 \left[ (n^2 - j^2) \frac{\pi}{8} \left(\frac{\lambda_{\text{th}}}{L}\right)^2 \frac{t}{\tau_{\text{th}}} \right] \left( \frac{n j \left((-1)^{n+j} - 1\right)}{(n^2 - j^2)^2} \right)^2 \quad (96) $$

where $Q \approx L/\lambda_{\text{th}}$ for $L \gg \lambda_{\text{th}}$ is given in Eq. (58) and the prime indicates $j \neq n$.

Eq. (96) can be explored numerically. Figure 3 shows the results for increasing values of the box length $L$. For finite values of $L$, the particle-in-the-box results match the classically expected result only for small times. One can interpret this result by a reflection of the particle at the infinitely high potential walls of the potential $V_L$ at the border of the box (see Eq. (43)), i.e. it stops to move freely after some time which scales with the length of the box. One sees, however, that $(\text{QCF})\delta_x^2(t) \to^{(\text{cl})} \delta_x^2(t)$, when $L \to \infty$. Numerical result excludes non-square integrable wave functions. We prove here that this result holds indeed also for the true continuum.

From Eq. (1) one has

$$ \begin{aligned} (\text{QCF})\delta_x^2(t) &= \int_{-\infty}^{\infty} dq \langle q | \hat{\rho}^{(T)} | \hat{d}(t)^2 | q \rangle = \int_{-\infty}^{\infty} dq \langle q | \hat{\rho}^{(T)} | \int_{-\infty}^{\infty} dq' | q' \rangle \langle q' | \hat{d}(t)^2 | q \rangle \\ &= \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq \rho^{(T)}(t; q, q') \langle q' | \hat{d}(t)^2 | q \rangle \\ &= \int_{-\infty}^{\infty} dq \frac{\lambda_{\text{th}}}{L_{\infty}} e^{-\lambda_{\text{th}}^2 q^2 / (4\pi)} \langle q | \hat{d}(t)^2 | q \rangle \quad \text{(from Eq. (51))} \\ &= \frac{\lambda_{\text{th}}}{2\pi} (2Dt)^2 \underbrace{\int_{-\infty}^{\infty} dq \, q^2 e^{-\lambda_{\text{th}}^2 q^2 / (4\pi)}}_{4\pi^2 / \lambda_{\text{th}}^3} \quad \text{(from Eq. (88))} \\ &= \frac{8\pi D^2}{\lambda_{\text{th}}^2 m^2} t^2 = v_{\text{th}}^2 t^2 \quad \text{(see Eq. (34))} \end{aligned} \quad (97) $$

which is the classical result, Eq. (4). In this derivation, the resolution of the identity in $q$-space was used.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** $(^{(QCF)}\delta_x^2(t)$ for a freely moving particle in a box. Black lines are numerical results from Eq. (96), for different values of $L/\lambda_{th}$. The blue line is the classical result $v_{th}^2 t^2$ from Eq. (4).</figcaption></figure>

We obtain the same result using the resolution of the identity in x-space:

$$ \begin{align} (\text{QCF}) \delta_x^2(t) &= \int_{-\infty}^{\infty} dq \langle q | \hat{\rho}^{(T)} | \hat{d}(t)^2 | q \rangle \nonumber \\ &= \int_{-\infty}^{\infty} dq \langle q | \hat{\rho}^{(T)} | \int_{-\infty}^{\infty} dx |x\rangle \langle x | \hat{d}(t)^2 | q \rangle \nonumber \\ &= \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dq \frac{e^{-\lambda_{\text{th}}^2 q^2 / (4\pi)}}{Q} \langle q|x \rangle \langle x|\hat{d}(t)^2|q \rangle \nonumber \\ &= \frac{\lambda_{\text{th}}}{L_{\infty}} \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dq e^{-\lambda_{\text{th}}^2 q^2 / (4\pi)} \frac{e^{-iqx}}{\sqrt{2\pi}} \frac{e^{iqx}}{\sqrt{2\pi}} (2Dqt)^2 \quad (\text{see Eq(89)}) \nonumber \\ &= \frac{\lambda_{\text{th}}}{2\pi} (2Dt)^2 \underbrace{\int_{-\infty}^{\infty} dq \, q^2 e^{-\lambda_{\text{th}}^2 q^2 / (4\pi)}}_{4\pi^2/(2\lambda_{\text{th}}^3)} = v_{\text{th}}^2 t^2 \quad (\text{see Eq(97)}) \tag{98} \end{align} $$
### 5.4. (TWP) $\delta_x^2$ for a free particle on a ring

For a free particle on a ring, Eq. (22) becomes

$$ (\text{TWP}) \delta_x^2(t) = \frac{L^2}{\pi^2 Q^2} \sum_n' \sum_j' e^{-\pi \left(\frac{\lambda_{\text{th}}}{L}\right)^2 (n^2 + j^2)} \frac{\sin^2\left[\frac{\pi}{2} \left(\frac{\lambda_{\text{th}}}{L}\right)^2 \frac{t}{\tau_{\text{th}}} (n^2 - j^2)\right]}{(n-j)^2} \quad (99) $$

(see appendix A); sums run over the whole range of integer numbers, primes indicate that terms with $j = n$ are discarded and where $Q$ is given by Eq. (58) ($Q \approx L/\lambda_{\text{th}}$ for $L \gg \lambda_{\text{th}}$). This quantity was further analyzed analytically and numerically in refs. 13 and 23. Figure 4 reproduces the main results.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** (TWP) $\delta_x^2(t)$ for a freely moving particle on a ring. Dashed line from Eq. (99): finite ring length $L = 10\lambda_{\text{th}}$; continuous line for an infinite ring length, Eq. (101); red line with slope 1 in the given units. The blue line is the classical result $v_{\text{th}}^2 t^2$ from Eq. (4).</figcaption></figure>

The MSD is a monotonous function of the ring length $L$:

$$ (\text{TWP}) \delta_x^2(t; L_1) \le (\text{TWP}) \delta_x^2(t; L_2), \quad \forall L_2 \ge L_1 > 0 $$

For finite ring lengths, (TWP) $\delta_x^2(t)$ stagnates at a plateau $\delta_x^2(L)$, the asymptotic value
of which is given analytically [13]. This asymptotic value for $L = 10\lambda_{th}$ in Figure 4 is approximately $7.2 \lambda_{th}^2/(2\pi)$.

The initial time behavior is quadratic, i.e. for times $t \ll \tau_{th}$ (see Eq. (31)) it corresponds always to half the classical result

$$ (\text{TWP}) \delta_x^2(t) \approx \frac{\lambda_{\text{th}}^2}{4\pi} \left(\frac{t}{\tau_{\text{th}}}\right)^2, \text{ for } t \ll \tau_{\text{th}} \\ \approx \frac{1}{2} v_{\text{th}}^2 t^2 = \frac{1}{2} \delta_x^2(t) \qquad (100) $$

where $v_{th}^2 = k_B T/m$.

For infinitely large rings, one obtains the simple analytical formula [13], expressed here in three equivalent variants:

$$ \lim_{L \to \infty} (\text{TWP}) \delta_x^2(t) = \frac{\hbar}{m} \left( \sqrt{t^2 + \tau_{\text{th}}^2} - \tau_{\text{th}} \right) \\ = \frac{\lambda_{\text{th}}^2}{2\pi} \left( \sqrt{\left(\frac{t}{\tau_{\text{th}}}\right)^2 + 1} - 1 \right) \\ = v_{\text{th}}^2 \tau_{\text{th}} \left( \sqrt{t^2 + \tau_{\text{th}}^2} - \tau_{\text{th}} \right) \qquad (101) $$

($\text{TWP}) \delta_x^2(t)$ vanishes identically in the classical limit $h = 0$.
5.5. (QCF) $\delta_x^2$ for a free particle on a ring

For a free particle on a ring, Eq. (15) becomes

$$ (\text{QCF}) \delta_x^2(t) = \frac{L^2}{\pi^2 Q} \sum_n \sum_j' e^{-\pi \left(\frac{\lambda_{\text{th}}}{L}\right)^2 n^2} \frac{\sin^2\left[\frac{\pi}{2} \left(\frac{\lambda_{\text{th}}}{L}\right)^2 \frac{t}{\tau_{\text{th}}} (n^2 - j^2)\right]}{(n-j)^2} \quad (102) $$

(see appendix A); sums run over all numbers in $\mathbb{Z}$, primes indicate $j \neq n$ and $Q$ is given by Eq. (58). Eq. (102) cannot be cast into a more compact form. Here, we analyze it further numerically for several values of the ring length $L$. The results are given in Figure 5.

<figure><img src="image_6.png" /><figcaption>**Figure 5.** $(\text{QCF}) \delta_x^2(t)$ for a freely moving particle on a ring. Black lines are numerical results from Eq. (102), for different values of $L/\lambda_{\text{th}}$. The blue line is the classical result $v_{\text{th}}^2 t^2$ from Eq. (4) for a particle moving in a continuum.</figcaption></figure>
Additional numerical tests with increasing values of L show the monotonic behavior

$$^{(QCF)}\delta_x^2(t; L_1) \leq ^{(QCF)}\delta_x^2(t; L_2), \forall L_2 \geq L_1 > 0.$$

With increasing L, the size of the numerical basis must be increased to ensure numerical convergence.

The main result from this analysis is that $^{(QCF)}\delta_x^2(t)$ defining the MSD of a freely moving particle on a ring diverges for $L \to \infty$ immediately for $t > 0$. This result irritates because, classically, one has $d(t) = x(t) - x(0) = (p/m)t$ both for the free particle moving in a continuum and on a ring. Hence the classical result depicted by the blue line in Figure 5 could be expected to hold. As shown in appendix D, however, $\hat{d}(t) \neq (\hat{p}/m)t$ for a free particle on a ring. Furthermore, in appendix E we show analytically, by evaluation of the positional quantum correlation function $C_x(t)$ on a ring of length L, that both $^{(QCF)}\delta_x^2(t)$ and $C_x(t)$ diverge for $L \to \infty$. $^{(QCF)}\delta_x^2(t)$ may return to zero at some selected later times, as discussed analytically in appendix E; these times compose a set of measure zero, however.

## 6. Discussion and conclusions

The mean square displacement (MSD) is an average squared position of a particle over time that is measured with respect to its position at a reference time. It is an important quantity in transport theory [19, 24], notably in dissipative systems [25, 26] and specifically in relation with diffusion. While this quantity has been extensively investigated and is today well understood in classical statistical mechanics, finding its corresponding quantum mechanical expression is not straightforward, in particular due to the delocalised nature of the quantum mechanical states under thermal conditions. Knowing this quantum expression is essential, since the MSD is related to diffusion as studied in scattering experiments involving single atoms or molecules, for instance in $^3$He spin echo experiments [27].

In the present work we systematically investigated and compared two quantum mechanical definitions for the MSD in the ballistic regime of thermalized, free particles of mass $m$. The first definition relates to the quantum correlation function (QCF) [19] and was first considered in the literature (see refs. 8–11, and work cited therein). The second definition was proposed by one of us [13], and is based on thermal wave packets (TWP) [14]. Both the $^{(QCF)}\delta_x^2(t)$ and $^{(TWP)}\delta_x^2(t)$ expressions for the MSD were evaluated here in two distinct topological spaces: the open interval topology and the ring or circle topology pertaining to periodic boundary conditions.

In the interval topology, the result for the QCF-MSD is the classically expected result: $^{(QCF)}\delta_x^2(t) = k_B T/m t^2$, which is in conformity with previous work [11]. In the ring topology, however, $^{(QCF)}\delta_x^2(t)$ does not yield the classical result for the ballistic motion. When the ring length becomes infinitely large, we showed that $^{(QCF)}\delta_x^2(t)$ diverges identically, except for some selected times.

In contrast, the TWP-MSD is a well behaved function in the ring topology: $^{(TWP)}\delta_x^2(t) = k_B T/(2m) t^2$ for short times; when the ring becomes infinitely large, one approaches the continuum situation in the ring topology, and $^{(TWP)}\delta_x^2(t) \propto t$ for long times, adopting thus the typical feature of Brownian diffusion. In the continuum of the interval topology, however, we showed in the present paper analytically that $^{(TWP)}\delta_x^2(t) \equiv 0$ identically.

Out of the four cases studied here, it would seem that only $^{(QCF)}\delta_x^2(t)$ yields adequate
results when evaluated in the interval topology. By adequate we mean here compatibil with results expected from classical mechanics. The criterion of compatibility with classical results is not a necessary condition for reality, however. Furthermore, the expressions for the TWP-MSD do not violate the correspondence principle. ($^{(TWP)}$)$\delta_x^2(t)$ has no classical counterpart, since it vanishes identically with a vanishing Planck constant. In ref. 13, it was shown how this quantity can in principle be extracted from $^3$He spin echo experiments. It is interesting that the TWP-MSD vanishes identically in the interval topology.

The remaining intriguing point is that $^{(QCF)}$δ_x^2(t)$ is essentially divergent for the ballistic particle moving on an infinitely large ring. One might argue that the ring topology is not an adequate topology to define the MSD in quantum mechanics, as the periodicity infers the position operator to be a sawtooth function (see appendix D). Indeed, as was pointed out by Judge and Lewis [28], the commutator of the position and momentum operator on a ring leads to an anomalous canonical Heisenberg uncertainty relation, which these authors attributed to the winding number of the particle on a ring; they wrote that this number cannot be reasonably accessed in quantum mechanics. As we showed in appendix D, the non-commutative behavior of momentum and sawtooth position operator leads to a complicated motion of the position operator on a ring. Thus, it should not be too surprising, if the QCF-definition of the MSD does not reproduce the classical result for a ballistic motion in the ring topology. It is interesting to recall that in ref. [12], the quantity $\langle : (\hat{x}(t) - \langle \hat{x}(t) \rangle_{\text{bath}})^2 : \rangle$ was investigated for a linear chain of coupled harmonic oscillators. While that quantity is only marginally related to the MSD of the present work, it is worth mentioning that it diverges for periodic boundary conditions on the chain, while it is finite for open end chains of increasing lengths [29].

The ring topology is relevant for numerical evaluations of quantum dynamics as plane wave bases of the kind used in Eq. (36) are often employed to represent states and operators [30]. Here one has to differentiate between distinct basis representations and distinct topological spaces. While results must not depend on particular bases chosen to represent the physical processes, the different topology of the space underlying a certain representation may indeed lead to physically well distinguishable results. Yet, as Fujikawa pointed out [31], one would desire to use plane waves to represent box-normalized functions. Fujikawa showed that Bloch-wave type linear superposition states of plane waves allow us to recover the normal Heisenberg uncertainty relation. The approach is interesting, however it is not possible to map it on the present problem, as the superposition states proposed by Fujikawa cease to be eigenstates of the free particle Hamiltonian, which are finally necessary to describe thermal states.

Periodic bases are successfully used in numerical calculations of many quantum dynamical properties, for instance, of adsorbates on crystal substrates. It remains an open question why such bases yield divergent results when calculating the mean square displacement of particles in the limiting case of a completely flat potential energy surface.

## Acknowledgement(s)

Authors thank the Université de Strasbourg and CNRS. R.M. wishes in particular to thank N. Moiseyev, E. Pollak and D. Tannor for fruitful discussions during a sabbatical leave in 2023. This work is dedicated to Prof. Attila G. Czászár on the occasion of his
65th birthday.

## Disclosure statement

Authors have no conflict of interest.

## Funding

This work was realized within the ANR funded project QDDA, which is herewith dully acknowledged.

## References

[1] E. Mátyus, G. Czakó and A.G. Császár, J. Chem. Phys. **130** (13), 134112 (2009).

[2] C. Fábri, M. Quack and A.G. Császár, J. Chem. Phys. **147** (13), 134101 (2017).

[3] C. Fábri, R. Marquardt, A.G. Császár and M. Quack, J. Chem. Phys. **150** (1), 014102 (2019).

[4] A.G. Császár, C. Fábri and T. Szidarovsky, in *Molecular Spectroscopy and Quantum Dynamics*, edited by R. Marquardt and M. Quack, Chap. 2 (Elsevier, St. Louis (Missouri), USA, 2020).

[5] A. Tajti, P.G. Szalay, A.G. Császár, M. Kállay, J. Gauss, E.F. Valeev, B.A. Flowers, J. Vázquez and J.F. Stanton, J. Chem. Phys. **121** (23), 11599-11613 (2004).

[6] T. Furtenbacher, I. Szabó, A.G. Császár, P.F. Bernath, S.N. Yurchenko and J. Tennyson, The Astrophysical Journal Supplement Series **224** (2), 44 (2016).

[7] T. Szidarovszky and A.G. Császár, The Journal of Chemical Physics **142** (1), 014103 (2015).

[8] G.W. Ford, J.T. Lewis and R.F. O'Connell, Phys. Rev. A **64**, 032101 (2001).

[9] G.W. Ford and J.T. Lewis, in *Probability, Statistical Mechanics and Number Theory*, edited by Gian-Carlo Rota, Vol. 9 (Academic Press, Orlando, 1986), pp. 169-194.

[10] D.K. Wójcik and J.R. Dorfman, Phys. Rev. Lett. **90**, 230602 (2003).

[11] C.W. Gardiner and P. Zoller, *Quantum Noise* Springer Series in Synergetics (Springer, Berlin, Heidelberg, New York, 2004).

[12] G.W. Ford, M. Kac and P. Mazur, Journal of Mathematical Physics **6** (4), 504-515 (1965).

[13] R. Marquardt, Molecular Physics **119** (17-18), e1971315 (2021).

[14] R.C. Tolman, *The principles of statistical mechanics* (Oxford University Press, Oxford (UK), 1938).

[15] F. Di Maiolo, D. Brey, R. Binder and I. Burghardt, The Journal of Chemical Physics **153** (18), 184107 (2020).

[16] O. Bindech and R. Marquardt, Eur. J. Phys. Spec. Top. (2023).

[17] R. Marquardt, Molecular Physics **119** (17-18), e1974110 (2021).

[18] A. Messiah, *Mécanique Quantique* (Dunod, Paris, 1969).

[19] R. Kubo, M. Toda and N. Hashitsume, *Statistical Physics II, Nonequilibrium Statistical Mechanics*, 2nd ed. (Springer, Heidelberg, 1992).

[20] C. Kittel and H. Krömer, *Thermal physics*, 2nd ed. (Freeman, San Francisco, 1980).

[21] B. Vacchini and K. Hornberger, Phys. Rep. **478** (4), 71-120 (2009).

[22] N. Moiseyev, *NHQM - non hermitian quantum mechanics* (Cambridge University Press, Cambridge (UK), 2011).

[23] R. Marquardt, Phys. Chem. Chem. Phys. **24**, 26519 (2022).

[24] R. Balescu, *Statistical Dynamics* (Imperial College Press, World Scientific Publishing Co., London, 1997).
[25] U. Weiss, *Quantum Dissipative Systems*, 3rd ed. (World Scientific Publisher Co., London, 2008).

[26] R. Zwanzig, *Non-equilibrium Statistical Mechanics* (Oxford University Press, Oxford, 2001).

[27] A.P. Jardine, H. Hedgeland, G. Alexandrowicz, W. Allison and J. Ellis, Prog. in Surf. Sci. **84**, 323–379 (2009).

[28] D. Judge and J. Lewis, Physics Letters **5** (3), 190 (1963).

[29] J. Florencio and M.H. Lee, Phys. Rev. A **31**, 3231–3236 (1985).

[30] M.H. Beck, A. Jäckle, G.A. Worth and H.D. Meyer, Physics Reports **324** (1), 1–105 (2000).

[31] K. Fujikawa, Progress of Theoretical Physics **124** (5), 747–759 (2010).
Appendix A. Matrix elements of the position and momentum operators on a ring

In the following $\langle m|\hat{O}|n\rangle = \int_0^L \phi_m^*(x) \hat{O} \phi_n(x) dx$, with $\phi_n(x)$ given by

$$ \phi_n(x) = e^{i q_n x / \sqrt{L}} \qquad (A.1) $$

Position, for $m \neq n$,

$$ \begin{aligned} \langle m | \hat{x} | n \rangle &= \frac{1}{L} \int_0^L x e^{i 2 \pi (n-m)} x/L \, dx = L \int_0^1 y e^{i 2 \pi (n-m)} y \, dy \\ &= L \left( \left[ \frac{y e^{i 2 \pi (n-m)} y}{2 \pi i (n-m)} \right]_0^1 + \frac{1}{2 \pi i (n-m)} \int_0^1 e^{i 2 \pi (n-m)} y \, dy \right) \\ &= \frac{L}{2 \pi i (n-m)} + L \underbrace{\frac{e^{i 2 \pi (n-m)} - 1}{(2 \pi i (n-m))^2}}_{0} \end{aligned} \qquad (A.2) $$

For $m = n$,

$$ \langle n | \hat{x} | n \rangle = \frac{1}{L} \int_{0}^{L} x \, dx = \frac{L}{2} \qquad (A.3) $$

Momentum:

$$ \begin{aligned} \langle m | \hat{p} | n \rangle &= \frac{2 \pi \hbar n}{L^2} \int_0^L e^{i 2 \pi (n-m)} x/L \, dx = \frac{2 \pi \hbar n}{L} \int_0^1 e^{i 2 \pi (n-m)} y \, dy \\ &= \frac{2 \pi \hbar n}{L} \underbrace{\frac{e^{i 2 \pi (n-m)} - 1}{2 \pi i (n-m)}}_{\delta_{nm}} \end{aligned} \qquad (A.4) $$
Note that, for the choice $-L/2 \le x \le L/2$, the matrix elements of the position operator become, for $m \ne n$,

$$ \begin{aligned} \langle m | \hat{x} | n \rangle &= \frac{1}{L} \int_{-L/2}^{L/2} x e^{i 2 \pi (n-m)} x/L \, dx = L \int_{-1/2}^{1/2} y e^{i 2 \pi (n-m)} y \, dy \\ &= L \left( \left[ \frac{y e^{i 2 \pi (n-m)} y}{2 \pi i (n-m)} \right]_{-1/2}^{1/2} + \frac{1}{2 \pi i (n-m)} \int_{-1/2}^{1/2} e^{2 \pi i (n-m)} y \, dy \right) \\ &= L \frac{\cos((n-m)\pi)}{2\pi i (n-m)} + L \underbrace{2i \frac{\sin((n-m)\pi)}{(2\pi i (n-m))^2}}_{0} \\ &= L \frac{(-1)^{n+m}}{2\pi i (n-m)} \end{aligned} \quad (A.5) $$

For $m = n$,

$$ \langle n | \hat{x} | n \rangle = \frac{1}{L} \int_{-L/2}^{L/2} x \, dx = 0 \quad (A.6) $$
Appendix B. Matrix elements of the position and momentum operators in a box

In the following $\langle m|\hat{O}|n\rangle = \int_0^L \phi_m^*(x) \hat{O} \phi_n(x) dx$, with $\phi_n(x)$ given by Eq. (44).

Position, for $m \neq n$,

$$ 
\begin{aligned}
\langle m | \hat{x} | n \rangle &= \frac{2}{L} \int_0^L x \sin(m \pi x / L) \sin(n \pi x / L) dx \\
&= 2L \int_0^1 y \sin(m \pi y) \sin(n \pi y) dy \\
&= \frac{4L}{\pi^2} \frac{mn}{(m^2 - n^2)^2} ((-1)^{m+n} - 1)
\end{aligned} 
\qquad (B.1) $$

For $m = n$,

$$ 
\begin{aligned}
\langle n | \hat{x} | n \rangle &= \frac{2}{L} \int_0^L x \sin^2(n \pi x / L) dx \\
&= 2L \int_0^1 y \sin^2(n \pi y) dy \\
&= \frac{L}{2}
\end{aligned} 
\qquad (B.2) $$

For $m \neq n$, matrix elements of the momentum operator are

$$ 
\begin{aligned}
\langle m | \hat{p} | n \rangle &= -i \frac{\hbar 2 \pi n}{L^2} \int_0^L \sin(m \pi x / L) \cos(n \pi x / L) dx \\
&= -i \frac{\hbar 2 \pi n}{L} \int_0^1 \sin(m \pi y) \cos(n \pi y) dy \\
&= i \hbar \frac{2 m n}{L} \frac{(-1)^{m+n} - 1}{m^2 - n^2}
\end{aligned} 
\qquad (B.3) $$

and for $m = n$

$$ \langle n | \hat{p} | n \rangle = 0 \qquad (B.4) $$
Appendix C. Calculation of the MSD using Zel'dovich states

Here we consider first using the Zel'dovich states to evaluate the expectation values;
the limits $L \to \infty$ is performed at the end of the calculation.

$$
\begin{align*}
\delta_x^2(t) &= \frac{\lambda_{\text{th}}^2}{4\pi^2} \int_{-\infty}^{\infty} dq''' \int_{-\infty}^{\infty} dq'' \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq \\
&\qquad \left\langle e^{i(\theta(q) - \theta(q'') + \theta(q') - \theta(q'''))} \right\rangle_{\theta} \\
&\qquad \times e^{-\frac{\lambda_{\text{th}}^2}{4\pi} (q'''^2 + q''^2 + q'^2 + q^2)} \Delta x_{L;q''q}(t) \Delta x_{L;q'''q'}(t) \\
&= \delta q^2 \frac{\lambda_{\text{th}}^2}{4\pi^2} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2}{2\pi} (q'^2 + q^2)} \Delta x_{L;qq}(t) \Delta x_{L;q'q'}(t) \\
&\quad + \\
&\qquad \delta q^2 \frac{\lambda_{\text{th}}^2}{4\pi^2} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2}{2\pi} (q'^2 + q^2)} \Delta x_{L;q'q}(t) \Delta x_{L;qq'}(t) \\
&= \delta_x^2(t)^{(\text{I})} + \delta_x^2(t)^{(\text{II})} \tag{C.1}
\end{align*}
$$

where $\delta q = \lim_{L \to \infty} 2\pi/L$.

The first integral in the foregoing equation yields

$$
\begin{align*}
\delta_x^2(t)^{(\text{I})} &= \frac{\lambda_{\text{th}}^2}{L^2} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2}{2\pi} (q'^2 + q^2)} \frac{L^2}{2\pi} (2Dt)^2 qq' \\
&= \frac{2}{\pi} \lambda_{\text{th}}^2 D^2 t^2 \left( \int_{-\infty}^{\infty} dq e^{-\frac{\lambda_{\text{th}}^2 q^2}{2\pi}} q \right)^2 \\
&= 0 \tag{C.2}
\end{align*}
$$
The second

$$
\begin{align*}
\delta_x^2(t)^{(\text{II})} &= \frac{\lambda_{\text{th}}^2}{L^2} \int_{-\infty}^{\infty} dq' \int_{-\infty}^{\infty} dq \, e^{-\frac{\lambda_{\text{th}}^2}{2\pi} (q'^2 + q^2)} \frac{L^2}{2\pi} e^{-L^2 (q' - q)^2} (Dt)^2 (q + q')^2 \\
&= 2 \frac{\lambda_{\text{th}}^2}{2\pi} (Dt)^2 \left( \int_{-\infty}^{\infty} dv \, e^{-\left(\frac{\lambda_{\text{th}}^2}{2\pi} + 2L^2\right) v^2} \right) \left( \int_{-\infty}^{\infty} du \, e^{-\frac{\lambda_{\text{th}}^2}{2\pi} u^2} u^2 \right) \\
&= \frac{\lambda_{\text{th}}^2}{\pi} (Dt)^2 \frac{\sqrt{2\pi}}{\sqrt{4\pi L^2 + \lambda_{\text{th}}^2}} \frac{\sqrt{2\pi^2}}{\lambda_{\text{th}}^3} \\
&= 2\pi^2 \lambda_{\text{th}}^2 \frac{\lambda_{\text{th}}^2}{8\pi} (v_{\text{th}} t)^2 \frac{1}{\lambda_{\text{th}}^3 \sqrt{4\pi L^2 + \lambda_{\text{th}}^2}} \\
&= {}^{(\text{cl})}\delta_x^2(t) \frac{\sqrt{\pi}}{8} \frac{\lambda_{\text{th}}}{L} + O(L^{-2}) \tag{C.3}
\end{align*}
$$

where $v_{\text{th}} = \sqrt{k_B T/m}$ is the thermal speed (see section 3.1) and ${}^{(\text{cl})}\delta_x^2(t) = v_{\text{th}}^2 t^2$ is the classical expression for the MSD of a free particle (ballistic motion).

In the limit $L \to \infty$, $\delta_x^2(t) \equiv 0$ identically, as already seen from the direct evaluation of the displacement $x_\theta(t) - x_\theta(0)$ in a thermal wave packet.
## Appendix D. Position and momentum operators on a ring

This section is closely related to the findings by Judge and Lewis [28]. The position operator on a ring is defined modulo the ring length L. Let $\hat{x}|x\rangle = x|x\rangle$ and $\hat{x}_L|x\rangle = x_L|x\rangle$ with $x_L = \mod(x, L)$, i.e. $x = nL + x_L$ with $n = \text{int}(x/L)$.

The following equation then defines the position operator on a ring of length L:

$$ \hat{x}_L = \hat{x} - nL\hat{1} \qquad (D.1) $$

The discontinuous function $x_L(x)$ has the sawtooth form depicted in Figure D.1.

<figure><img src="image_7.png" /><figcaption>Figure D.1. Position on a ring of length L.</figcaption></figure>

Therefore, the derivative of $x_L(x)$ is given by

$$ \frac{dx_L}{dx} = \begin{cases} 1, & x \neq nL, \forall n \in \mathbb{Z} \\ -L\delta(x - nL), & \exists n \in \mathbb{Z}, x = nL \end{cases} \qquad (D.2) $$

In position space, the momentum operator is still defined as $\hat{p} = -i\hbar\frac{\partial}{\partial x}$, so that the canonical commutator becomes

$$ [\hat{x}_L, \hat{p}] = \begin{cases} i\hbar\hat{1}, & x \neq nL, \forall n \in \mathbb{Z} \\ -i\hbar L \delta(x - nL) \hat{1}, & \exists n \in \mathbb{Z}, x = nL \end{cases} \qquad (D.3) $$

In the basis of eigenstates of the free particle on a ring, $\hat{H} = \hat{p}^2/(2m)$, $\langle x|n \rangle = e^{i q_n x / \sqrt{L}} (q_n = 2\pi n / L, n \in \mathbb{Z})$, the commutator yields
$$
\begin{align}
\langle m | [\hat{x}_L, \hat{p}] | n \rangle &= i\hbar \delta_{nm} + O(\epsilon) - \frac{i\hbar L}{L} \underbrace{\int_{-\epsilon/2}^{\epsilon/2} dx \, \delta(x) e^{i(q_n - q_m)x}}_{=1} \nonumber \\
&= \begin{cases} -i\hbar, & m \neq n \\ i\hbar - i\hbar = 0, & m = n \end{cases} \tag{D.4}
\end{align}
$$

Indeed, the same result is obtained from evaluation of the matrix elements of the
position and momentum operators in the aforementioned basis (see section A):

$$
\begin{align}
\langle m | [\hat{x}_L, \hat{p}] | n \rangle &= \hbar q_n x_{mn} - \hbar q_m x_{mn} \nonumber \\
&= \begin{cases} \hbar \frac{q_n - q_m}{i(q_n - q_m)} = -i\hbar, & m \neq n \\ 0, & m = n \end{cases} \tag{D.5}
\end{align}
$$

The consequence for the time evolution of the position operator of a freely moving
particle on the ring is essential:

$$
\begin{align*}
\frac{d\hat{x}_L}{dt} &= \frac{i}{\hbar} [\hat{H}, \hat{x}_L] = -\frac{i}{2m\hbar} (\hat{x}_L \hat{p}^2 - \hat{p}^2 \hat{x}_L) \\
&= -\frac{i}{2m\hbar} ([\hat{x}_L, \hat{p}]\hat{p} + \hat{p}[\hat{x}_L, \hat{p}]) \\
&= -\frac{i}{2m\hbar} \begin{cases} 2i\hbar\hat{p}, & x \neq nL, \forall n \in \mathbb{Z} \\ -i\hbar L (\hat{p}\delta(x-nL) + \delta(x-nL)\hat{p}), & \exists n \in \mathbb{Z}, x = nL \end{cases} \\
&= \begin{cases} \frac{\hat{p}}{m}, & x \neq nL \\ \frac{L}{2m} (\hat{p}\delta(x-nL) + \delta(x-nL)\hat{p}), & x = nL \end{cases} \tag{D.6}
\end{align*}
$$

$$
\frac{d\hat{p}}{dt} = \frac{i}{\hbar} [\hat{H}, \hat{p}] = 0 \qquad (\text{D.7})
$$

Eq. (D.6) shows that the sawtooth position operator is related to a flux of particles
at the boundaries. When one tries to integrate Eq. (D.6) over time one sees that $\hat{x}_L(t)$
cannot simply be reduced to $\hat{x}_L(0) + \hat{p}/m \, t$ for all values of $x_L$, so that $\hat{d}(t)$ is not
simply $\hat{p}/m \, t$.
## Appendix E. Positional quantum correlation function of the free particle on a ring

Eq. (14) states $^{(QCF)}\delta_x^2(t) = C_x(0) - C_x(t)$, with

$$ 
\begin{aligned}
C_x(t) &= \mathrm{Tr} \left( \hat{\rho}^{(T)} [\hat{x}(t) \hat{x}(0) + \hat{x}(0) \hat{x}(t)] \right) = 2 \mathcal{R} \mathrm{Tr} \left( \hat{\rho}^{(T)} \hat{x}(t) \hat{x}(0) \right) \\
&= \frac{2}{Q} \sum_n \sum_j e^{-\beta E_n} |x_{nj}|^2 \cos \left( \frac{E_n - E_j}{\hbar} t \right)
\end{aligned}
\quad (\text{E.1})
$$

For the free particle on a ring of length $L$ this becomes (see Eqs. (30) and (31), as well as appendix D):

$$ C_x(t) = \frac{\lambda_{\text{th}} L}{2\pi^2} \sum_n \sum'_j e^{-\pi \left(\frac{\lambda_{\text{th}}}{L}\right)^2 n^2} \frac{\cos\left(\pi \left(\frac{\lambda_{\text{th}}}{L}\right)^2 \frac{t}{\tau_{\text{th}}}(n^2 - j^2)\right)}{(n-j)^2} \quad (\text{E.2}) $$

where the prime indicates that $n \neq j$ in the sums (the diagonal matrix element of the position operator in the basis of planes waves is a constant and, without loss of generality, we may take the conventional result $x_{nn} = 0$).

For $t \neq 0$, the natural bounds of the cosine function impose

$$ -C_x(0) < C_x(t) < C_x(0) \quad (\text{E.3}) $$

and hence

$$ ^{(\text{QCF)}}\delta_x^2(t) > C_x(0)(1-a(t)) \quad (\text{E.4}) $$

with some real function $a(t)$ satisfying $|a(t)| < 1$.

For $t = 0$ the (primed) sum over $j$ can be carried out analytically and yields

$$ \sum_{j \neq n} \frac{1}{(n-j)^2} = \frac{\pi^2}{3} \quad (\text{E.5}) $$

and one thus has

$$ C_x(0) = \frac{\lambda_{\text{th}} L}{2\pi^2} \sum_n e^{-\pi \left(\frac{\lambda_{\text{th}}}{L}\right)^2 n^2} \frac{\pi^2}{3} \approx \frac{L^2}{6} \quad (\text{E.6}) $$

where the approximation stems from the replacement of the sum by a Riemann sum and holds the better, the larger $L$. The result in Eq. (E.6) is twice the variance $v_x = \mathrm{Tr}(\hat{\rho}^{(T)} \hat{x}^2)$ (in the case $\mathrm{Tr}(\hat{\rho}^{(T)} \hat{x}) = 0$ considered here).

One thus sees that, for any $t \neq 0$, $^{(QCF)}\delta_x^2(t) > L^2$, i.e. this quantity diverges for the free particle on an infinitely large ring for all times but those times, at which $C_x(t) = C_x(0)$, i.e. when it is zero. But these times compose a set of measure zero.