Eﬀect of a moving mirror on the free fall of a quantum particle in a homogeneous gravitational ﬁeld J. Allam1and A. Matzkin1

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Eﬀect of a moving mirror on the free fall of a quantum particle in a

homogeneous gravitational ﬁeld

J. Allam1and A. Matzkin1

1Laboratoire de Physique Théorique et Modélisation,

CNRS Unité 8089, CY Cergy Paris Université,

95302 Cergy-Pontoise cedex, France

Abstract

We investigate the eﬀect of time-dependent boundary conditions on the dynamics of a quantum

bouncer – a particle falling in a homogeneous gravitational ﬁeld on a moving mirror. We examine

more particularly the way a moving mirror modiﬁes the properties of the entire wavefunction of

a falling particle. We ﬁnd that some eﬀects, such as the fact that a quantum particle hitting a

moving mirror may bounce signiﬁcantly higher than when the mirror is ﬁxed, are in line with

classical intuition. Other eﬀects, such as the change in relative phases or in the current density

in spatial regions arbitrarily far from the mirror are speciﬁcally quantum. We further discuss how

the eﬀects produced by a moving mirror could be observed in link with current experiments, in

particular with cold neutrons.

arXiv:2210.11306v1 [quant-ph] 20 Oct 2022

I. INTRODUCTION

The quantum bouncer – a quantum particle falling in a uniform gravitational ﬁeld

bounded by a perfectly reﬂecting mirror – is one of the paradigmatic examples of quan-

tum mechanics. Mentioned in some textbooks [1, 2], the quantum bouncer has been used

as a model to investigate wavepacket dynamics and the quantum-classical correspondence

[3, 4], to derive semiclassical propagators [5, 6], and, on the experimental side, to identify

the quantized eigenstates of cold neutrons falling on a mirror in the Earth’s gravitational

ﬁeld [7, 8].

In this work, we will be interested in the dynamics of a quantum particle obeying the

Schrödinger equation with a linear potential (due to a homogeneous gravitational ﬁeld)

falling on a moving mirror. Such a problem belongs to the class of systems subjected to

time-dependent boundary conditions. Quantum systems with time-dependent boundary

conditions are interesting from a mathematical [9, 10], foundational [11, 12] or practical

[13, 14] perspective. Analytical solutions are known only for some special systems [15]. Even

the simplest case – an inﬁnite well with a moving wall – needs to be solved numerically

[16, 17]. Concerning experiments, setups with neutrons falling on a moving mirror have

been implemented in order to develop a gravity resonance spectroscopy technique [18] and

test exotic theories of gravity [19].

Our aim here will be to investigate some eﬀects induced by moving boundary conditions

on the dynamics of a quantum bouncer. Indeed, although boundary conditions change the

Hamiltonian only in a small spatial region, the quantum-mechanical wavefunction changes

everywhere, not only in the neighborhood of that small region. This implies that measurable

eﬀects (like the current density or the diﬀerence in relative phases at a given point) due to

moving boundaries can in principle be observed everywhere as long as the wavefunction does

not vanish. Such eﬀects were recently investigated in cavities with a moving wall [17] or in

conﬁned time-dependent oscillators [12, 20]. The original motivation from a fundamental

point of view was to look for a novel type of single-particle non-locality [11]. In this paper

we will be extending these studies to a particle in free-fall bouncing on a mirror.

To this end, we will ﬁrst brieﬂy recall the basic features of the Schrödinger equation

with time-dependent boundary conditions (Sec. II) as well as the main issues that appear

when dealing with free fall and moving boundaries (Sec. 3). We investigate in Sec. 4 the

evolution of quantum properties in the presence of moving boundaries. This will be done

by comparing the dynamical evolution of a given initial state in the presence of ﬁxed and

moving boundaries. We will compare the short-time as well as the long-time dynamics for

diﬀerent types of initial states. We will particularly focus on the evolution of the current

density, the phase,and the probability density of a bouncing wavepacket. We will discuss

our results in Sec. V, in particular on the prospects for observing experimentally the eﬀects

investigated in this work. A short Conclusion is provided in Sec. VI.

II. SCHRÖDINGER EQUATION WITH TIME-DEPENDENT BOUNDARY CON-

DITIONS

Before getting to the problem of a quantum particle bouncing on a moving mirror (Sec.

III B), let us brieﬂy look at the simplest system with time-dependent boundary conditions: a

quantum particle placed in an inﬁnite well with one of the walls (say the right edge) moving

according to some function L(t). Such a system is deﬁned (see eg [21]) by the Hamiltonian

H=P2

2m+V(1)

V(x) = 





0for 0≤x≤L(t)

+∞otherwise. (2)

where mis the mass of the particle and L(t). The solutions of the Schrödinger equation must

obey the boundary conditions Ψ(0, t) = Ψ(L(t), t) = 0, so that formally a diﬀerent Hilbert

space needs to be deﬁned at each t[9]. In such problems it is important to distinguish the

instantaneous eigenstates of H,

φn(x, t) = p2/L(t) sin [nπx/L(t)] (3)

that verify

H|φn(t)i=En(t)|φn(t)i(4)

(where En(t) = n2~2π2/2mL2(t)are the instantaneous eigenvalues), from the function basis

solutions of the Schrödinger equation

i~∂tψn(x, t) = Hψn(x, t).(5)

Indeed, the eigenstates φn(x, t)do not obey the Schrödinger equation. If the evolution of

an initial wavefunction, say Ψ(x, 0),is sought in terms of linear superposition, then the

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摘要：

EectofamovingmirroronthefreefallofaquantumparticleinahomogeneousgravitationaleldJ.Allam1andA.Matzkin11LaboratoiredePhysiqueThéoriqueetModélisation,CNRSUnité8089,CYCergyParisUniversité,95302Cergy-Pontoisecedex,FranceAbstractWeinvestigatetheeectoftime-dependentboundaryconditionsonthedynamicsofaquan...

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Eﬀect of a moving mirror on the free fall of a quantum particle in a homogeneous gravitational ﬁeld J. Allam1and A. Matzkin1

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