Dynamical symmetrization of the state of identical particles

Armen E. Allahverdyan,1 Karen V. Hovhannisyan,2, 1 and David Petrosyan1, 3

1A. Alikhanyan National Science Laboratory (YerPhI), 0036 Yerevan, Armenia2The Abdus Salam International Centre for Theoretical Physics (ICTP), 34151 Trieste, Italy

3Institute of Electronic Structure and Laser, FORTH, GR-70013 Heraklion, Crete, Greece

We propose a dynamical model for state symmetrization of two identical particles produced inspacelike-separated events by independent sources. We adopt the hypothesis that the pair of non-interacting particles can initially be described by a tensor product state since they are in principledistinguishable due to their spacelike separation. As the particles approach each other, a quantumjump takes place upon particle collision, which erases their distinguishability and projects the two-particle state onto an appropriately (anti)symmetrized state. The probability density of the collisiontimes can be estimated quasi-classically using the Wigner functions of the particles’ wavepackets,or derived from fully quantum mechanical considerations using an appropriately adapted time-of-arrival operator. Moreover, the state symmetrization can be formally regarded as a consequenceof the spontaneous measurement of the collision time. We show that symmetric measurementsperformed on identical particles can in principle discriminate between the product and symmetrizedstates. Our model and its conclusions can be tested experimentally.

I. INTRODUCTION

The symmetrization postulate of quantum mechanics states that any joint state of identical particles (i.e., havingall their non-dynamic features, such as mass, charge, spin, etc., the same) should be either symmetric (for bosons)or antisymmetric (for fermions) under permutations of the particles [1–5]. Common sense though suggests that wecould, in principle, attach labels and distinguish identical, non-interacting particles emanating from different, largelyseparated sources, at least until their coordinate probability densities start to overlap. Indeed, assume that two distantsources SL and SR produce nearly-simultaneously two identical particles 1 and 2 in states |L〉 and |R〉, respectively,in spacelike-separated events. Then, if a particle is detected close to a source, say SL, shortly after the creation, wecan conclude with high degree of confidence that this is the same particle 1 that was emitted by SL. The hypothesisthat independently generated, spatially separated identical particles can be considered distinguishable, with the jointwavefunction represented by a tensor product of the wavefunctions of each particle, has also been discussed in thepast [6–8]. It was then argued that transition to the symmetrized state might occur once the spatial wavefunctionsof the particles start to overlap, reducing their distinguishability [7, 9]. It has recently been suggested that such atransition during bound state formation may be related to the excess energy transferred to an environment [10].

Let us recall two common arguments against the hypothesis that the initial state of largely separated particlescan be represented by a tensor product of single-particle states. The first argument is field-theoretical [9]: particlescorrespond to excitations of a quantum field, and particle creation and annihilation correspond to a transition betweenthe Fock states of the field effected by appropriate bosonic or fermionic field operators. This reasoning is, however,circular: the symmetrization postulate is not derived from second quantization; rather, the symmetrization postulateis used to derive the properties (commutation relations) of the second-quantized bosonic or fermionic field operators[5, 11]1.

The second argument relies on the vast experimental evidence that identical particles within a small distance fromeach other, or those that have interacted with each other in the past, are in appropriately symmetrized states. Yet, forspatially separated, non-interacting particles, the results of localized in space measurements (i.e., coordinate, but notmomentum) are the same for both product and symmetrized states [4]. One may then argue that product states, evenif they were possible, are redundant in the theory. These arguments, however, ignore the possibility of a transitionbetween product and symmetrized states of non-interacting particles prepared far apart and approaching each other.Then, the difference between product and symmetrized states can be detected via symmetric measurements on theparticles, even if their states do not (yet) overlap in space, as discussed below.

Before proceeding, we note that measurements on identical particles are insensitive to their permutations, sincethe interaction of a measuring apparatus with identical particles is, by definition, symmetric under particle permuta-tions [4, 8, 9]. Hence, all allowed measurements on identical particles should be described by permutation-invariantoperators [4, 9]. Such measurements produce permutation-invariant results for any state, symmetrized or not. In

1 In this context, recall the message of the spin-statistics theorem from relativistic field theory [11, 12]: integer-spin (half-integer-spin)fields cannot be quantized via anticommutators (commutators), but can be quantized via commutators (anticommutators). Hence,integer-spin (half-integer-spin) fields can be represented as bosons (fermions). This statement however does not exclude that, e.g.,half-integer spins can be quantized via some other algebraic (paraparticle) structure [11, 12].

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FIG. 1. Schematic representation of our 1D model. Two identical particles, 1 and 2, emanating at time t = 0 from different,spatially separated sources, have the coordinate wavefunctions ψL(x1, t) = 〈x1|L(t)〉 and ψR(x2, t) = 〈x2|R(t)〉 (blue) movingtowards each other along the x axis. The two-particle wavefunction Ψp(x1, x2, t) = ψL(x1, t)ψR(x2, t) factorizes. As their prob-ability densities start to overlap (gray), the particles collide (red circle) at some time tc, which results in the (anti)symmetrizedtwo-particle wavefunction Ψs(x1, x2, t > tc) ∝ ψL(x1, t)ψR(x2, t) + ηψR(x1, t)ψL(x2, t) (η = ±1).

fact, some textbooks do not account for this aspect and “derive” the state symmetrization from the symmetry of the(position) measurement statistics [5, 13]. The constraint of permutation-symmetric observables for identical particleshas important consequences for the definition of their reduced (marginal) states, as we discuss in Appendix A.

In this paper, we adopt the initial product state hypothesis and show that permutation-symmetric measurements canin principle differentiate between the product and symmetrized states of the particles. We then propose a simple andintuitive one-dimensional (1D) dynamical model for symmetrization of the wavefunction of a pair of initially separatedidentical particles upon their collision, illustrated in Fig. 1. Inspired by the quantum jump approach to continuouslymeasured quantum systems [14–18], in our model, we apply the symmetrization operator to the two-particle state,which projects the initial product state onto a fully (anti)symmetric superposition. The physical intuition behindthe application of the symmetrization operation is that, once the particles collide, they lose their individuality, whicherases the possibility to distinguish their labels. Note that this happens operationally already for classical identicalparticles. We model the interaction between the particles similarly to how it is done for ideal gases: the particles arefree except when they are in an immediate neighborhood of each other.

Our symmetrization quantum jump occurs when the particles collide with each other, and we determine the proba-bility density of the collision times from semiclassical and fully quantum-mechanical considerations. For semiclassicalcalculations of the collision time distribution, we assume Gaussian wavepackets for single particles and use the result-ing non-negative Wigner functions [4, 19]. The semiclassical probability distribution of the collision times turns outto be nearly identical to that obtained with the fully quantum treatment of the collision time, which we determine byintroducing an appropriate “time of arrival” operator [20–22]. This operator is, however, not an ordinary self-adjointoperator corresponding to a standard projective measurement, but it can be described by a positive operator-valuedmeasure (POVM) with the post-measurement state chosen arbitrarily. This permits us to formally regard the sym-metrization quantum jump as a transition induced by a spontaneous measurement of the collision time.

The paper is organized as follows. In Sec. II we review the symmetric measurements performed on a system ofidentical particles and show that such measurements can in fact discern the difference between the symmetrized andproduct states. In Sec. III we present our model for state symmetrization upon collision of the particles. In Sec. IV wederive the probability density of collision times from the semiclassical and fully quantum perspectives. We summarizeour model and the results in Sec. V. Some of the mathematical details are deferred to Appendices A, B, C, D, E, F.

II. MEASUREMENT PROBABILITIES FOR IDENTICAL PARTICLES

Here we discuss which measurements are allowed for identical particles and how to employ them to discern thedifference between product and symmetrized states. Clarifying these issues is important for understanding the physicsof identical particles and avoiding some common misconceptions. As mentioned above, any Hamiltonian for a systemof identical particles, including their interactions with measuring apparata, is symmetric under particle permutations.Hence, all allowed measurements on identical particles are described by permutation-symmetric operators [4, 8, 9], andthis argument is independent on the state of the system (recall that in quantum mechanics measurement operatorsare defined independently from states). We emphasize that the hypothesized asymmetric product state |L〉 ⊗ |R〉of identical particles is prepared as a result of particle generation, which is a process outside the formalism of non-relativistic quantum mechanics. Moreover, non-symmetric states cannot be prepared via projective measurements,since all such measurements are described by symmetric projectors.

Consider a single-particle operator O decomposed into a complete set of N projectors Pk as O =∑Nk=1 αkPk, with

PkPl = δklPk,

N∑k=1

Pk = I. (1)

3

For a pair of particles, the permutation-symmetric measurement of operator O ⊗ O =∑k≤l αkαlPkl then involves

N(N + 1)/2 projectors

Pkl ≡ {Pk ⊗ Pl + Pl ⊗ Pk}Nl<k, Pkk ≡ {Pk ⊗ Pk}Nk=1,

N∑k≤l

Pkl = I⊗ I.(2)

It follows that, for any density operator σ of the system, the measurement probabilities are permutation-symmetric:

tr(σPkl) = tr(πσπ Pkl), (3)

where π is the particle permutation operator.

A. Coordinate measurements

Given a state |Ψ〉 of two identical particles living in a 1D space, the joint probability density for the two coordinatesreads

DΨ(x1, x2) = 〈Ψ|Px1x2|Ψ〉, (4)

with

Px1x2≡ |x1〉〈x1| ⊗ |x2〉〈x2|+ |x2〉〈x2| ⊗ |x1〉〈x1|, Pxx = |x〉〈x| ⊗ |x〉〈x| (5)

being a continuous analogue of the projectors in Eq. (2). This representation stems from the decomposition x⊗ x =∫∫x1≤x2

dx1dx2 x1x2Px1x2 and the corresponding normalization condition∫∫

x1≤x2

dx1dx2 Px1x2 = I⊗ I.

Consider the two-particle product state

|Ψp〉 = |L〉 ⊗ |R〉 (6)

with the first (second) particle produced by SL (SR), and the symmetric (η = 1) or antisymmetric (η = −1) state

|Ψs〉 =1√2

(|L〉 ⊗ |R〉+ η|R〉 ⊗ |L〉). (7)

Using Eq. (4), we obtain

DΨp(x1, x2) = |ψL(x1)|2 |ψR(x2)|2 + |ψL(x2)|2 |ψR(x1)|2, (8)

DΨs(x1, x2) = 〈Ψs|Px1x2 |Ψs〉 = 〈Ψp|Px1x2 + ηPx1x2 π|Ψp〉= DΨp(x1, x2) + η[ψ∗L(x1)ψL(x2)ψR(x1)ψ∗R(x2) + c.c.], (9)

where the coordinate wavefunctions are defined as

ψL,R(x) ≡ 〈x|L,R〉. (10)

Thus, both DΨp(x1, x2) and DΨs

(x1, x2) are invariant under permuting x1 and x2, even though |Ψp〉 is a non-symmetric product state. When the wavefunctions do not overlap in space, ψ∗L(x)ψR(x) ' 0 ∀x, the second termon the right-hand side of Eq. (9) vanishes, and therefore DΨp

(x1, x2) ' DΨs(x1, x2), i.e., the non-symmetric product

state |Ψp〉 of Eq. (6) leads to the same joint coordinate density as the (anti)symmetric state |Ψs〉 of Eq. (7). Moreover,for non-overlapping wavefunctions, also one of the terms on the r.h.s. of Eq. (8) vanishes. For example, if x1 andx2 are chosen such that |ψL(x1)| 6= 0 and |ψR(x2)| 6= 0, then the second term in Eq. (8) vanishes, showing thatthere is no difference between using the symmetrized measurement operator in Eq. (5) or the non-symmetric operator

P(ns)x1x2 = |x1〉〈x1| ⊗ |x2〉〈x2|.When, however, states |L〉 and |R〉 do overlap in space, i.e., 〈L|x〉〈x|R〉 6' 0 [in contrast to a possible 〈L|R〉 =∫dxψ∗L(x)ψR(x) ' 0], then all the three expressions are in general different, and DΨ(x1, x2) = |ψL(x1)|2 |ψR(x2)|2

obtained with the non-symmetric operator P(ns)x1x2 is ruled out for identical particles due to its manifest asymmetry.

4

B. Simplest symmetric measurements

Consider the simplest case of N = 2 in Eq. (2) with the single-particle projectors P and P ′ = I− P . The symmetricmeasurement on two identical particles of Eq. (2) now has three projectors,

P ⊗ P , P ′ ⊗ P ′, P ⊗ P ′ + P ′ ⊗ P , (11)

corresponding to the detection (i.e., measurement outcome 1) of both, none, or one of the particles, respectively. Themeasurement probabilities for states (6) and (7) are given by

〈Ψp|P ⊗ P |Ψp〉 = 〈L|P |L〉〈R|P |R〉, (12)

〈Ψs|P ⊗ P |Ψs〉 = 〈LR|P ⊗ P |LR〉+ η〈LR|P ⊗ P |RL〉= 〈L|P |L〉〈R|P |R〉+ η|〈L|P |R〉|2, (13)

and the other probabilities 〈P ′ ⊗ P ′〉, 〈P ⊗ P ′〉 and 〈P ′ ⊗ P 〉 are easily obtained using the above equations.

Note that the interference term ∝ η in Eq. (13), which allows P ⊗ P to distinguish |Ψs〉 from |Ψp〉, will also appearwhen considering mixed states that contain both symmetrized and product states. As an example, consider a mixedstate of the system given by the density operator

σ = (1− p)|Ψp〉〈Ψp|+ p|Ψs〉〈Ψs|, (14)

where p denotes the probability that the state is symmetrized. Using the above relations, we obtain

tr[σP ⊗ P ] = 〈L|P |L〉 〈R|P |R〉+ pη|〈L|P |R〉|2. (15)

This example will also be useful in Sec. III.We may choose P to correspond to a particle detector of width 2` placed at some position x,

P =

∫ x+`

x−`dx |x〉〈x|. (16)

Then the measurements described by Eqs. (12) and (13) can reveal the difference between |Ψp〉 and |Ψs〉 even for〈L|R〉 = 0, provided the two particle wavefunctions overlap, |〈L|x〉〈x|R〉| 6= 0, for some coordinates x. This issue willbe further discussed in Sec. 4IV E.

C. Measurements via coordinate superposition states

There is a class of measurements that can distinguish between the states |Ψp〉 and |Ψs〉 even for vanishing overlap〈L|x〉〈x|R〉 = 0∀x. Consider the projector

PC = |C〉〈C|, 〈C|C〉 = 1, (17)

where |C〉 involves a coherent superposition of coordinates, in contrast to P of Eq. (16) which is an incoherent mixtureof coordinate eigenstates.

Let us assume that the single-particle wavepackets ψL,R(x) = 〈x|L,R〉 separated by distance r are centered around

positions ∓r/2 with the localization length rL,R � r. We take 〈x|C〉 ∝ r−1/2 to be nearly constant for x ∈ [− r2 ,r2 ]

and quickly falling to zero for |x| > r2 , such that 〈C|C〉 = 1 is normalized. Substituting PC into Eq. (13) we obtain

〈Ψs|PC ⊗ PC |Ψs〉 = (1 + η) |〈R|C〉〈C|L〉|2 = (1 + η) O(rLrR

r2

), (18)

where O corresponds to the standard asymptotic big-O notation, while

〈C|L,R〉 ' 1√r

∫ r/2

−r/2dxψL,R(x). (19)

The measurement probability in Eq. (18) is small due to the assumed r � rL, rR, but the correlation factor may still

be detectable. Thus, in the case of fermions (η = −1), we have for the antisymmetric state 〈Ψs|PC ⊗ PC |Ψs〉 = 0

exactly, whereas for the product state 〈Ψp|PC ⊗ PC |Ψp〉 = O(rLrR/r2).

5

We can further increase the measurement probability of Eq. (18) if we use for |C〉 a superposition of two states,

|C〉 =1√2

(|A1〉+ |A2〉), (20)

with the condition 〈Ai|Ak〉 = δik (for i, k = 1, 2) to guarantee that 〈C|C〉 = 1 is normalized and P 2C = PC is a

projector. For simplicity, we also assume that 〈A1|R〉 = 0 and 〈A2|L〉 = 0. With such a PC = |C〉〈C|, from Eqs. (12)and (13) we obtain

〈Ψp|PC ⊗ PC |Ψp〉 =1

4|〈A1|L〉|2|〈A2|R〉|2, (21)

〈Ψs|PC ⊗ PC |Ψs〉 =1

4(1 + η)|〈A1|L〉|2|〈A2|R〉|2. (22)

Now, if |〈A1|L〉〈A2|R〉| ∼ 1, the measurement probabilities of Eqs. (21) and (22) will be sizable. Hence, the interferenceterm ∝ η in Eq. (22) will contribute to the measurement probabilities, which can therefore reveal the differencebetween |Ψp〉 and |Ψs〉. We emphasize again that these are valid measurements even for vanishing spatial overlap〈L|x〉〈x|R〉 ' 0 ∀x, but such measurements will be impossible to realize when the particles are very far apart, sincein the scenario of Eq. (19) 〈C|L,R〉 → 0, and in the scenario of Eq. (20) it will be difficult to realize a projector ontoa superposition of well-separated states [23]. Other complications may arise from considering characteristic times ofsuch measurements that are not instantaneous, which is, however, beyond the scope of this work.

III. SYMMETRIZATION OF A TWO-PARTICLE STATE

We now turn to the dynamics of the system. We assume that at time t = 0 two identical particles, 1 and2, are produced by two different spatially well-separated sources SL and SR in states |L〉 and |R〉, respectively.Due to a large interparticle distance, their wavefunctions ψL,R(x, t) = 〈x|L(t), R(t)〉 have vanishing initial overlapψ∗L(x, 0)ψR(x, 0) = 0 ∀x and hence 〈L|R〉 = 0. Although the particles are identical, they are distinguishable at earlytimes by the very fact of being produced far apart from each other: the particle labels 1 and 2 are meaningful andremain so at least for some time. We thus adopt the initial product state hypothesis and write the two particle stateas a product state |Ψ(t)〉 = |Ψp(t)〉 = |L(t)〉⊗ |R(t)〉 as per Eq. (6). We furthermore assume that the particles do notinteract unless they are at the same location. Until then, the state of each particle evolves independently and theirindividual dynamics are governed by the same single-particle Hamiltonian H, leading to

|L(t)〉 = e−iHt/~|L〉, |R(t)〉 = e−iHt/~|R〉. (23)

Hence, the state overlap is conserved in time, 〈L(t)|R(t)〉 = 〈L|R〉 = 0. Yet, as the particles approach each other,their probability densities start to overlap: |ψL(x, t)|2|ψR(x, t)|2 6= 0 for some interval of values of x. Eventually, theparticles collide at some time tc, which erases their distinguishability. This amounts to a transition to the symmetrizedstate |Ψ(t)〉 = |Ψs(t)〉 = 1√

2(|L(t)〉 ⊗ |R(t)〉+ η|R(t)〉 ⊗ |L(t)〉) of Eq. (7), with η = ±1 for bosons/fermions.

The collision time tc is a random variable, with the probability density ρ(tc) determined in Sec. IV. We describethe transition from product state |Ψ(t < tc)〉 = |Ψp(t)〉 to symmetrized state |Ψ(t > tc)〉 = |Ψs(t)〉 by a quantum

jump at time t = tc realized by the projector Π = 12 (1 + ηπ), with π being the permutation operator:

|Ψp〉 →Π|Ψp〉√〈Ψp|Π|Ψp〉

= |Ψs〉. (24)

Note that the permutation operator π commutes with any Hamiltonian for identical particles. Hence, once the stateis symmetrized upon particle collision, it will no longer change by the subsequent application of the projector Π.

Even though we deal with a closed, non-dissipative system, the spontaneous symmetrization of Eq. (24) is formallyidentical to a quantum jump in a continuously monitored dissipative system [14–17]. In a usual quantum measurement,the experimenter is free to choose the time at which the measurement is performed. Such a measurement is realizedby coupling the system to a measuring apparatus which disrupts the coherent evolution of the closed system for acertain period of time, controlled by the experimenter, during which the measurement takes place. When measuringtime, however, the time of registering the measurement outcome is random and coincides with the measurementoutcome itself. The measurement is realized by the collision itself, and takes place without the presence of an externalapparatus, i.e., the state of the system becomes symmetrized whether or not we query the transition time.

6

Thus, in a single realization, or quantum trajectory, the state of a pair of particles evolves according to the randombut pure state

|Ψ(t)〉 = θ(tc − t)|Ψp(t)〉+ θ(t− tc)|Ψs(t)〉, (25)

where θ(t) is Heaviside step function. For an ensemble of particles prepared in state |Ψp(0)〉 at time t = 0, theprobability of transition (24) to happen during the time interval [0, t] is

pc(t) =

∫ t

0

dtc ρ(tc). (26)

Hence, the ensemble-averaged density operator for the system is given by

σ(t) =(1− pc(t)

)|Ψp(t)〉〈Ψp(t)|+ pc(t)|Ψs(t)〉〈Ψs(t)|. (27)

Now, whenever pc(t) → 1, the symmetrization is complete, σ = |Ψs〉〈Ψs|; if pc(t) = 0 for all times, as for initiallyseparated particles propagating away from each other, the state remains a tensor product, σ = |Ψp〉〈Ψp|; while ingeneral, 0 < pc < 1, we have a mixed state of Eq. (27).

Single quantum trajectories and the ensemble-averaged evolution of the system can also be simulated numericallyusing the stochastic wavefunction approach [14–17]. Namely, to simulate a quantum trajectory (a single realizationof experiment) starting from the product state |Ψ(t = 0)〉 = |Ψp〉, we draw from a uniform distribution a randomnumber γ ∈ [0, 1] and compare it with the collision probability of Eq. (26). The transition (24) from the productstate |Ψp〉 to the symmetrized state |Ψs〉 occurs at time t′c when pc(t

′c) = γ. Subsequently, the two-particle state

continues to evolve as a symmetrized state, |Ψ(t > t′c)〉 = |Ψs〉. The density operator of the system, σ(t), is obtainedby averaging over M � 1 independently simulated trajectories |Ψ(m)(t)〉:

σ(t) = limM→∞

1

M

M∑m=1

|Ψ(m)(t)〉〈Ψ(m)(t)|. (28)

IV. COLLISION TIME OF TWO PARTICLES

Our next task is to determine the probability density of the collision times ρ(tc), given the initial states of theparticles each evolving under the free Hamiltonian. In subsection IV B we derive ρ(tc) using a quasi-classical approach,while in subsection IV C we present the fully quantum treatment, followed by the numerical results in subsectionsIV D and IV E.

A. Single-particle dynamics

The Schrodinger equation for a single-particle wavefunction evolving under the free propagation Hamiltonian H =1

2m p2 is easily solved in the momentum representation, i∂tψ(p, t) = p2

2mψ(p, t) (~ = 1), leading to

ψ(p, t) = e−itp2/(2m)χ(p), (29)

where χ(p) is a normalized function,∫

dp|χ(p)|2 = 1. In subsection IV B we discuss in the quasi-classical regimewith non-negative Wigner functions corresponding to the single-particle wavefunctions [19]. We therefore considerGaussian wavefunctions ψL,R(p, t) = 〈p|L(t), R(t)〉 and take |χ(p)|2 to be a normalized Gaussian function

χ(p) =

√a

π14

exp

[−b

2

2+ p(ba− ic)− a2p2

2

], (30)

with constants a > 0 and b defined via the mean and variance of the momentum through

〈p〉 ≡∫

dp p|χ(p)|2 =b

a, 〈p2〉 − 〈p〉2 =

1

2a2, (31)

7

while c is the initial center of mass coordinate of the wavepacket. Recalling the δ-function normalized momentumeigenfunctions, 〈x|p〉 = 1√

2πeipx, we obtain from Eq. (30) the wavefunction in the coordinate representation:

ψ(x, t) =1√2π

∫dp eipxψ(p, t) =

1

π14

√a(1 + iτ)

exp

[−b

2

2− (ξ − ib)2

2(1 + iτ)

]=

1

π14

√a(1 + iτ)

exp

[− (ξ − bτ)2

2(1 + τ2)+i

2

2bξ + τ(ξ2 − b2)

1 + τ2

],

(32)

where ξ ≡ (x− c)/a and τ ≡ t/(ma2) are the dimensionless coordinate and time, respectively.As the particles approach each other, eventually their spatial probability densities start to overlap. Yet, their

wavefunction overlap is conserved in time, 〈L(t)|R(t)〉 = 〈L|R〉, and remains small if initially 〈L|R〉 � 1. Indeed,using Eq. (30), we can verify that the overlap is given by

〈L|R〉 =

∫dpχ∗L(p)χR(p)

=

√2aLaRa2L + a2

R

exp

[− (aLbR − aRbL)2 + (cR − cL)2

2(a2L + a2

R)+ i

(cL − cR)(aRbR + aLbL)

a2L + a2

R

],

(33)

where aL,R, bL,R, and cL,R are the parameters of the Gaussian wavepackets in Eq. (30). We thus see that |〈L|R〉| � 1

either for initially largely separated particles, (cR−cL)2

2(a2L+a2R)� 1, or for particles moving in the opposite directions,

(aLbR−aRbL)2

2(a2L+a2R)� 1.

B. Quasi-classical collision time

Consider two classical point-like particles with trajectories x1,2(t) = x1,2(0) +p1,2(0)m t. The particles collide at time

tc = mx1(0)− x2(0)

p2(0)− p1(0), (34)

which makes them practically indistinguishable thereafter. Given now the initial factorized distributions DL(x, p) andDR(x, p) of coordinates and momenta of the particles 1 and 2, the probability density of collision times tc is given by

ρcl(tc) =

∫∫∫∫dx1 dp1 dx2 dp2 δ

(tc −m

x1 − x2

p2 − p1

)DL(x1, p1)DR(x2, p2)

=

∫∫∫dp1 dx2 dp2

|p1 − p2|m

DL

(x2 −

p1 − p2

mtc, p1

)DR(x2, p2), (35)

and is normalized as∫

dtc ρcl(tc) = 1 provided that both DL,R(x, p) are normalized.To describe the collision of quantum particles, we can use in the above equation instead of DL,R(x, p) an ap-

propriate quasi-probability distribution. A well-behaved (non-negative) quasi-probability distribution for Gaussianwavefunctions of the particles is the Wigner function [4, 19]

W (x, p, t) =1

π

∫dq ψ∗(p+ q, t)ψ(p− q, t) e−2ixq.

Using the wavefunctions ψ(p, t) of Eq. (29) at time t = 0, we then obtain the probability density of collision times tc,

ρcl(tc) =1

π

∫∫∫dq dp1dp2

|p1 − p2|m

χ∗L(p1 + q)χL(p1 − q)χ∗R(p2 − q)χR(p2 + q) e2iqtcp1−p2m

=1

πm

aRaLa2R + a2

L

∫du |u| exp

[−a2Ra

2L

(u+ bR

aR− bL

aL

)2a2R + a2

L

]exp

[−(tcum + cL − cR

)2a2R + a2

L

].

(36)

This integral over the relative momentum u ≡ p2−p1 can be expressed through the error function. Note that the firstGaussian in the second line of Eq. (36) is centered at momentum difference u ' bL

aL− bR

aR, while the second Gaussian

is centered around the initial interparticle distance utc/m ' (cR − cL).

8

C. Quantum collision time

We now present a fully quantum derivation of the collision time, following the ideas of Refs. [20–22] on the quantumarrival time. As a quantum generalization of the classical collision time in Eq. (34), we define a collision time operator

tc = −m2

{x⊗ I− I⊗ x, 1

p⊗ I− I⊗ p

}, (37)

where { , } is the anticommutator. Note that tc is symmetric with respect to the particle permutation: π tcπ = tc.In the momentum representation, denoting p⊗ I = p1, I⊗ p = p2, we have x⊗ I = i∂p1 , I⊗ x = i∂p2 . Introducing

the relative and center of mass momenta, u = p2 − p1 and v = 12 (p1 + p2), we have for any state |φ〉

〈u, v|tc|φ〉 = −im{

1

u, ∂u

}φ(u, v). (38)

This expression does not depend on v due to translational invariance. Denoting the tensor product of u space and vspace by ◦ (reserving ⊗ for the tensor product of the original single-particle Hilbert spaces), we can recast Eq. (38) as

tc = t ◦ I(v), (39)

where I(v) is the identity operator in the v space, and

t = −im{

1

u, ∂u

}= −im

(2

u∂u −

1

u2

)(40)

is precisely the quantum time-of-arrival operator [20–22, 24] that lives in the u space.Due to singularity of t at u = 0, for each eigenvalue tc, there are two linearly independent eigenvectors |tc,+〉 and

|tc,−〉 which we choose as [25]

〈u|tc,±〉 =θ(±u)√

4πm

√|u| eitcu

2/(4m); (41)

see Appendix B for a detailed discussion. The functions 〈u|tc,+〉 and 〈u|tc,−〉 correspond, respectively, to relativemomenta u < 0 and u > 0. Note that, although we start counting from t = 0, tc ranges from −∞ to∞ to account forboth future collisions of the particles moving towards each other, and (possible) past collisions of the particles thatare moving away from each other. The eigenprojector of t corresponding to the eigenvalue tc is then

Y (tc) = |tc,+〉〈tc,+| + |tc,−〉〈tc,−|. (42)

The eigenprojectors are complete in the u space:∫

dtcY (tc) = I(u). The eigenprojector of tc corresponding to the

eigenvalue tc can be written as

P (tc) = Y (tc) ◦ I(v), (43)

and we immediately see that ∫dtcP (tc) = I

(u) ◦ I(v) = I⊗ I, (44)∫dtc〈p′1, p′2|P (tc)|p1, p2〉 = δ(p1 − p′1) δ(p2 − p′2). (45)

Importantly, even though tc is formally Hermitian and its eigenprojectors satisfy the completeness relation (44), itis not a self-adjoint operator [26–28] 2 and the eigenvectors of t (and therefore of tc) are not orthogonal:

〈t1,±|t2,±〉 =1

2π

∫ ∞0

du eiu(t2−t1+i0) =1

2πP[

1

t1 − t2

]+i

2δ(t1 − t2), (46)

2 Although this operator is not self-adjoint, it can be made Hermitian (also referred to as “symmetric” in the mathematical literature) ina suitable Hilbert space; see Ref. [28] for an accessible review on the difference between Hermitian and self-adjoint operators.

9

where we used Eq. (41) and the Sokhotski-Plemelj theorem, and P denotes the principal value map. That tc is nota self-adjoint operator is a manifestation of the fact that there can be no time observable in quantum mechanics[24, 25, 27, 29] (see also Appendix C).

Even though the eigenresolution of tc does not define a projective measurement due to Eq. (46), the fact thatoperators |tc,±〉〈tc,±| ◦ I(v) are Hermitian, positive-semidefinite, and sum up to the identity, as per Eqs. (42)–(44),means that they constitute a POVM [4]. Hence, we can adopt the view that the measurement of t, and thus tc,realizes the POVM defined by its eigenresolution [22, 24, 30]. An important difference between the usual projectivemeasurements and general POVMs is that, for the latter, there is no unique prescription for identifying the post-measurement states, i.e., there is unitary freedom. This is also true for the measurement of tc. As explained inAppendix D, this freedom can be used to choose the post-measurement state to be |Ψs(tc)〉, so that the symmetrizationjump in Eq. (24) can be thought of as the post-measurement state after the collision time tc is measured.

Using the two-particle product state in Eq. (6) in the momentum representation, 〈p1, p2|Ψ(t)〉 = ψL(p1, t)ψR(p2, t),the probability density of the collision times is given by

ρt(tc) = 〈Ψ(t)|P (tc)|Ψ(t)〉

=

∫∫∫∫dp1dp2 dp′1dp′2 〈Ψ(t)|p′1, p′2〉〈p′1, p′2|P (tc)|p1, p2〉〈p1, p2|Ψ(t)〉

=∑s=±

∫∫∫dv dudu′ ψ∗L

(v − u′

2, t)ψ∗R

(v +

u′

2, t)ψL

(v − u

2, t)ψR

(v +

u

2, t)〈u′|Y (tc)|u〉.

(47)

Consistently with the discussion above [see Eq. (C1)], the collision time distribution depends on the initial time t asρt(tc) = ρt=0(tc + t): if the system is prepared in state |Ψ(t)〉 at time, say, t > 0 instead of t = 0, the collision willhappen earlier.

D. Gaussian wavepackets

We now assume that two Gaussian wavepackets described by Eq. (29), initially (at t = 0) separated by a distanced ≡ cR − cL > 0, are moving towards each other. Omitting the subscript of ρ in Eq. (47) (ρt=0 → ρ), we have

ρ(tc) =1

2π3/2m

aLaR√a2L + a2

R

exp

[− (aLbR − aRbL)2

a2L + a2

R

]×∫∫ ∞

0

dudu′√uu′ exp

[ itc4m

(u′2 − u2)−A2(u′2 + u2) +A1uu′]

cosh[ i

2(cR − cL)(u′ − u) +A0(u+ u′)

],

(48)

where

A0 =aLaR(aLbR − aRbL)

a2L + a2

R

, A1 =(a2R − a2

L)2

8(a2L + a2

R), A2 =

a4R + 6a2

Ra2L + a4

L

16(a2L + a2

R).

For a = aL = aR, Eq. (48) simplifies to

ρ(tc) =a e−

12 (bR−bL)2

4πm√

2π

(∣∣∣∣ ∫ ∞0

du√u exp

[u2

4

( itcm− a2

)+iu

2d+

au

2(bR − bL)

]∣∣∣∣2+

∣∣∣∣ ∫ ∞0

du√u exp

[u2

4

( itcm− a2

)− iu

2d− au

2(bR − bL)

]∣∣∣∣2).

(49)

In Fig. 2(a) we show the probability densities ρcl(tc) and ρ(tc) of the collision time obtained from the semiclassicaland fully quantum treatments, which are nearly identical for initially separated Gaussian wavepackets with vanishingoverlap |〈L|R〉| ' 0. There, the blue dashed and solid lines correspond to the particles with the same averagemomentum difference bL

aL− bR

aR, but for the former case the momentum uncertainty 1√

2aL,Ris larger, and therefore

the dashed curve is lower and more smeared. For the red curve, the average momentum is again the same, but themomentum uncertainty is even larger than the mean. Hence ρ(tc) ' ρcl(tc) has nonvanishing values even for negativetimes tc < 0, while the integrated collision probability pc '

∫∞0

dtc ρ(tc) = 0.7469 is smaller than one. This meansthat there is a finite probability 1 − pc & 0.25, given by Eq. (26), that the two particles move away from each otherand never collide, i.e., the state σ(t) of Eq. (27) remains mixed even for large times.

10

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

tc

ΡΡ

cl

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.02

0.04

0.06

0.08

t(b)

FIG. 2. (a) Probability densities ρ(tc) and ρcl(tc) of the collision time tc, as obtained from Eqs. (36) and (49) for two Gaussianwavepackets with large initial separation cR−cL = 5, and aR,L = 1, bL = −bR = 3.33 (blue solid line); aR,L = 0.3, bL = −bR = 1(blue dashed line); and aR,L = 0.1, bL = −bR = 0.333 (red solid line). For all curves the difference between the quantum ρ(tc)and semiclassical ρcl(τ) probability densities is indiscernible. (b) The magnitudes of the first (black line) and second (blue line)terms on the r.h.s. of Eq. (50), for a pair of Gaussian wavepackets with cR − cL = 2.5, aR,L = 0.5, and bL = −bR = 1.5, asmeasured by a coordinate detector of Eq. (16) placed in the two particle collision region, x = (cL + cR)/2 and ` = 0.25. Thecollision probability pc(t) of Eq. (26) with ρ(tc) of Eq. (49) is well localized in the interval of times t ∈ [0.1, 2]. We have setm = 1 and ~ = 1 in both figures.

E. Coordinate detection

We finally illustrate the dynamics of the measurement probability (15), namely

tr[σ(t)P ⊗ P ] = 〈L(t)|P |L(t)〉 〈R(t)|P |R(t)〉+ ηpc(t)|〈R(t)|P |L(t)〉|2, (50)

assuming that |L(t)〉 and |R(t)〉 are identical, initially well-separated Gaussian wavepackets propagating toward each

other, while the coordinate detector P of Eq. (16) is placed in the collision region. Mathematically this is expressed

by the fact that ρ(tc) is peaked around times t when both 〈L(t)|P |L(t)〉〈R(t)|P |R(t)〉 and pc(t)|〈R(t)|P |L(t)〉|2 attaintheir maxima (see Appendix E for analytical expressions for these quantities). In Fig. 2(b) we show the dynamics ofthe two terms on the r.h.s. of Eq. (50). As expected, the first term is several times larger than the second, correlationterm, which, however, is still well pronounced. Note that a proper choice of the detector width ` in Eq. (16) isimportant for this conclusion, since for `→ 0 the detector reduces to a point and produces no signal, while for `→∞we also have |〈R(t)|P |L(t)〉|2 = |〈R(t)|L(t)〉|2 = |〈R(0)|L(0)〉|2 = 0.

V. CONCLUSIONS

In summary, we have proposed a self-consistent model for the dynamics of identical particles produced independentlyby spatially separated sources and therefore described initially by a product state. The symmetrization of the two-particle state occurs dynamically as the particles collide with each other which erases their distinguishability. Oursimple two-particle 1D model provides an intuitively plausible symmetrization picture. We show that, for Gaussianwavepackets, the collision probabilities can be accurately described via a quasi-classical approach employing Wignerfunctions of the particles. Quantum mechanically, the transition between the product and symmetric states canbe formally regarded as a consequence of spontaneous measurement of the (non-self-adjoint) collision time operator.Semiclassical extension of the model to 3D collisions of short-range interacting particles is straightforward, as discussedin Appendix F, but a fully quantum generalization of the collision time operator to 3D space is a known problem andwill require further research.

Our model is consistent with the bulk of experimental observations of the (anti)symmetrization effects, such as,e.g., the Pauli principle or Bose-Einstein condensation, which consider identical particles that had sufficient time tointeract and collide with each other. For such particles, we can safely assume that the symmetrization transition hadalready taken place before the system became subject to interrogation. The main purpose of our model is to provide

11

a clear physical picture of the transient regime between the particles being very far (when their state is still a tensorproduct) and sufficiently close (when their state is already symmetrized), by employing several physically plausibleassumptions.

It is worth emphasizing that our results are not interpretational—they can be tested experimentally. Once such testsare preformed, the hypothesis that independently generated, spatially separated identical particles can be considereddistinguishable can be supported or laid to rest.

ACKNOWLEDGMENTS

We thank Janet Anders for interesting discussions on identical particles, and Roger Balian and Theo Neuwenhuizenfor useful remarks. A.E.A. thanks Theo Neuwenhuizen for his role in motivating this research. This work wassupported by the SCS of Armenia, grants No. 18T-1C090 and No. 20TTAT-QTa003.

Appendix A: Reduced states of identical particles

Reduced (sometimes also called “marginal”) states of multipartite quantum systems are defined through localobservables. Consider a compound system in a Hilbert spaceH = H1⊗H2 consisting of two distinguishable subsystems1 and 2 living, respectively, in H1 and H2. Given a state of the total system, σ, the reduced state of, say, the firstsubsystem is defined as a positive-semidefinite operator σ1 with tr σ1 = 1 such that

tr(σ1O) = tr(σ O ⊗ I2) ∀ O ∈ L(H1), (A1)

where L(H1) is the algebra of observables O on H1 and I2 is the identity operator in H2.For identical particles, the set of observables is restricted to permutation-symmetric operators, i.e., only such

operators can be measured. This in turn implies that tr(σ O ⊗ I2) is devoid of any physical meaning since any

apparatus measuring a single-particle observable O in fact measures O⊗ I + I⊗ O; in other words, it measures O forboth particles. Hence, the single-particle states are defined through

tr(σ1O) + tr(σ2O) = tr(σ [O ⊗ I + I⊗ O]) ∀ O ∈ L(H1) ∼= L(H2), (A2)

where H1∼= H2 for identical particles (the symbol ∼= means isomorphic).

When σ is permutation-symmetric, the standard approach is to require that σ1 = σ2, in which case Eq. (A2)uniquely determines the reduced state. For example, if the joint state is ∝ |L〉 ⊗ |R〉 + η|R〉 ⊗ |L〉, with some〈L|R〉 = 0 and η = ±1, then the reduced states are

σ(sym)1 = σ

(sym)2 =

|L〉〈L|+ |R〉〈R|2

. (A3)

Note that the operational definition (A2) only necessitates that σ1 + σ2 = |L〉〈L|+ |R〉〈R|, which leaves some freedomin choosing the reduced states, e.g., as σ1 6= σ2. Motivated by the intuitive notion of particles, Ref. [31] used thisfreedom to devise an interpretation of the quantum mechanics of indistinguishable particles in which σ1 = |L〉〈L|and σ2 = |R〉〈R|. This interpretation, however, breaks down for bosons whenever 〈L|R〉 6= 0 (for fermions, theinterpretation requires amendments but does not fall apart overall). Moreover, specific assignments for reduced statesdo not entail any observational consequences, since, due to their symmetry, measurements cannot verify whether

σ(sym)1 = σ

(sym)2 as in Eq. (A3) or σ1 = |L〉〈L| and σ2 = |R〉〈R|.

When the joint state is not permutation-symmetric, there is no reason to assume that σ1 = σ2. For example,when the joint state is |L〉 ⊗ |R〉, Eq. (A2) implies that σ1 + σ2 = |L〉〈L| + |R〉〈R| for arbitrary states |L〉 and |R〉,not necessarily orthogonal. Now it is most sensible to assign σ1 = |L〉〈L| and σ2 = |R〉〈R| which is a prescriptionthat we follow. Operationally, this prescription is justified by the fact that product states such as |L〉 ⊗ |R〉 produce

independent probabilities with respect to measurements of the form O ⊗ O; such an example is provided by theposition measurement in Eq. (12).

Appendix B: Eigenvalue degeneracy

To find the eigenstates of operator t, we look for the solutions of the equation t|φ〉 = tc|φ〉, where tc denotes theeigenvalue of t. In the u-representation, according to Eq. (40), this equation takes the form

φ′(u) =φ(u)

2u+itcu

2mφ(u). (B1)

12

We seek the solutions among the continuous, piecewise differentiable functions φ(u) = 〈u|φ〉. Solving Eq. (B1) foru < 0 and u > 0 gives

φ(u) =

{C−√−u eitcu2/(4m), u < 0

C+√u eitcu

2/(4m), u > 0, (B2)

where C− and C+ are some constants [cf. Eq. (41)]. The solutions of Eq. (B1) are unique both for u < 0 and u > 0.Therefore, if Eq. (B1), along with the continuity and piecewise differentiability of φ(u), provides a way to connect C−and C+, the eigenstate will be unique. Otherwise, we shall extend the solutions (B2) to (resp.) u < 0 and u > 0 asin Eq. (41), and the eigenstate will be double degenerate.

The only point at which the two solutions meet is u = 0. We see that φ(0−) = φ(0+) = 0, so the continuitycondition does not provide sufficient information on C− and C+, since φ(0) = 0 for any C− and C+. As for φ′(u), theeigenvalue equation (B1) leads to

φ′(0−) = limε1→0−

C−√ε1, φ′(0+) = lim

ε2→0+

C+√ε2, (B3)

which again does not connect C− to C+ in any conceivable way. Indeed, if we, e.g., impose φ′(0−) = φ′(0+), we seethat C− = C+ when ε1 = ε2 → 0, C− = 2C+ when ε1 = 4ε2 → 0, and so on, which means that C− and C+ areindependent. In a sense, the singularity of t at u = 0 is too strong to allow to uniquely connect the eigensolutionsfrom both sides. The independence of C− and C+ means double degeneracy, and the functions in Eq. (41) constitutean orthogonal basis in the eigensubspace of the eigenvalue tc.

Appendix C: Nonexistence of collision time observable

The physical reason for non-existence of a self-adjoint time operator is that the possible energetic states of a system(the spectrum of the Hamiltonian) has to be limited from below: if there were a self-adjoint time operator, it couldbe used to translate energy (as momentum is used to translate coordinate), yielding physical states with arbitrarilylow energies [22, 29].

To illustrate this statement, let us assume that there exists a collision time observable tc, and consider its expectationvalue obtained from measurements on the two-particle system in some initial state |Ψ〉: 〈Ψ|tc|Ψ〉. If we start observingthe system not at time t = 0, but at a later time t > 0, the collision time should be reduced by t. We therefore imposethe time-translation rule 〈Ψ(t)|tc|Ψ(t)〉 = 〈Ψ|tc|Ψ〉 − t, which is equivalent to

eit(H⊗I+I⊗H)tce−it(H⊗I+I⊗H) = tc − t. (C1)

Since this expression should hold for any t, it is equivalent to

[H ⊗ I + I⊗H, tc] = i, (C2)

which, as mentioned above, is incompatible with H ⊗ I + I⊗H having a spectrum bounded from below. Therefore,the collision time operator tc cannot be a self-adjoint operator that satisfies the Born’s rule.

Appendix D: Features of POVM

POVMs represent (generalized) quantum measurements and determine the probabilities of measurement outcomesthrough the Born rule [4]. This observation is used in the literature [24] to postulate that the outcome probabilitiesof measuring t defined in Eq. (40) are determined by the POVM comprised of the operators |tc, s〉〈tc, s| (s = ±),which are Hermitian, positive-semidefinite, and sum up to the identity:

∑s

∫dtc|tc, s〉〈tc, s| = I

(u). Hence, in view of

Eq. (39), the POVM describing the measurement of tc will be comprised of operators

|tc, s〉〈tc, s| ◦ I(v), s = ±. (D1)

A POVM, while being able to determine the probabilities of measurement outcomes, does not give a uniqueprescription (operator) to identify the post-measurement states. More specifically, if I ≥ Mx ≥ 0 is an element of aPOVM, then the most general reduction rule for it can be written as

σ → AxσA†x

tr(σMx), (D2)

13

where σ is a density operator and the otherwise arbitrary operator Ax satisfies A†xAx = Mx [32]. In our case [Eq. (D1)],the most general such decomposition is

|tc, s〉〈tc, s| ◦ I(v) =[|tc, s〉〈φs,tc | ◦ I(v)

]V †s,tc · Vs,tc

[|φs,tc 〉〈tc, s| ◦ I(v)

], (D3)

where |φs,tc 〉 are arbitrary normalized states, 〈φs,tc |φs,tc〉 = 1, and Vs,tc are arbitrary unitary operators (V †s,tc Vs,tc =

I(u) ◦ I(v)). Note that, in view of their normalization, states |φs,tc 〉 cannot coincide with |tc, s〉 because of Eq. (46).Thus, the general state-reduction rule described by Eq. (D2) for POVM elements given by Eq. (D1) is

σ →Vs,tc

[|φs,tc 〉〈tc, s| ◦ I(v)

]σ[|tc, s〉〈φs,tc | ◦ I(v)

]V †s,tc

tr(σ[|tc, s〉〈tc, s| ◦ I(v)

]) . (D4)

Introducing the v-space operator 〈tc, s|σ|tc, s〉 =∑n,n′ [〈tc, s| ◦ 〈n|] σ [|tc, s〉◦ |n′〉] · |n〉〈n′|, where {|n〉}n is an arbitrary

basis in the v-space, we can rewrite Eq. (D4) as

σ →Vs,tc

[|φs,tc 〉〈φs,tc | ◦ 〈tc, s|σ|tc, s〉

]V †s,tc

tr(〈tc, s|σ|tc, s〉

) . (D5)

When initial state σ is a pure state, Eq. (D5) simply reduces to

σ → |ξs,tc 〉〈ξs,tc |, (D6)

where

|ξs,tc 〉 = Vs,tc[|φtc,s〉 ◦ |ς〉

], (D7)

with

|ς〉 =∑n

[〈tc, s| ◦ 〈n|

]|ξs,tc 〉√

tr(〈tc, s|σ|tc, s〉

) |n〉; (D8)

the state |ς〉 is normalized and does not depend on the choice of the basis {|n〉}n.

Since Vs,tc is arbitrary, |ξs,tc 〉 can be an arbitrary pure state that does not depend on s. In particular, it can bechosen to be |Ψs〉 of Eq. (7), so the quantum jump in Eq. (24) can be made consistent with the post-measurementstate change (D6) induced by the collision time operator tc.

Note that Eq. (D5) does not violate the repeatability principle: if the same measurement is carried out twice, thenthe second measurement should confirm the result of the first one. This is because there exists an actual projectivemeasurement in a larger Hilbert space the reduction of which to the Hilbert space of the two particles induces thePOVM in Eq. (D1) and the transition in Eq. (D5) [4]

Appendix E: Detection of Gaussian wavepackets

We assume the Gaussian wavepackets of Eqs. (29), (30), (31), (32) with d ≡ cR − cL > 0, a ≡ aR = aL, andb ≡ bL = −bR. The relevant quantities in Eq. (50) are then given by the following explicit expressions:

〈R(t)|P |L(t)〉 =

∫ (cR+cL)/2+`

(cR+cL)/2−`dxψ∗R(x, t)ψL(x, t)

= e−b2− d2

4a2

∫ `

a√

1+τ2

− `

a√

1+τ2

dx√π

exp

[−(x− i

τd2a + b√

1 + τ2

)2]

=1

2e−b

2− d2

4a2

[erf

(`− i(ab+ τ(d/2) )

a√

1 + τ2

)− erf

(−`+ i(ab+ τ(d/2) )

a√

1 + τ2

)],

〈R(t)|P |R(t)〉 =1

2

[erf

(`+ abτ − (c/2)

a√

1 + τ2

)− erf

(−`+ abτ − (d/2)

a√

1 + τ2

)],

〈L(t)|P |L(t)〉 =1

2

[erf

(`− abτ + (d/2)

a√

1 + τ2

)− erf

(−`− abτ + (d/2)

a√

1 + τ2

)],

where erf(x) ≡ 2√π

∫∞0

ds e−s2

is the error function defined with convention erf(x)→ 1 for x→∞, and τ = t/(ma2)

is the dimensionless time. For `→∞ we get 〈R|P |L〉 → e−b2− d2

4a2 which is consistent with Eq. (33).

14

Appendix F: 3D collisions

In the main text, we considered a 1D model, where the physical picture of inter-particle collisions is rather clearand can be analyzed in both semiclassical and the fully quantum pictures. In particular, we neglected inter-particleinteractions having in mind that including a short-range inter-particle potential is straightforward. It amounts toredefining the collision operator from coinciding coordinates to the coordinate difference equal to the characteristiclength l of a short-range inter-particle potential. In 3D the situation is more complicated for at least two reasons:First the inter-particle interaction cannot be neglected, since the point coordinates of (classical) particles genericallynever coincide in 3D, i.e., the set of initial conditions for which this happens is of measure zero. This means that in3D a finite characteristic range l of the interaction potential should always be present. Second, the classical expressionfor the collision time presented below is conditional and complicated, and its quantization is as yet unclear. Hence,presently, we can estimate the collision time in 3D only semiclassically.

Considering two classical particles with a free dynamics up to the relative distance l, we have

~xk(t) = ~xk(0) +~pk(0)t

m, k = 1, 2, (F1)

and the collision time is determined from

|~x1(t)− ~x2(t)| = l, (F2)

which implies a spherically symmetric inter-particle potential with range l > 0. Solutions of the quadratic equation(F2) are analyzed assuming that |~x1(0)− ~x2(0)| > l at the initial time t = 0. Out of the two solutions of Eq. (F2), weshould select the one where t is positive and closest to zero. Denoting by ~x12 = ~x1(0)−~x2(0) and ~p12 = ~p1(0)− ~p2(0),we obtain that the collision happens under the conditions

− ~x12 · ~p12 > |~p12|√~x2

12 − l2 > 0, (F3)

while the collision time is

tc = − 1

m~p 212

[~x12 · ~p12 +

√(~x12 · ~p12)2 − ~p 2

12(~x 212 − l2)

]. (F4)

Note that the first inequality in Eq. (F3) ceases to hold for l→ 0, which confirms the necessity of a finite interactionrange l.

Equations (F3) and (F4) suffice for making semiclassical estimates of the collision times using Wigner functions [cf.Eq. (35)], but their quantization requires further research.

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