## Abstract

The unexpected connection is unravelled between the collapse of the wave function on the appearance of particle and the quantum time-of-arrival problem in one dimension. To do so, a theory of quantum first time of arrival is developed in the interacting case for arbitrary arrival point in one dimension based on a self-adjoint and canonical coarse graining of a time-of-arrival operator that derives the classical time-of-arrival observable. The appearance of particle in quantum mechanics is then considered in the light of this theory. It is found that the appearance of particle arises as a combination of the collapse of the initial wave function into one of the eigenfunctions of the time-of-arrival operator, followed by a unitary Schrödinger evolution of the eigenfunction.

## 1. Introduction

There are two dynamics in standard quantum mechanics (SQM): the well-understood and uncontroversial continuous, unitary evolution of quantum states according to the Schrödinger equation; and the ill-understood and controversial discontinuous, non-unitary evolution of the same states during quantum measurements (Bhom 1952*a*,*b*; Everett 1957; Griffiths 1984; Ghirardi *et al*. 1986; Zurek 2003; Schlosshauer 2004). The abrupt (discontinuous) and irreversible (non-unitary) evolution during quantum measurements is known as wave function collapse. It is the consensus that the collapse occurs at the moment the measurement is made, and the wave function collapses randomly into one of the eigenfunctions of the observable being measured. Now one distinct aspect of the wave function collapse is the appearance of particles from the wave description of matter. The particles are distinct among manifestations of the collapse because particles are reidentifiable, localized entities with permanent properties. These properties of particles are at odds with their quantum mechanical description as spread-out waves evolving according to the Schrödinger equation, and their appearance as localized objects presents a problem.

Schrödinger recognizes this problem early on and points out that ‘emerging particle from decaying nuclei is described as a spherical wave that impinges continuously on a surrounding luminescent screen over its full expanse[;] the screen, however, does not show a more or less constant uniform surface glow, but rather lights up at one instant at one spot’ (Schrödinger 1980*b*). Schrödinger finds this idea of abrupt, instantaneous and spatially non-local collapsing of the entire spread-out wave function into a function of point support at the moment of the appearance of the particle ‘ridiculous’ (Schrödinger 1952) and altogether denies the existence of particles. However, such collapse of the wave function seems inevitable if we need to reconcile quantum description with our experience with measuring instruments. And out of this inevitability of the collapse a question arises: Is the collapse of the wave function on the appearance of particle fundamental? That is, does the collapse occur at the moment of the appearance of particle and hence cannot be broken down into a series of casually separated processes? The consensus within SQM is that the collapse is fundamental. In this paper, we consider this question in the light of the quantum arrival problem within SQM (Bohm 1952*a*; Kijowski 1974; Holland 1993; Grot *et al*. 1996; Delgado & Muga 1997; Leavens 1998; Kochański & Wódkiewicz 1999; León *et al*. 2000; Muga & Leavens 2000; Muga *et al*. 2002; Galapon *et al*. 2002*a*,*b*, 2004, 2005*a*,*b*; Galapon 2004; Hegerfeldt *et al*. 2004) and find a different answer.

The appearance of particle, as posed by Schrödinger, is evidently a quantum arrival problem: finding a quantum mechanical mechanism for the localization of a unitarily evolving wave function at a definite point in space at a definite time at the registration of the particle. As the problem stands, the quantum time-of-arrival problem (QTOAP)—the problem of finding the time-of-arrival distribution at some given arrival points in space for a given initial state—is a promising framework to meet the problem head on. That is so because the QTOAP deals with the appearance of the particle at the arrival point at the arrival time. Any solution then to the QTOAP should not only be able to predict the time-of-arrival distributions but must also explain how the particle appears at the moment of arrival. The QTOAP, however, is wrought with controversy and there are as many solutions as there are independent researchers in the field (Muga & Leavens 2000), not to mention non-trivial assertions that the problem is not meaningful or possesses no solution at all (Allcock 1969*a*–,*c*; Yamada & Takagi 1991*a*,*b*, 1992; Halliwell & Zafiris 1998). Moreover, available QTOA theories have only addressed the time-of-arrival distribution aspect of the problem (Bohm 1952*a*,*b*; Kijowski 1974; Holland 1993; Grot *et al*. 1996; Delgado & Muga 1997; Leavens 1998; Kochański & Wódkiewicz 1999; Baute *et al*. 2000; León *et al*. 2000; Muga & Leavens 2000; Muga *et al*. 2002; Hegerfeldt *et al*. 2004), so that connection to wave function collapse at the appearance of particle has never been made until now.

In the following, we make the connection between the QTOAP and the problem of particles in one dimension within the confines of SQM. We do so by a generalization of our earlier solution to the one-dimensional free QTOAP (Galapon *et al.* 2005*b*) for arbitrary arrival point, for arbitrary interaction potential via spatial confinement followed by a limiting procedure for arbitrarily large confining lengths. We will find that the resulting quantum time-of-arrival theory suggests that the collapse of the wave function on the appearance of the particle is not fundamental: the collapse occurs much earlier than the appearance of the particle and that the subsequent localization of the wave function on the appearance of the particle arises from the unitary Schrödinger equation.

The paper is organized as follows. Section 2 summarizes the construction of time-of-arrival operator for arbitrary interaction potential. Section 3 discusses the coarse graining of the time-of-arrival operators by spatial confinement and the dynamics of their coarse-grained versions. Section 4 investigates the behaviour of the eigenfunctions of the confined time of arrival (CTOA) operators in the limit of arbitrarily large confining lengths. Section 5 discusses the emergent picture of the appearance of particles in the light of quantum arrival theory. Section 6 lays down the complete formalism in the computation of the time-of-arrival distribution for arbitrary arrival point for arbitrary interaction potential in one dimension. Section 7 provides the conclusion. Appendix A details the numerical implementation of the algorithm used in computing the time-of-arrival distribution.

## 2. Quantum time-of-arrival operator in the interacting case

Generally in SQM, to determine the distribution of a given observable one needs only the spectral resolution of an operator representation of the observable in question. Then, naturally to address QTOAP within SQM one needs to construct the appropriate time-of-arrival operator for the given system. However, the consensus is that no such operator can be constructed in the most general case of arbitrary arrival point and of arbitrary interaction potential. In one dimension, the most quoted reason is that the classical time of arrival at point *x*—given by , where *H* is the Hamiltonian, *V* is the interaction potential, *μ* is the mass of the particle and (*q*, *p*) are, respectively, the position and the momentum at *t*=0—does not admit a sensible quantization because it is generally not everywhere real and single valued in the entire phase space (Peres 1995; León *et al*. 2000; Muga & Leavens 2000). For this reason, it is believed that if a theory of quantum arrival existed it could not rest on the spectral resolution of a time-of-arrival operator.

But if we go by the standard formulation, we must insist on finding an operator —a time-of-arrival operator—that must, foremost, reduce to the classical time of arrival *T*_{x}(*q*, *p*). This is the minimum requirement that must satisfy to be identifiable as a time-of-arrival operator. Motivated to breaking the circularity of quantization when invoking the correspondence principle and to sidestepping the well-known existence of obstruction to quantization in important spaces such as the Euclidean space, we introduced in Galapon (2004) the idea of supraquantization—the derivation of a quantum observable corresponding to a classical observable without explicit reference of quantization—and found such an operator appropriately reducing to *T*_{x}(*q*, *p*) in the classical limit.

In coordinate representation, the operator is the integral operator , with the kernel given by(2.1)in which we have referred to *T*(*q*, *q*′) as the kernel factor and is determined by the interaction potential *V*(*q*). To simplify our discussion, let us for the moment consider arrival at the origin, *x*=0, and consider later arbitrary arrival points. For such a case, the kernel factor is the solution to(2.2)subject to the conditions *T*(*q*, *q*)=*q*/2 and *T*(*q*, −*q*)=0. The operator is the solution to time–energy canonical commutation relation in the form , where *ϕ*, *φ* are infinitely differentiable functions with compact supports, subject to the condition that must reduce to in the limit *ℏ*→0.

The classical time of arrival derives from via the Weyl–Wigner transform of its kernel: . For linear systems, we have ; while for nonlinear systems, , where *t*_{0}(*q*, *p*) is the expansion of *T*_{0}(*q*, *p*) about the classical free time of arrival, and referred to as the local time of arrival in Galapon (2004). Generally, *t*_{0}(*q*, *p*) converges only in some neighbourhood of the arrival point. In its region of convergence, *t*_{0}(*q*, *p*) is real and single valued and is equal to *T*_{0}(*q*, *p*). That is, *t*_{0}(*q*, *p*) is the classical first time of arrival for initial states lying in the neighbourhood of the arrival point. This indicates that our operator is a quantum first time-of-arrival operator and derives *T*_{0}(*q*, *p*) via the unique extension of *t*_{0}(*q*, *p*) in the larger classical phase space.

We now devote the rest of the paper to studying the physical content of , and show that it comprises a solution to the QTOAP and Schrödinger problem in one dimension.

## 3. Coarse graining of the time-of-arrival operator

A major obstacle in understanding the physics of is its current inaccessibility to analysis; for example, solving for the eigenvalue problem of may be intractable for arbitrary potential. We then approach the unravelling of its physical content by successive coarse graining—successive approximation of with discrete observables1. To do that, we need first to explicitly write in terms of the position and momentum operators, and , respectively. must be written in them such that, in coordinate representation, the kernel of is given by equation (2.1). It turns out that this can be accomplished by Weyl quantizing . To proceed, let us, in the mean time, consider everywhere analytic potentials. For such cases, is an expansion in *q*^{n}*p*^{−m} for positive integers *n* and *m*. The explicit operator form of is then obtained by replacing *q*^{n}*p*^{−m} within . In this form, the canonical relation translates to . Moreover, in this form, it is clear that is just the Weyl quantization of the local time of arrival *t*_{0}(*q*, *p*) for linear systems; while, for nonlinear systems, it is the quantization of *t*_{0}(*q*, *p*) plus quantum corrections required to satisfy the time–energy commutation relation. We emphasize that, due to the existence of obstruction to quantization in Euclidean space, cannot be constructed, except for linear systems, via direct quantization of the classical time of arrival.

Now a coarse graining of is constructed by confining the system in a large box of length 2*l* centred at the arrival point. The coarse-grained version of is then obtained by projecting its explicit operator form in the Hilbert space , under the condition that the Hamiltonian is purely kinetic for vanishing interaction potential. This condition projects the momentum operator into the ring of momentum operators , with _{γ} having the domain consisting of absolutely continuous functions *ϕ*(*q*) in with square integrable first derivatives, which further satisfy the boundary condition . Since depends on the momentum operator, the coarse graining of is then the set of operators {_{γ}} with each _{γ} corresponding to the momentum _{γ}.

In coordinate representation, each _{γ} in the Hilbert space is the integral operatorUsing the coordinate representation of the operators _{m,n} in for a given *γ*, the kernel of _{γ} can be shown to be given by(3.1)(3.2)with *T*(*q*, *q*′) the kernel factor in equation (2.1), and , in which *ζ*=(*q*−*q*′) and *η*=(*q*+*q*′)/2, and *H*(*x*) is the Heaviside step function (Galapon 2006). Using equations (3.1) and (3.2) together with equation (2.2), our earlier restriction on the potential can already be lifted to include a more general interaction potential. Observe that equation (3.2) explicitly reduces to equation (2.1) as *l* approaches infinity, which must be the case. Owing to this we will exclusively use _{0} in our investigation in the limit of arbitrarily large confining lengths.

Now comparing kernels (3.1) and (3.2) with those constructed in Galapon (2006), we find that they are just the CTOA operators for a given interaction potential. The CTOA operators for a given confining length then constitute the coarse graining of . For continuous potentials, the CTOA operators are compact non-degenerate self-adjoint operators. Their compactness implies that they have pure discrete spectrum, with corresponding complete square integrable eigenfunctions. It is their compactness that makes them a coarse graining of the operator . The eigenfunctions can be written as , where the sign indicates the sign of the corresponding eigenvalue, with , *n*=1,2, …; moreover, they are ordered according to . We have referred to an eigenfunction as non-nodal (nodal) when it does not vanish (it vanishes) in the interval [−*l*,*l*] (Galapon 2004; Galapon *et al*. 2004).

The spectral properties of the CTOA operators have the surprising property that they are intimately tied with the internal unitary dynamics of the system: the eigenfunctions evolve according to Schrödinger's equation such that the *event* of the position expectation value assuming the arrival point, and the *event* of the position uncertainty being minimum occur at the same instant of time equal to their corresponding eigenvalues, with the uncertainty decreasing with the magnitude of the eigenvalue. We referred to this property as unitary arrival of the eigenfunctions at the arrival point (Galapon *et al*. 2004; Galapon 2006). The unitary arrival of the eigenfunctions at their respective eigenvalues is consistent with the interpretation that is a first time-of-arrival operator. However, the arrival is not sharp in the sense that the evolving eigenfunctions may have considerable spatial supports even at their eigenvalues; this is true for the large eigenvalue–eigenfunctions. Nevertheless, the property already hints that may account for the localization of the wave function in space and time on the appearance of particle. Indeed that is what we will find as the coarse graining gets more refined: as the confining length increases indefinitely. We proceed numerically.

## 4. The limit for arbitrarily large confining lengths

Let us consider specifically the behaviour of the harmonic oscillator CTOA operator for the arrival point *x*=0 in the limit of arbitrarily large confining lengths. The corresponding kernel factor is ; this is found by solving equation (2.2) with *V*(*q*)=*ω*^{2}*μ*/2 (Galapon 2006). (Throughout all our computations we set *ℏ*=*μ*=*ω*=1.) Figure 1 shows the representative behaviour of the evolving eigenfunctions of the harmonic oscillator CTOA operators for finite *l*. Now let us investigate the behaviour of _{0} for large values of *l*. For any given time *τ* we can find an *l*_{0}>0 large enough such that |*τ*| is less than the norm or the absolute value of the largest eigenvalue of the CTOA operator; that is, there exists an eigenvalue of _{0} greater than *τ*. For any given length *l*>*l*_{0} of spatial confinement, we can find a pair of nodal and non-nodal eigenfunctions *P*_{k} such that their eigenvalues *τ*_{k} are closest to *τ*. Consider a sequence of monotonically increasing *l* values, *l*_{1}, *l*_{2}, *l*_{3}, …, with *l*_{0}<*l*_{1}<*l*_{2}<*l*_{3}<⋯. Then, there will be a *k*_{1} corresponding to *l*_{1} such that is closest to *τ*, and a *k*_{2} corresponding to *l*_{2} such that is closest to *τ*; and so on. Our computation indicates that as *l*→∞, . That is, the pair of eigenfunctions *P*_{k} becomes degenerate.

Our computation likewise indicates that the eigenvalues in the neighbourhood of *τ* get denser as the confining length tends towards infinity, so that, for a given *τ*, for a given *l*_{r} converges to *τ*. This indicates that the spectrum of the CTAO operator tends to the continuum, which implies that the eigenfunctions will eventually be thrown out of the Hilbert space—they become non-square integrable. Now, we know from our earlier results (Galapon *et al*. 2004) that the width of is minimum at the eigenvalue of . What is the behaviour of as *l*_{k} increases indefinitely? Figure 2 shows that for a given fixed time *τ*, tends to the Dirac delta with support at the arrival point as *l*_{r} tends to infinity. Then, the CTOA eigenfunctions evolve to a singular support at the arrival point at their respective eigenvalues in the limit; in particular, tends to the position eigenfunction at the arrival point! These results have been shown earlier to be true for the free particle as well (Galapon *et al*. 2005*b*). Moreover, computations on other potentials, including the quartic oscillator, show the same dynamical behaviour for arbitrarily large *l* values. These all together suggest that the behaviours of the CTOA operators in the limit are the same.

## 5. The appearance of particles

If indeed this is the case, then their limiting dynamical behaviour gives us the picture of quantum arrival within the SQM as that of unitarily evolving CTOA eigenfunction to a localized support at the arrival point. If we accept the dogma that particles are wave packets of singular support, then the CTOA eigenfunctions for the arbitrarily large values of *l* are particles at their respective eigenvalues. This endorses a mechanism for the localization of the wave function in space and in time at the registration of the particle: consider a quantum particle prepared in some initial state *ψ*_{0}. Without loss of generality, we can assume that *ψ*_{0} has a compact support. We can then enclose the system by a box of very large length (for all practical purposes infinite), with the support of *ψ*_{0} laying completely in the box. Since the CTOA eigenfunctions are complete we can decompose *ψ*_{0} in terms of these eigenfunctions. Now if we presuppose that particle detectors somehow respond only to a localized wave packet or localized energy, then registration or arrival of the particle at the arrival point at time *τ* can be interpreted as the detection of the component eigenfunction whose eigenvalue is *τ* that is unitarily arriving (essentially collapsing for arbitrarily large *l*) at the arrival point. This implies that the appearance or arrival of the particle is a combination of a collapse of the initial wave function into one of the eigenfunctions of the time-of-arrival operator right after the preparation of the initial state followed by a unitary evolution of the eigenfunction. That is, the collapse of the wave function on the appearance of particle is not fundamental but decomposable into a series of casually separated processes.

Clearly, our interpretation contrasts with the standard interpretation of the collapse of the spatial wave function on the appearance of the particle. In SQM, when a quantum object is prepared in some initial state *ψ*_{0} and when an observable of the object is measured at a later time *T*, then the state at the moment of measurement, which is , where *U*_{t} is the time-evolution operator, collapses randomly into one of the eigenfunctions of the observable. Now the consensus is that the appearance of particle is a position measurement, so that the appearance at point *q*_{0} at some time *T* is the projection of the evolved wave function to the eigenfunction *δ*(*q*−*q*_{0}) of the position operator. But according to our quantum arrival description, the collapse occurs much earlier than the appearance of the particle, with the initial state collapsing to one of the eigenfunctions of the time-of-arrival operator right after the preparation and with the particle appearing later at the moment the eigenfunction has evolved to a state of localized support at the arrival point. The appearance of particles then (at least within the context of quantum arrival) does not arise out of position measurement but rather out of time measurement.

One may, however, question the validity of our interpretation when there was no initial intention to observe the arrival of the particle. If from the very start the instrument has been set-up to detect the arrival of the particle, it can be argued that the set-up is already a measurement that has caused the initial state to collapse into one of the CTOA eigenfunctions, then evolve until observed. But this reasoning appears untenable when the decision to observe arrival is deferred, because the initial state has been evolving according to the Schrödinger equation, assuming that no other observation is made. But not quite. Quantum mechanics is inherently non-local in time (Wheeler 1978; Scully & Drühl 1982; Kwiat *et al*. 1992; Aharonov & Zubairy 2005); and that means the ‘description of the past must bear actions of the present’ (Greene 2004).

This temporal non-locality of quantum mechanics is aptly illustrated by Wheeler's delayed-choice gedanken experiment, which is depicted in figure 3: a photon is incident on a 50–50 beam splitter BS1, and a choice is available on whether to insert the beam splitter BS2 or not once the photon has already passed through BS1. Now when BS2 is not present, either detector D1 or D2 clicks, indicating that the photon has taken either path P1 or P2, respectively; since BS1 is 50–50, we expect that D1 and D2 have equal probabilities of registering the arrival of the photon. On the other hand, when BS2 is present, the phase shift between the two paths can be fixed such that, say, D1 has 100% of detecting the photon; this requires the photon taking the two paths simultaneously and interfering with itself destructively at D2 and constructively at D1. The absence of BS2 elicits the particle property of the photon; while its presence elicits the wave property of the same photon. These properties are mutually exclusive descriptions of the photon.

Now if from the very start BS2 is already absent, we already know that the photon will have to take either one of the paths, and manifest particle behaviour; on the other hand, if from the very start BS2 is already present, we already know that the photon will take both paths, and manifest wave behaviour. In either case, we have an unequivocal description of the history of the photon before registering in one of the detectors. But what can be said of the photon's history when the decision to insert or not the second beam splitter is delayed even after the photon has passed through the first beam splitter? SQM predicts that the description of the photon depends on whether BS2 is present or not, independently of when BS2 is introduced—even after the photon has long passed through BS1. The recent experiment by Jacques *et al*. (2007) has realized Wheeler's gedanken experiment and has shown that the behaviour of the photon is consistent with this prediction of quantum mechanics. In Wheeler's words, ‘We, now, by moving the [beam splitter] in or out have an unavoidable effect on what we have a right to say about the already past history of that photon’ (Wheeler 1984). That is the description of the past is not complete without regard to present actions.

Thus, the collapse right after the preparation (when arrival measurement is to be made) and the Schrödinger evolution right after the preparation (when some other measurement is to be made) are two mutually exclusive potentialities that are simultaneously true for the system, which one is realized depends on the decision what to do with the system at the moment. Temporal non-locality then replaces the spatial non-locality inherent in the spontaneous localization of the wave function at the appearance of the particle in the standard interpretation.

## 6. Quantum time-of-arrival distribution

### (a) The formalism

With the above interpretation, we can now naturally use without ambiguity the standard rules of quantum mechanics in computing the probabilities and densities of quantum arrivals at arbitrary arrival point *x* for any given initial state *ψ*_{0}(*q*) under an interaction potential *V*(*q*). In principle, we should use to compute for the probability, but to facilitate the proper interpretation of the resulting probability we replace the mean time with its coarse graining _{γ}=0 for some very large confining length *l*. Meanwhile, let us assume that *ψ*_{0}(*q*) has a compact support. We then construct the operator _{0} for the given *l*, with its kernel given by equation (3.2) and kernel factor solved through equation (2.2). In general, solving for the kernel factor *T*(*q*, *q*′) is non-trivial. Fortunately, it can be expanded in the form , where the leading term is given by(6.1)in which _{0}*F*_{1} is a specific hypergeometric function, with *ζ*=(*q*−*q*′) and *η*=(*q*+*q*′)/2. For linear systems, only the leading term contributes. But for nonlinear systems, all terms contribute; however, in the classical limit the higher order terms have the leading contribution so that *T*_{0}(*q*, *q*′) may sufficiently approximate *T*(*q*, *q*′) (Galapon 2004, 2006).

Once _{0} is constructed, compute , where and are, respectively, the eigenfunctions and eigenvalues of _{0}. The overlap is the probability that the initial state will collapse into the *s*th eigenfunction right after the preparation. With this interpretation of the overlap, , in the limit of infinite *l*, can be naturally interpreted as the probability that one of the components of the eigenfunctions with corresponding eigenvalues less than or equal to *t* shall have unitarily evolved to a localized wave function at the arrival point *x*. If our detector is what we have presupposed above, then is the probability of detection or arrival at *x* after some time *t*. Given , the time-of-arrival probability density is found by differentiation with respect to time, . The peaks of determine the most likely times of arrival at the given arrival point. Now, if the initial state has infinite tails, we can always approximate it with arbitrary accuracy by a function *ψ*_{l} whose support lies entirely in the interval [−*l*,*l*] such that *ψ*_{l}→*ψ*_{0} as *l*→∞. Then, is computed as above. The whole process can be implemented numerically by choosing the confining length to be large enough. The probability density can then be obtained by numerical interpolation and differentiation (see appendix A for details).

Now, when the arrival point is different from the origin, our results above can be carried over by a mere translation of the origin to the arrival point. This affects a change from the original potential *V*(*q*) to , and from the original initial state *ψ*_{0}(*q*) to . We then confine the system with a large box with length 2*l* centred at *x*. The box must contain the support of . Then, we proceed as described above.

In principle, if we can solve the eigenvalue problem for , we do not need to go through the above limiting procedure. If _{t} is the spectral decomposition of , not necessarily projection valued, then the probability of arrival at time *t* is given by . However, it may only be for the free particle that _{t} can be solved explicitly (Galapon *et al*. 2005*b*), so that we are forced to go about the above coarse graining of to compute for the arrival probability in the interaction case. But more than a calculational tool, the coarse graining has enabled us to give an unambiguous interpretation for .

Note that the theory allows us to compute for the time-of-arrival distribution anywhere in the configuration space—even at classically forbidden regions. Of course, that is the essence of quantum tunnelling. But does our formulation give insight as to how a quantum particle initially prepared without sufficient energy to surmount a potential barrier surmounts it nevertheless? Since the detection of the particle at some point in the configuration space is, according to our interpretation, a measurement of the quantum observable and since is conjugate to the system Hamiltonian, the detection inevitably perturbs the energy of the quantum particle. That is, by the uncertainty principle, precise measurement of the arrival of the particle translates to a large uncertainty in its energy. This resulting broad distribution of energy makes available sufficient energy to the particle to surmount the potential hill. This follows naturally if we insist on the conservation of energy.

### (b) Example: the harmonic oscillator

As an example, let us consider the still untouched harmonic oscillator time-of-arrival problem. It is sufficient for us to consider cases where we can compare with the classical case, in particular arrival at the origin. We choose our initial states to be particular Gaussians of the form . We compare the time-of-arrival distribution computed using our algorithm above with the classical time of arrival . Figure 4*a* shows the computed time-of-arrival distribution for a fixed average position *q*_{0} and for varying average momentum *p*_{0}. Evidently, the time-of-arrival distribution becomes localized with increasing average momentum. Figure 4*b* shows the logarithm of the difference between the classical time of arrival and the most likely quantum time of arrival against the average momentum of the Gaussian state. The difference decreases with increasing average momentum, or the quantum time-of-arrival distribution becomes increasingly localized at the classical time of arrival. This implies that the quantum first time-of-arrival distribution approaches the classical distribution for arbitrarily large momenta or for high energy oscillators. For small momenta or small energies, the most likely time of arrival is shorter than the classical time of arrival, so that quantum oscillators are, on the average, faster than their classical counterparts.

One desirable property of the time-of-arrival distribution is covariance; that is, translation in time should not affect the distribution. Covariance has been a primary requirement on time operators, and it is the lack of covariance of the CTOA operators (they being compact) that their introduction has been initially doubted upon. Figure 5 gives evidence of covariance of the distribution for times smaller than the period of the harmonic oscillator time-evolution operator. The given initial state is evolved through different times. These evolved states are used as the initial states in the computation for the TOA distribution. If the distribution is covariant, then the resulting distributions must be translations of each other. This is evident in the figure. Thus, while the CTOA operators are non-covariant, covariance may naturally emerge in the limit of infinite confining length.

## 7. Conclusion

We have developed a theory of quantum arrival in one dimension for arbitrary arrival point, for arbitrary interaction potential. It is already the most standard theory that we can conjure within SQM: It is a theory based on a self-adjoint and canonical coarse graining of a time-of-arrival operator that derives the classical time of arrival observable in the classical limit; moreover, it is a theory based on the collapse-supplemented Schrödinger equation, with probabilities computed in the standard way. It then provides a new opportunity of studying quantum time of arrival, which may give us new insights into other areas involving time in quantum mechanics, such as dynamical aspects of quantum tunnelling. But more than this opportunity is the novel insight the theory provides on the problem of particles. It suggests that the appearance of particle arises as a combination of the collapse of the initial wave function into one of the eigenfunctions of the time-of-arrival operator, followed by the unitary Schrödinger evolution of the eigenfunction. This implies that particles do not arise out of position measurements but out of time-of-arrival measurements, and that the collapse of the wave function on the appearance of particle is not fundamental but decomposable into a series of casually separated processes.

Schrödinger, in his 1953 Geneva lecture, concludes, ‘Well, what are these corpuscles, really? [It] may be permissible to say that one can think of particles as more or less temporary entities within the wave field whose form and general behavior are nevertheless so clearly and sharply determined by the laws of waves’ (Schrödinger 1980*a*). The emergent description of the appearance of particle out of our time-operator-based theory of quantum arrival is consistent with this expectation of Schrödinger—the particle appears out of an evolving eigenfunction via Schrödinger's wave equation, and appears temporarily at the moment the eigenfunction assumes a well-localized support. Schrödinger might have been happy to learn that such a description existed; but, at the same time, appalled at the thought that such a description still appealed to collapse of the wave function. Nevertheless, he might still have found the collapse at least consolatory because it has brought the idea of particles nearer to his uncompromising preconceptions.

## Acknowledgments

This work was supported by UP-OVCRD Outright Research Project No. 070703 PNSE and UP System Grant, and also benefited from a recent collaboration with F. Delgado, I. Egusquiza and Prof. J.G. Muga. The author especially acknowledges the numerous discussions with Prof. J. G. Muga, which have contributed to the development of the theory.

## Footnotes

↵Coarse graining by discrete observables is usually necessary in quantum measurements of observables with continuous spectrum (Busch

*et al*. 1995).- Received July 7, 2008.
- Accepted July 31, 2008.

- © 2008 The Royal Society