Physics Letters A 379 (2015) 2445–2451

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Physics Letters A
www.elsevier.com/locate/pla

A self-adjoint arrival time operator inspired by measurement models

J.J. Halliwell ∗ , J. Evaeus, J. London, Y. Malik
Blackett Laboratory, Imperial College, London SW7 2BZ, UK

a r t i c l e i n f o a b s t r a c t

Article history: We introduce an arrival time operator which is self-adjoint and, unlike previously proposed arrival
Received 22 April 2015 time operators, has a close link to simple measurement models. Its spectrum leads to an arrival time
Received in revised form 9 July 2015 distribution which is a variant of the Kijowski distribution (a re-ordering of the current) in the large
Accepted 29 July 2015
momentum regime but is proportional to the kinetic energy density in the small momentum regime,
Available online 31 July 2015
Communicated by P.R. Holland
in agreement with measurement models. A brief derivation of the latter distribution is given. We make
some simple observations about the physical reasons for self-adjointness, or its absence, in both arrival
time operators and the momentum operator on the half-line and we also compare our operator with the
dwell time operator.
© 2015 Elsevier B.V. All rights reserved.

1. Introduction constructed and it coincides with that postulated by Kijowski [11],

namely
The arrival time problem in quantum mechanics is the question
of determining the probability that an incoming wave packet, for
a free particle, arrives at the origin in a given time interval [1–5].  K (τ ) = |ψ|φτ |2
Classically, for a particle with initial position x and momentum p, 1 1 1
the arrival time is given by the quantity = ψτ || p̂ | 2 δ(x̂)| p̂ | 2 |ψτ  (4)
m
mx
τ =− . (1)
(where (τ )dτ is the probability of arriving at the origin between
p
τ and τ + dτ ), for which there is some experimental evidence [12].
The quantum problem is most simply solved using the spectrum
This is related by a simple operator re-ordering to the quantum–
of an operator corresponding to this quantity, such as that first
mechanical current at the origin,  Ĵ (t ), the result expected on
studied by Aharonov and Bohm [6],
  classical grounds, where the current operator is given by
m 1 1
T̂ AB = − x̂ + x̂ . (2)
2 p̂ p̂ 1  
Ĵ (t ) = p̂ δ(x̂(t )) + δ(x̂(t )) p̂ , (5)
A heuristic result due to Pauli [7] (significantly updated by Galapon 2m
[8,9]) indicates that an object such as this, which is conjugate
to a Hamiltonian with a semi-bounded spectrum, cannot be self- with x̂(t ) = x̂ + p̂t /m. This picture, the standard one, is nicely sum-
adjoint. Indeed we find that its eigenstates, which in the momen- marized in Ref. [13] and some developments of it and explorations
tum representation (with x̂ → i h̄∂/∂ p) are given by of the underlying mathematics are described in Refs. [8,9,14,15].
 1 The purpose of the present paper is to make two contributions
| p| 2 p2 to the standard picture presented above. The first is to discuss
φτ ( p ) = e i 2mh̄ τ , (3)
2π mh̄ three simple self-adjoint modifications of the Aharonov–Bohm op-
erator, discuss the relationship with the momentum operator on
are complete but not orthogonal. There is a POVM associated with
the half-line, and identify the underlying physically intuitive rea-
these states [10] from which an arrival time distribution can be
sons why some of these operators are self-adjoint and some not.
The second and main result is to present a new self-adjoint arrival
* Corresponding author. time operator which has a much closer link to models of measure-
E-mail address: j.halliwell@imperial.ac.uk (J.J. Halliwell). ment than any previously studied operators.

http://dx.doi.org/10.1016/j.physleta.2015.07.040
0375-9601/© 2015 Elsevier B.V. All rights reserved.
2446 J.J. Halliwell et al. / Physics Letters A 379 (2015) 2445–2451

2. Self-adjoint arrival time operators relinquished information is essentially the specification of whether
the particle is incoming (x and p with opposite signs) or outgoing
The lack of self-adjointness of Eq. (2) is not necessarily a prob- (x and p with the same sign). Of course in practice we are usu-
lem but nevertheless a number of efforts have been made to ally interested in the arrival time for a given state, for which this
restore it. Here, we take a simple approach and note that self- information is already known, at least semiclassically, so from this
adjointness may be achieved by a number of simple modifications point of view the difference between the Aharonov–Bohm operator
of the states Eq. (3). The states and its self-adjoint variants may not be important. Nevertheless it
 1 is of interest to uncover the underlying origins of self-adjointness
| p| 2 p 2
or its absence and the above properties give some useful clues.
φτ ( p ) = e i ( p ) 2mh̄ τ , (6)
2π mh̄ Physically speaking, self-adjointness or its absence are about
precision. A self-adjoint operator has orthonormal eigenstates and
where ( p ) is the sign function, are orthogonal and complete and an associated projection operator onto a range of its spectrum. Pro-
so are eigenstates of a self-adjoint operator. This operator, first jections onto different ranges have zero overlap. An operator that
considered by Kijowski [11] and subsequently explored at length is not self-adjoint has non-orthogonal eigenstates and has at best
by Delgado and Muga [16], may be written a POVM onto a range of its spectrum. Two POVMs localizing onto
  different ranges will have a small overlap which means there is
m 1 1
T̂ KDM = − x̂ + x̂ , (7) intrinsic imprecision in the specification the ranges they localize
2 | p̂ | | p̂ |
onto. To understand the origin of the lack of self-adjointness in
and is a quantization of the classical expression −mx/| p |. It is also the Aharonov–Bohm operator we would like to find a physically
usefully written in the representation using the pseudo-energy, intuitive understanding of where this imprecision comes from.
ξ = p | p |/2m, where we have
3. The momentum operator on the half-line
∂
T̂ KDM = −ih̄ (8)
∂ξ To understand the above issue, we turn to the frequently-
1 studied situation of the momentum operator on the half-line x > 0
(which acts on states (ξ ) = (m/ p ) φ( p )), and is self-adjoint
2
[19–21]. There, the momentum operator cannot be made self-
since ξ takes an infinite range. This is in contrast to the Aharonov–
adjoint since it generates translations into negative x. However, p̂ 2
Bohm operator which, in the energy representation, has the form
can be made self-adjoint, with suitable boundary conditions, and
   
∂ ∂ therefore, by the spectral theorem, | p̂ | can be made self-adjoint.
T̂ AB = −ih̄ ⊕ −ih̄ (9) Hence just by relinquishing information about the sign of p̂ a self-
∂E ∂E
adjoint operator is obtained. The obstruction to self-adjointness on
where the two parts of the direct sum refer to the positive and the half-line therefore lies in the sign function of p̂. Differently put,
negative momentum sectors. Its lack of self-adjointness is due the problem is that the operator θ(x̂)θ( p̂ )θ(x̂) cannot be made self-
to the fact that E > 0, as is frequently noted. (See for example, adjoint. For similar reasons, we also note that the position operator
Ref. [17].) on the positive momentum sector cannot be self-adjoint. A POVM
A second modification of the Aharonov–Bohm operator is to su- for the momentum operator on the half-line may be constructed,
perpose opposite values of τ in Eq. (3) and then note that the but this is not directly relevant to what we do here [21].
1
subsequent states, which are proportional to | p | 2 sin( p 2 τ /2mh̄) We propose that there is a simple physical way of under-
are orthogonal and are the eigenstates of the self-adjoint opera- standing the underlying imprecision linked to this lack of self-
tor adjointness. Suppose we tried to measure the momentum. Let us
 therefore consider a simple measurement model of momentum
T̂ MI = 2
T̂ AB , (10) on the half-line x > 0 using sequential position measurements,
from which information about momentum can be deduced. Sim-
considered by de la Madrid and Isidro [18]. This is a quantization ilar approaches to calculating the time-of-flight momentum have
of m|x|/| p |. A third modification is to note that the orthogonal- been given elsewhere [22] and we make use of these results, but
1
ity of the states | p | 2 sin( p 2 τ /2mh̄) is not affected by restriction to adapted to the case of propagation in the region x > 0. We sup-
positive or negative momenta so we may consider these two sec- pose we have an initial incoming state ψ at time t 0 consisting
tors separately and as a consequence the operator of a spatially very broad gaussian, close to a plane wave, of mo-
mentum p 0 < 0 and we ask if it passes through a spatial region
T̂ 3 = θ( p̂ ) T̂ MI θ( p̂ ) − θ(− p̂ ) T̂ MI θ(− p̂ ), (11) [x̄1 − , x̄1 + ] in x > 0 at time t 1 and at a later time t 2 through
is self-adjoint. This operator, which does not seem to have been a spatial region [x̄2 − , x̄2 + ]. The probability for these two
noted previously, is a quantization of the classical expression measured results is
m|x|/ p.
p (x̄1 , t 1 , x̄2 , t 2 )
These three examples all side-step the Pauli theorem since they
do not have canonical commutation with the Hamiltonian. Fur- = ψ| g † (t 1 , t 0 ) P x̄1 g † (t 2 , t 1 ) P x̄2 g (t 2 , t 1 ) P x̄1 g (t 1 , t 0 )|ψ (12)
thermore, they all give probability distributions which are simple
where
variants on the Kijowski distribution, the expected result, as is eas-
ily deduced from their eigenstates. x̄+
From these three examples we make the following sim- P x̄ = dx|xx| (13)
ple observation. The Aharonov–Bohm operator arises from the
x̄−
quantization of the classical expression −mx/ p and is not self-
adjoint. However, quantizing any of the three classical expres- is a projector onto the range [x̄ − , x̄ + ] and g (t 1 , t 0 ) denotes
sions −mx/| p |, m|x|/| p | or m|x|/ p leads to a self-adjoint operator. the propagator in the region x > 0. The precise form of the prop-
Hence, self-adjoint modifications of the Aharonov–Bohm operator agator depends on the boundary conditions on the states imposed
are easily obtained by relinquishing just one or two bits of in- at x = 0. There is a one-parameter family that leads to a self-
formation, namely the sign of x, or p, or the signs of both. The adjoint Hamiltonian, of the form
J.J. Halliwell et al. / Physics Letters A 379 (2015) 2445–2451 2447

ψ(0) + α ψ  (0) = 0 (14) boundaries, as there is in the half-line case. This is consistent with
self-adjointness.
where α is real [19]. For simplicity, we focus on the simplest case Given these features of the momentum on the half-line, we may
α = 0 in which the states are required to vanish at x = 0 and in now draw a direct connection with the Aharonov–Bohm operator.
this case the propagator in x > 0 is given by the simple method of In exploring the quantization of the classical arrival time −mx/ p,
images expression it is not at all obvious at the classical level why this object leads
 1 to an operator which is not self-adjoint. To explore this, one might
m 2
take as a simple starting point the simple expression θ(t + mx/ p ),
g (x1 , t 1 |x0 , t 0 ) = θ(x1 )θ(x0 )
2π h̄(t 1 − t 0 ) which may be used to address the simple question of whether the
     arrival time is greater or less than a given time t. The arrival time
im(x1 − x0 )2 im(x1 + x0 )2
× exp − exp . is then constructed using
2h̄(t 1 − t 0 ) 2h̄(t 1 − t 0 )
∞  
(15) mx d mx
− = dt t θ t+ (18)
Semiclassically, the first term clearly corresponds to the direct clas-
p dt p
−∞
sical path between the initial and final spacetime points and the
second term corresponds to a reflected path (or equivalently, to the Quantizing θ(t + mx/ p ) directly is problematic since −mx/ p does
path coming from the image point −x0 ). Other choices of bound- not turn into a self-adjoint operator so it is natural to consider
ary conditions yield different propagators but all have the same alternative but equivalent classical expressions obtained by scaling
feature of including both direct and reflected paths. out p to relate it to the expression θ(x + pt /m), which clearly can
be turned into a self-adjoint operator. (The latter expression is also
To determine the momentum from the probability Eq. (12) we
a natural starting point for measurement inspired models as we
consider the conditional probability of finding the particle near x̄2
shall see.) However, the sign of p is important in this scaling and
at time t 2 , given that it was near x̄1 at time t 1 , which is given by
we in fact obtain
p (x̄1 , t 1 , x̄2 , t 2 )      
p (x̄2 , t 2 |x̄1 , t 1 ) = . (16) mx pt pt
p (x̄1 , t 1 ) θ t+ = θ( p )θ x + + θ(− p )θ −x − (19)
p m m
By comparing with earlier cases for the whole real line [22], it This is just θ(x)θ( p ) + θ(−x)θ(− p ), shifted along the classical
is reasonably easy to see what happens without doing any fur- equations of motion, precisely the object that cannot be made self-
ther calculation. This probability distribution will have two peaks, adjoint as we have seen. For example, with a particular choice of
one at the expected value for an incoming state with momen- operator ordering the quantization of the right-hand side is
tum p 0 , namely x̄2 = x̄1 + ( p 0 /m)(t 2 − t 1 ). The other peak is at    
x̄2 = x̄1 − ( p 0 /m)(t 2 − t 1 ) corresponding to the parts of the in- i i
exp Ht θ(x̂)θ( p̂ )θ(x̂) − θ(−x̂)θ(− p̂ )θ(−x̂) exp − Ht
coming state that already reached the origin prior to t 1 and been h̄ h̄
reflected back. Hence the measured momentum probability will be (20)
concentrated about both p 0 and − p 0 without a possibility of dis-
tinguishing between the two. The magnitude of the momentum is A different operator ordering would lead to the expressions of
measured unambiguously but the sign of the momentum is intrin- the form θ( p̂ )θ(x̂)θ( p̂ ) which are also not self-adjoint, as noted
sically ambiguous, due to reflection. already. Hence, the quantization of the classical arrival time ex-
Of course the case described above is the extreme case of an pression −mx/ p boils down to the expression θ(x̂)θ( p̂ )θ(x̂), pre-
incoming plane wave with no spatial localization so the state ar- cisely the underlying non-self-adjoint object encountered in the
rives at the first measurement at a very wide spread of times. The momentum operator on the half line. The same remarks concern-
opposite extreme is a wave packet tightly peaked in both posi- ing reflection therefore apply.
By contrast, for the classical expression −mx/| p |, similar steps
tion and momentum which arrives at the first measurement at a
yield
precise time and the probability Eq. (12) is strongly peaked about
   
the direct momentum p 0 with negligible peak about the reflected mx | p |t
one. But for a more general state, there always will be some peak- θ t+ =θ x+ (21)
| p| m
ing about the reflected momentum and hence ambiguity in the
momentum measurement. This is the physically intuitive reason The right-hand side, when quantized, is simply θ(x̂) unitarily
underlying the imprecision in the sign of the momentum operator evolved in time with the pseudo-Hamiltonian ξ̂ = p̂ | p̂ |/2m, so is
on the half-line associated with its lack of self-adjointness. self-adjoint.
This intuitive link between reflection and lack of self-adjoint- In summary, the key difference between the Aharonov–Bohm
ness is further substantiated in another closely related but dif- operator and the three self-adjoint variants of it is that the
ferent case, namely the momentum operator in a finite spatial Aharonov–Bohm operator involves specification of whether the
interval [a, b]. In this case the momentum operator is in fact self- states are incoming or outgoing. This involves specifying the signs
adjoint, for suitable periodic boundary conditions on the state, the of the momenta on the two half-lines x > 0 and x < 0. Restric-
simplest of which is tion to the half-line in quantum mechanics creates quantum–
mechanical reflection which renders the sign of the momentum
ψ(a) = ψ(b) (17) imprecise. This is the physical reason why we would expect that
the Aharonov–Bohm operator is not self-adjoint.
(see for example Ref. [19]). One can again ask how this reconciles
with measurement of momentum in this interval, using sequential 4. Difficulties with the standard arrival time operators
position measurements – is there any imprecision due to reflec-
tion effects? The point here is that the boundary conditions mean Aside from the self-adjointness issue, all four of the operators
the configuration space has the topology of the cylinder and the discussed in Section 2 suffer from a number of problems. Firstly,
semiclassical interpretation of the propagator is that it consists of their relation to actual measurements, or at least to simple mea-
straight line paths on the cylinder. There is no reflection at the surement models is not obvious since there is no physical system
2448 J.J. Halliwell et al. / Physics Letters A 379 (2015) 2445–2451

that couples to the Aharonov–Bohm operator or any of the above We may use Eq. (24) to see the origin of the low momentum
variants of it. Secondly all the above arrival time operators yield regime formula Eq. (22) as follows. The formula Eq. (22) is the ap-
an arrival time distribution which is the Kijowski distribution or propriate one in the case of very frequent measurement in which
variants thereof for all ranges of momenta. Simple measurement there is significant reflection off the detector. To model this, we
models (such as those based on a complex potential [12,23] or suppose that prior to t = 0, the system has been subjected to fre-
stopwatch [24,25]) agree with this for large momenta. But cru- quent projections onto x < 0, so that the state at t = 0 has the
cially, for small momenta, and more specifically for arrival time form
measurements more precise than h̄m/ p 2 (the energy time [26,27]),
reflection off the detector becomes significant and measurement |ψ = P̄ (tn ) · · · P̄ (t 2 ) P̄ (t 1 )|φ0  (25)
models typically yield a distribution proportional to the kinetic en-
ergy density for a sequence of times 0 > tn > · · · > t 2 > t 1 and for some ini-
tial state |φ0  in the distant past. If the interval between these
(τ ) = N ψτ | p̂ δ(x̂) p̂ |ψτ , (22) projections is sufficiently small (in comparison to the energy time
h̄m/ p 2 ) the evolution in Eq. (25) will in fact be well-approximated
where N is a model-dependent normalization factor [12,23–25,28]. by the restricted propagator for the half-line x < 0, given by
This behaviour does not arise in any of the standard arrival time Eq. (15) adapted to the case x < 0, so there will be reflection off
operators at small momentum. the origin [26]. At t = 0 the state will therefore have the approxi-
Thirdly, even after normalizable states are constructed from the mate form ψ(x) = θ(−x)(φ(x) − φ(−x)), for some φ(x), so ψ(x) is
eigenstates of T̂ AB and its three modifications (by superposing over zero at x = 0 but has non-zero derivative. The arrival time distri-
1
narrow ranges of τ , as described in Ref. [25]), they all go like p 2 bution is then
for small p, which means that they have infinite ( x)2 , so have
poor spatial localization properties, contrary to the intuitive notion (τ ) = ψ| Ĵ (τ )|ψ
of what an arrival time state should look like. The last two prob- ih̄
lems are issues around small momentum behaviour and could be =− [ψ ∗ (0, τ )ψ  (0, τ ) − ψ(0, τ )ψ  ∗ (0, τ )], (26)
2m
solved by an arrival time operator whose eigenstates go like p for
small p. The main result of the remainder of this paper, and the where the dash denotes spatial derivative. This is zero at τ = 0 so
solution to these three problems, is the construction of an arrival we have to expand the wave function for small τ . We make use of
time operator directly inspired by simple measurement models. the free particle propagator and write
We briefly mention the work of Grot, Rovelli and Tate, who re-
moved the singularity at p = 0 in T̂ AB by a somewhat artificial 0  1
m 2 imy 2
regularization procedure [29]. This produced a self-adjoint opera- ψ(0, τ ) = dy e 2τ h̄ ψ( y , 0), (27)
2π ih̄τ
tor but the low momentum behaviour produces a spatial spread −∞
even more severe than the examples above [25], reiterating the
need for a physically motivated handling of the low momentum where we have used the fact that ψ(x > 0, 0) = 0. Making the
1 1
regime. See also Ref. [30] for other criticisms of traditional arrival change of variables y = zτ 2 , expanding ψ( zτ 2 , 0), and recalling
time operators. that ψ(0, 0) = 0, we thus obtain

5. Measurement models  1 0
mτ 2
 imz2
ψ(0, τ ) ≈ ψ (0, 0) dz z e 2h̄ . (28)
2π ih̄
To motivate our proposed new arrival time operator and also −∞
to substantiate the arrival time distribution in the low momentum
regime Eq. (22) we consider two measurement models for arrival Evaluating the integral we then find
time.  3
A simple measurement model for the probability of a particle 1 h̄ 2 1
 J (τ ) ≈ τ 2 |ψ  (0, 0)|2 . (29)
crossing the origin in a given time interval [0, τ ] involves spatial 2π
1
2 m
measurements onto x > 0 and x < 0 described by projectors P =
θ(x̂) and P̄ = θ(−x̂), and we simply check to see if the particle is This is of the desired form, Eq. (22) since
on opposite sides of the origin at the initial and final times. The
ψ| p̂ δ(x̂) p̂ |ψ = h̄2 ψ  (0, 0)
2
probability for crossing is then (30)

(A similar derivation was given in Ref. [31] with different aims.)

p (0, τ ) = ψ| P̄ P (τ ) P̄ |ψ + ψ| P P̄ (τ ) P |ψ. (23)
Hence the low momentum regime result arises simply because the
Since d P (t )/dt = Ĵ (t ), where Ĵ (t ) is the current operator Eq. (5), very frequent measurement causes the wave function to vanish at
this may be rewritten the origin so it is necessary to expand the average current around
this for small times.
τ τ A second simple model for measuring the arrival time, con-
p (0, τ ) = dt ψ| P̄ Ĵ (t ) P̄ |ψ − dt ψ| P Ĵ (t ) P |ψ, (24) sidered by numerous authors [24,25], consists of a stopwatch –
a system with coordinate y and zero Hamiltonian which couples
0 0
to the particle through the interaction p y θ(−x). Classically, for a
thus indicating the appearance of the current operator in arrival particle approaching from the left it therefore causes a shift in the
time probabilities derived from a measurement model (as it does stopwatch variable y for the entire time the particle is in x < 0,
in models, e.g. Ref. [24]). This is not in fact the Kijowski distri- stopping when the particle reaches the origin, with final value
bution (integrated over a time range) Eq. (4). However, estimates
T
involving gaussian states show that it is very close either for states
p 
strongly peaked in momentum, or for time intervals large com- y ( T ) − y (0) = dt θ −x − t , (31)
pared to the energy time, h̄m/ p 2 . m
0
J.J. Halliwell et al. / Physics Letters A 379 (2015) 2445–2451 2449

where T is taken to be very large. This is easily seen to be equal The remaining time integral may be evaluated by sandwiching be-
classically to −mx/ p for p > 0. In the quantum–mechanical analy- tween momentum states. We have
sis of this system we can see the form of the interaction between ∞  
particle and stopwatch via the S-matrix, p̂t
dt  p 1 |δ x̂ + | p2 
⎛ ⎞ m
T   −∞
i p̂
S = T exp ⎝− λ dt p̂ y (t )θ −x̂ − t ⎠ (32) ∞  
h̄ m it
0 =  p 1 |δ(x̂)| p 2  dt exp ( p 21 − p 22 )
2mh̄
−∞
where T is the usual time ordering operator and λ is the cou-
pling constant. The stopwatch has zero free Hamiltonian so p̂ y (t ) = 2m δ( p 21 − p 22 )
is independent of t. For weak couplings we therefore see that the m
combination of the particle variables that the stopwatch “sees” is = (δ( p 1 − p 2 ) + δ( p 1 + p 2 ))
| p1 |
precisely the quantization of the right-hand side of Eq. (31). It does
not couple to anything like the Aharonov–Bohm operator. (1 + R̂ )
=  p1 | | p2  (37)
It is also of interest to rewrite the right-hand side of Eq. (31) to | p̂ |
indicate its connection to the current. We have,
where we have introduced the reflection operator R̂, defined by
T T R̂ | p  = | − p . We thus find that our time operator may now be
p  written
dt θ −x − t = dt t J (t ) (33)  
m m 1 1
0 0 T̂ = − x̂ (1 + R̂ ) + (1 + R̂ )x̂ . (38)
2 | p̂ | | p̂ |
where J (t ) = ( p /m)δ(x + pt /m) is the classical current and we
have dropped a boundary term since θ(−x − pt /m) is zero for large It may be written in terms of the KDM operator Eq. (7) as
t with positive p. We could also contemplate a similar integration ih̄m
by parts for the corresponding quantum operators but there it is T̂ = T̂ KDM + R̂ (39)
2 p̂ | p̂ |
less obvious that the boundary term may be dropped. Neverthe-
less, both sides of Eq. (33) give two alternative starting points for Like the KDM operator, it is self-adjoint, as we shall confirm. The
the construction of arrival time operators, classically equivalent to extra term, involving reflection, arises because of the time integral
−mx/ p, but potentially different in the quantum case and arguably in Eq. (37), which is precisely how reflection effects arise in scat-
more relevant to measurement models. The expression on the left- tering theory in the usual S-matrix expansion.
hand side is difficult to handle as a quantum operator so we focus We also note that our operator bears a close comparison with
on a version of the right-hand side in what follows. the dwell time operator. This is the operator describing the time
spent by a particle in a spatial region [0, L ], defined by
6. New operator and its properties ∞
i i
T̂ D = dt e h̄ Ht P L e − h̄ Ht , (40)
Motivated by the above observations and in particular by the
−∞
key role the current plays in measurement models, we look for a L
new time operator defined in terms of the current operator Ĵ (t ), where H = p̂ 2 /2m and P L = 0 dx|xx| is a projector on the region
Eq. (5). We begin by noting the classical result indicated already, [0, L ] [32]. The dwell time operator commutes with the Hamilto-
namely nian so is unaffected by the Pauli theorem and indeed it is self-
adjoint. We would expect self-adjointness on physical grounds. In a
∞ measurement model for dwell time (see for example Ref. [24]), one
mx
− = dt t J (t ), (34) might consider a clock model in which the Hamiltonian (which we
| p|
−∞ take to be self-adjoint) includes a coupling between the clock and
the operator P L and the clock variables would then “see” the ex-
where J (t ) is the classical current, here valid for all values of x pression Eq. (40) in an S-matrix expression, analogous to Eq. (32).
and p. Quantization of the left-hand side yields the KDM operator The dwell time operator may also be written more explicitly as
Eq. (7). Here, however, we instead take the right-hand side as the  
mL i sin ( p̂L /h̄)
starting point for quantization and this leads to a different opera- T̂ D = 1 + e − h̄ p̂L R̂ , (41)
tor, namely | p̂ | ( p̂L /h̄)
∞ in which one can clearly see reflection effects. Classically, the dwell
time can be written as the difference between two arrival times,
T̂ = dt t Ĵ (t ), (35)
at x = L and x = 0
−∞
mL m(x − L ) mx
=− + (42)
whose properties we study. To be clear, there are two motivations | p| | p| | p|
for examining this expression. Firstly, it has a potentially closer
but not in general in the quantum case Eq. (41). However, it is
connection to measurements than the KDM operator, as indicated.
true both for large momenta, pL h̄, where the quantum case
But secondly, and independently of this, it may be simply regarded
essentially coincides with the classical one as one might expect,
as an exploration of the consequences in the quantum theory of
but more interestingly also for low momenta, pL h̄, where one
taking a different but classically equivalent starting point.
can see that our new arrival time operator satisfies
To evaluate the integral in Eq. (35) we first note the operator
mL 
identity
T̂ D ≈ 1 + R̂
    | p̂ |
p̂t p̂t p̂t
δ x̂ + = −x̂ δ x̂ + (36) i i
= e h̄ L p̂ T̂ e − h̄ L p̂ − T̂ . (43)
m m m
2450 J.J. Halliwell et al. / Physics Letters A 379 (2015) 2445–2451

 
Hence the reflection term occurring in our new arrival time op- ∂2 A 2 ∂ A τ 2 2 A
φ ( p) − φ ( p) + p φ τ ( p ) = 0. (52)
erator is closely related to a similar term appearing in the more ∂ p2 τ p ∂p τ mh̄
familiar dwell time operator.
It may be shown that this equation has two linearly indepen-
Returning to our new time operator, we consider its covariance
dent solutionsin terms of Bessel functions, an antisymmetric
 one,
properties. Note that it has the same commutation relation with 1 p2 τ 3 p2 τ
the Hamiltonian as the KDM operator, namely p| p| 2 J 3 and a symmetric one | p | 2 J − 3 2mh̄ which is ir-
2mh̄
4 4
relevant so is dropped. Inserting this antisymmetric solution into
[ H , T̂ ] = ih̄ ( p̂ ) (44) Eq. (51) and using properties of the Bessel functions [33] we ob-
tain φτS ( p ) and the full normalized solution is then found to be
since the extra term in Eq. (39) commutes with H . Applying the
unitary Hamiltonian evolution operator to the eigenvalue equation, 1     
τ2 3 p2 τ 1 p2 τ
φτ ( p ) = √ | p| 2 J − 1 + ip | p | 2 J 3 . (53)
T̂ |φτ  = τ |φτ  (45) 8mh̄ 4 2mh̄ 4 2mh̄
(Here, a possible overall factor of i is fixed by noting that the
we find that
solution must satisfy φ ∗ ( p ) = φ(− p ).) These eigenstates may be
 
i i shown, at some length, to be orthonormal and complete and thus
T̂ e − h̄ Ht |φτ  = (τ + ( p̂ )t ) e − h̄ Ht |φτ  . (46)
T̂ is a self-adjoint operator. Using the asymptotic forms of the
i Bessel functions, we obtain approximations for the eigenstates in
For the KDM case, we can solve this equation for e − h̄ Ht |φτ  by the large and low momentum regimes respectively,
splitting into positive and negative momentum states, thereby ⎧ 1 
finding that the positive momentum states are shifted forwards ⎪
⎪ − πi ( p) | p| 2 p2 τ p2 τ
in time and the negative momentum states backwards (as one can
⎨e 8 2π mh̄
exp i ( p ) 2mh̄ 2mh̄
1
φτ ( p ) ∼ 1 (54)
see also in the explicit eigenvalues, Eq. (6)), so there is covariance ⎪
⎪ 1  τ4 | p| p2 τ
in the positive and momentum sectors separately. However, for our ⎩ 3 3 2mh̄
1.
2 4 (mh̄) 4
new operator T̂ , the presence of the reflection term mixes up the
positive and negative momentum sectors and it can be shown that As anticipated, the eigenstates do not shift in a simple way under
the spectrum does not transform in any simple way under unitary time translations, except in the large momentum regime, where
time shifts. This difference between T̂ and the KDM operator may they have the same covariance properties as the KDM states.
also be seen by considering their commutation with the pseudo- The arrival time probability following from our new operator
energy ξ̂ = p̂ | p̂ |/2m. We have for a time interval [τ1 , τ2 ] is now given by
τ2
[ξ̂ , T̂ KDM ] = ih̄ (47)
p (τ1 , τ2 ) = dτ |ψ|φτ |2 . (55)
which implies the simple shift properties stated above. However, τ1
the commutator with T̂ has an extra term,
  We compare this directly with the measurement model, Eq. (24),
1 so we set τ1 = 0 and consider the interval [0, τ2 ]. If τ2 is large
[ξ̂ , T̂ ] = ih̄ 1 + R̂ (48)
2 compared to the energy time mh̄/ p 2 , the time integral in Eq. (55)
takes contributions from both large and small momentum regimes
from which no simple covariance properties follow, as one can in Eq. (54). It is easy to verify that the large momentum regime
show. These results mean that we do not expect any obvious co- will give the dominant contribution. The eigenstates in that regime
variance properties in the spectrum of T̂ . However, since the extra are of the form (6) and thus the probability distribution is similar
term is clearly quantum–mechanical in nature, we would expect to that for T̂ KDM , i.e. a variant of the Kijowski distribution [34].
covariance behaviour similar to the KDM case in the semiclassical Note, however, there is a difference in that there is a phase de-
regime. pending on ( p ) which implies that the terms in the distribution
representing interference between positive and negative momenta
7. Spectrum and arrival time probabilities will have a different phase to that in the distribution derived from
T̂ KDM . This difference is not surprising given the presence of a re-
Consider now the spectrum of the arrival time operator Eq. (38). flection term in our operator. We thus get broad agreement with
The eigenvalue equation in momentum space is Eq. (24).
   ∂ If the time τ2 is short compared to the energy time in Eq. (55),
mh̄ ∂ 1 1
−i 1 + R̂ + 1 + R̂ φτ ( p ) the low momentum regime applies. The eigenstates are of the form
2 ∂ p | p| | p| ∂p φτ ( p ) ≈ C | p |, where C is read off from Eq. (54) and we easily ob-
= τ φτ ( p ). (49) tain
1
The eigenstates can be written as the sum of their symmetric and 2 π |τ | 2
antisymmetric parts φτ ( p ) = φτS ( p ) + φτA ( p ) and the eigenvalue |ψ|φτ | ≈ 2 3 1
ψ|| p̂ |δ(x̂)| p̂ ||ψ, (56)
equation reduces to a coupled system of first order equations 2  3 m 2 h̄ 2
4
 
∂ 1 iτ A which is the kinetic energy density, Eqs. (22), (29), the physically
φτS ( p ) = φ ( p ), (50)
∂p | p| mh̄ τ expected result in this regime. The numerical prefactors are not
exactly the same as in Eq. (29) – they differ by about 20 percent
and but the overall factors in Eq. (56) are fixed by the completeness
∂ A iτ | p| S relation of the eigenstates φτ ( p ). There is no reason to expect per-
φτ ( p ) = φ ( p ). (51) fect agreement since the two formulae have different origins. So
∂p mh̄ τ
again we get broad agreement with the measurement model.
Solving for the antisymmetric part of the state we obtain the sec- Hence our new arrival time operator Eq. (38) gives the physi-
ond order differential equation cally expected results for both large and small momentum regimes,
J.J. Halliwell et al. / Physics Letters A 379 (2015) 2445–2451 2451

and in particular, gives the behaviour anticipated by measurement half-line – it relates to the imprecision arising due to reflection
models in the low momentum regime, behaviour not captured when x is restricted to a half-infinite range.
by any of the standard arrival time operators described in Sec-
tion 2. Acknowledgements
However, it is important to note that approximate agreement
of the probability Eq. (55) with the low momentum arrival time We are grateful to Gonzalo Muga for useful conversations over
formula Eq. (22) is achieved only for an interval [0, τ2 ] with τ2 a long period of time. This work was supported in part by EPSRC
small. This is all that is required since we are comparing with Grant No. EP/J008060/1.
the formula derived from measurement, Eq. (24), which concerns
a time interval close to zero. The initial time τ = 0 has a distin- References
guished status in this model since the state is prepared then (and
in the stopwatch model it is the initial time at which stopwatch [1] G.R. Allcock, Ann. Phys. 53 (1969) 253;
and particle are in a factored state). Hence this feature is physi- G.R. Allcock, Ann. Phys. 53 (1969) 286;
G.R. Allcock, Ann. Phys. 53 (1969) 311.
cally expected. It is also mathematically consistent since, as noted,
[2] J.G. Muga, R. Sala Mayato, I.L. Egusquiza (Eds.), Time in Quantum Mechanics,
the spectrum of our operator does not have any obvious covari- Springer, Berlin, 2002.
ance properties under time translation so there is no inconsistency [3] J.G. Muga, A. Ruschhaupt, A. del Campo (Eds.), Time in Quantum Mechanics,
in the model possessing a distinguished point in time. For more vol. 2, Springer, Berlin, 2009.
general time intervals [τ1 , τ2 ] we get the low momentum result [4] J.G. Muga, C.R. Leavens, Phys. Rep. 338 (2000) 353.
[5] J.G. Muga, S. Brouard, D. Macias, Ann. Phys. 240 (1995) 351.
only for time intervals where both end-points are close to τ = 0.
[6] Y. Aharonov, D. Bohm, Phys. Rev. 122 (1961) 1649.
For times τ1 , τ2 which are very close, but large, the probability [7] W. Pauli, Handbuch der Physik, vol. 24, Springer, Berlin, 1933, pp. 83–272.
Eq. (55) is still an integral over a range of time of a distribution [8] E. Galapon, article in Ref. [3].
similar to the Kijowski distribution, not the low momentum for- [9] E.A. Galapon, Proc. R. Soc. Lond. A 487 (2002) 1;
mula. E.A. Galapon, Proc. R. Soc. Lond. A 487 (2002) 2671;
E.A. Galapon, Proc. R. Soc. Lond. A 458 (2002) 451.
These features mean that the regimes in this model are not in
[10] R. Giannitrapani, Int. J. Theor. Phys. 36 (1997) 1575.
general defined by the size of the interval |τ2 − τ1 | compared to [11] J. Kijowski, Rep. Math. Phys. 6 (1974) 361.
the energy time, as one might have thought from some measure- [12] J.A. Damborenea, I.L. Egusquiza, G.C. Hegerfeldt, J.G. Muga, Phys. Rev. A 65
ment models, except for the special case of the interval [0, τ2 ]. To (2002) 052104;
obtain a result of this form would require a more elaborate opera- G.C. Hegerfeldt, D. Seidel, J.G. Muga, Phys. Rev. A 68 (2003) 022111;
A. Ruschhaupt, J.G. Muga, G.C. Hegerfeldt, article in Ref. [3].
tor containing an explicit parameter modeling the precision of the
[13] I.L. Egusquiza, J.G. Muga, A.D. Baute, article in Ref. [2].
measurement. This is of interest to construct and will be pursued [14] E.A. Galapon, R.F. Caballar, R.T. Bahague Jr., Phys. Rev. Lett. 93 (2004) 180406;
elsewhere. E.A. Galapon, F. Delgado, J.G. Muga, I. Egusquiza, Phys. Rev. A 72 (2005) 042107.
Furthermore, as indicated earlier, there are two different moti- [15] G.C. Hegerfeldt, J.G. Muga, J. Munoz, Phys. Rev. A 82 (2010) 012113;
vations for looking at the new time operator and the second mo- G.C. Hegerfeldt, J.G. Muga, J. Phys. A 43 (2010) 505303.
[16] V. Delgado, J.G. Muga, Phys. Rev. A 56 (1997) 3425.
tivation is the different question of exploring the consequences of
[17] I.L. Egusquiza, J.G. Muga, Phys. Rev. A 61 (2000) 012104.
taking an alternative but equivalent classical starting point, with- [18] R. de la Madrid, J. Isidro, Adv. Stud. Theor. Phys. 2 (2008) 281.
out necessarily making any claims about a connection to measure- [19] G. Bonneau, J. Faraut, G. Valent, Am. J. Phys. 69 (2001) 322.
ment models. In that case, our new operator shows that a different [20] F. Rellich, Math. Ann. 122 (1950) 343;
classical starting point leads to a probability distribution |ψ|φτ |2 T.E. Clark, R. Menioff, D.H. Sharp, Phys. Rev. D 22 (1980) 3012;
E. Farhi, S. Gutmann, Int. J. Mod. Phys. A 5 (1990) 3029;
which is a variant of the Kijowski distribution for large momenta.
N. Linden, M. Perry, Nucl. Phys. B 357 (1991) 289;
At small momenta, the alternative starting point yields improved J. Twamley, G.J. Milburn, New J. Phys. 8 (2006) 328.
behaviour in the eigenstates, since the states go like | p | instead [21] Y. Shikano, A. Hosoya, J. Math. Phys. 49 (2008) 052104.
1
of p 2 . This means that arrival time states obtained by superposing [22] J.J. Halliwell, Phys. Rev. D 48 (1993) 2739;
J.J. Halliwell, Phys. Rev. D 48 (1993) 4785.
small ranges of τ have finite ( x)2 , and hence better localization
[23] J.G. Muga, D. Seidel, G.C. Hegerfeldt, J. Chem. Phys. 122 (2005) 154106;
properties than the four standard results discussed in Section 2. So J. Echanobe, A. del Campo, J.G. Muga, Phys. Rev. A 77 (2008) 032112;
interpreted purely as an alternative classical starting point our new J.J. Halliwell, Phys. Rev. A 77 (2008) 062103.
operator gives useful and physically reasonable results. [24] J.M. Yearsley, D.A. Downs, J.J. Halliwell, A.K. Hashagen, Phys. Rev. A 84 (2011)
022109.
[25] J. Oppenheim, B. Reznik, W.G. Unruh, Phys. Rev. A 59 (1999) 1804.
8. Summary and conclusion
[26] J.J. Halliwell, J.M. Yearsley, Phys. Rev. A 79 (2009) 062101;
J.J. Halliwell, J.M. Yearsley, J. Phys. A, Math. Theor. 43 (2010) 445303.
In the construction of arrival time operators, different classical [27] Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, W.G. Unruh, Phys. Rev. A 57
starting points lead to inequivalent quantum operators with dif- (1998) 4130.
ferent physical predictions. The Aharonov–Bohm operator and its [28] C. Anastopoulos, K. Savvidou, J. Math. Phys. 47 (2006) 122106;
C. Anastopoulos, K. Savvidou, Phys. Rev. A 86 (2012) 012111.
variants do not capture the expected physical behaviour in the low
[29] N. Grot, C. Rovelli, R.S. Tate, Phys. Rev. A 54 (1996) 4676.
momentum regime, suggesting that a new starting point is called [30] B. Mielnik, G. Torres-Vega, Concepts Phys. 2 (2005) 81.
for. In this paper we constructed an arrival time operator taking a [31] D. Sokolovski, Phys. Rev. A 86 (2012) 012105.
different starting point based on the current, inspired by measure- [32] For a review of the dwell time operator see, for example, J. Munoz,
ment models and classically equivalent to −mx/| p |, thus obtaining I.L. Egusquiza, A. del Campo, D. Seidel, J.G. Muga, in Ref. [3], and references
therein. See also J.G. Muga, S. Brouard, R. Sala, Phys. Lett. A 167 (1992) 24;
an arrival time probability giving the expected physical behaviour
J. Munoz, D. Seidel, J.G. Muga, Phys. Rev. A 79 (2009) 012108.
in both large and small momentum regimes. In addition, we have [33] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover
shed some light on the intuitive origins of self-adjointness or its Publications, New York, 1972.
absence in both arrival times and the momentum operator on the [34] V. Delgado, Phys. Rev. A 57 (1998) 762.