Angular time delay in quantum mechanical scattering Jochen Zahn Institut für Theoretische Physik Universität Leipzig

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Angular time delay in quantum mechanical scattering

Jochen Zahn

Institut für Theoretische Physik, Universität Leipzig

Brüderstr. 16, 04103 Leipzig, Germany

jochen.zahn@itp.uni-leipzig.de

July 18, 2023

Abstract

We apply Brunetti and Fredenhagen’s concept of the time of occurrence of an event in

quantum mechanics [1] to the example of scattering oﬀ a spherical potential. Thereby, we

re-derive the expression of Froissart, Goldberger, and Watson for the angular time delay

[2], clarifying some conceptual issues with their derivation. We also present an elementary

re-derivation of the “space shift” (essentially the impact parameter) deﬁned in the quantum

mechanical context by the same authors. We clarify the relation of both quantities to their

classical counterparts in the context of the WKB approximation. As an example, we apply

the concepts to scattering at a hard sphere. We ﬁnd pronounced peaks in the both the time

delay and the space shift at the minima of intensity in the forward diﬀraction region for

short wavelength scattering and discuss whether these could in principle be observable.

1 Introduction

The problem of time observables in quantum mechanics is a topic nearly as old as quantum

mechanics itself, cf. the Introduction of [3] for a historical overview. In the traditional formulation

of quantum mechanics, time is not an observable, but a parameter. In other words, the formalism

answers questions like: “When measuring observable Aat time ton a system prepared in the state

Ψat time t0, what is the probability of measuring the value a?” However, not all experimental

situations are of this type. This is the case in particular for scattering experiments, where the

outgoing particles are registered by detectors which are constantly turned on. In such a situation,

the question “When does the detector click?” [1] seems to be meaningful and experimentally

accessible, at least in principle. It is thus probably no coincidence that the ﬁrst deﬁnition of a

quantity closely related to a time observable was given in the context of scattering, namely the

Eisenbud-Wigner time delay [4]

tdelay,ℓ = 2∂Eδℓ(E)(1)

with δℓthe scattering phase of the ℓth partial wave.1The relation (1) is derived in [5] using

stationary phase arguments.

It is important to note that the physical interpretation of the time delay of a partial wave is

a priori unclear, as an incoming wave consists of a superposition of partial waves and a typical

detector discriminates between diﬀerent scattering angles, not angular momenta (it can however

be justiﬁed in a high energy approximation, see Section 6below). However, the relation (1)

is straightforwardly adjustable to the case of one-dimensional scattering at a potential barrier,

where 2δℓshould be replaced by the phase of the transmission amplitude. In that case, the

1Throughout, we are using units in which ℏ= 1.

arXiv:2210.13018v2 [quant-ph] 17 Jul 2023

interpretation is clear: It is the time delay w.r.t. the situation in the absence of the potential.

Furthermore, as the derivation is based on stationary phase arguments, it should be interpreted

as an expectation value.

For the case of three-dimensional scattering, a time delay for the full scattering process, not

just partial waves, was given by Goldberger, Froissart, and Watson [2]:2

tdelay(θ) = ∂Earg A(E, θ).(2)

Here A(E, θ)is the scattering amplitude at energy Eand angle θ. This should be interpreted

as the time delay as measured by the detector at scattering angle θ. Again, this expression is

derived using stationary phase arguments.

Driven by the availability of attosecond light pulses, the topic of time delay in photoemission

has attracted considerable attention in recent years, cf. [8,9] for example. Also its angular

dependence has been studied extensively in recent years, both theoretically and experimentally,

see [10,11,12,13,14], for example. The angular time delay (2) was also suggested as a tool to

identify resonances in reactive molecular collisions [15]. Recently, the partial wave time delay

(1) was used to discuss causality (violation) in gravitational eﬀective theories [16].

The interpretation of (2) as a time delay has been critized [17] on two grounds: First, the

derivation based on the stationary phase arguments does not take into account the reshaping of

the wave packet due to scattering. Second, the expression can not be interpreted as a time delay,

because in the absence of a scattering potential, no scattering to angle θoccurs, so it is unclear

w.r.t. which reference process it is deﬁned. The main objective of this paper is to argue that

these objections can be refuted.3We use a formalism proposed by Brunetti and Fredenhagen [1]

for the deﬁnition of positive operator valued measures describing the distribution of occurrence

times of general “eﬀects”, described by projectors (or, more generally, positive operators). This

formalism provides an idealized description of the measurement, without the need to introduce

a dissipative mechanism. The determination of the full distribution of occurrence times requires

the knowledge of the wave function ψ(E), which can in general not be assumed. However, if

either a reference eﬀect or a reference dynamics is available, then the diﬀerence of the ﬁrst

moments of the distribution, i.e., the time delay, only depends on |ψ(E)|2, i.e., the relative

phases between the ψ(E)at diﬀerent energies Ebecome irrelevant. In particular, for ψ(E)

peaked at some energy E, it suﬃces to compute the time delay at this energy. No stationary

phase arguments are needed.

Regarding the second objection (absence of a reference process), two approaches are possible.

One could take the detector at a reference angle θ0as a reference eﬀect, in which case one obtains

tθ0

delay(θ) = tdelay(θ)−tdelay(θ0),(3)

where the expressions on the r.h.s. are deﬁned by (2). The second option is to use a reference

dynamics, in which case a natural choice is scattering at a point scatterer, i.e., a hard sphere of

radius Rin the limit R→0.4With this reference dynamics, one obtains the time delay (2), as

shown below.

In [2], also an expression for the “space shift”, i.e., the displacement of the outgoing particle

from the axis of symmetry (essentially the impact parameter), is given. In fact, this expression is

derived alongside (2), i.e., using wave packets and stationary phase arguments. We will present

2See also [6,7], which contained the deﬁnition somewhat implicitly, and without derivation.

3We note that in the alternative treatment proposed in [17], which is based on the dwell time of the particle

inside a ball around the scatterer, the time delay (2) also occurs, in an integral over the angles, weighted with

the diﬀerential cross section. This also suggests the interpretation of (2) as the time delay at scattering angle θ,

but does not prove it.

4This was, somewhat implicitly, also the reference process used in [2].

a considerably simpler re-derivation of this result, which evades the use of wave packets and also

indicates how this quantity can in principle be measured.

As both the angular time delay and the space shift have counterparts in classical scattering,

it is important to clarify the relation of the quantum mechanical concepts to the classical ones.

We show that in the WKB approximation, both quantities assume their classical values, at least

when classically only a single impact parameter contributes to scattering at a given angle θ.

When evaluating angular time delay and space shift for scattering at a hard sphere, we ﬁnd,

in the long wavelength regime, essentially angle independent time delay and phase shift, as

expected for s wave dominated scattering. At short wavelengths, the quantum results approach

the classical ones, but there are pronounced oscillations both of time delay and phase shift in

the forward diﬀraction region.

The article is structured as follows: In the next section, we describe in detail the formalism

of Brunetti and Fredenhagen [1]. In particular, we show that in the limit of an inﬁnitesimally

thin detector, one recovers Kijowski’s distribution [18]. In Section 3, we use the framework to

discuss scattering in one dimension. In particular, we exemplify the two possible ways to deﬁne

a time delay, with reﬂection time delay (via a reference eﬀect) and transmission time delay (via a

reference dynamics). The case of scattering at a spherical potential is discussed in Section 4. In

particular, we recover the expression (2) for the time delay of scattering at scattering angle θ. In

Section 5we present an elementary derivation (and provide an interpretation) of the “space shift”.

In Section 6, we show that in the WKB approximation, both the time delay and the space shift

reduce to their classical counterparts, provided that classically only a single impact parameter

contributes to scattering at angle θ. In Section 7, we apply the formalism to scattering at a hard

sphere and qualitatively discuss the results. We conclude with a summary and an outlook.

2 Time of occurrence of an event

In quantum mechanics, one describes a detector, i.e., a measuring device answering yes-no

questions, by a projector P. The prototypical example is a projector PV|x⟩=χV(x)|x⟩on

a region Vin space, describing a detector sensitive to a particle in that region (more general

detectors also sensitive to the direction of the momentum of the particle will be discussed in

the concrete examples in the next two sections). One may then ask: “At what time does the

detector click?”. A formalism to answer this question without the need to model the speciﬁcs

of the detector (for example by some dissipative mechanism, cf. [19,20] for reviews) has been

proposed by Brunetti and Fredenhagen [1]. Let Pt=eiHtP e−iHt be the time-evolution of P

with the Hamilton operator H(we are working in the Heisenberg picture). One interprets

P(R) := ZR

Ptdt(4)

as the observable for the total time the particle spends in the detector. Its expectation value

coincides with the notion of dwell time, cf. [21] and the review [22], with the diﬀerence that we

are here concerned with the dwell time inside the detector, not inside some region containing

the scatterer. The Hilbert space now decomposes in the direct sum

H=Hﬁn ⊕ H0⊕ H∞,(5)

where states in Hﬁn have a ﬁnite expectation value for P(R), whereas it vanishes (diverges) for

states in H0(H∞). Elements of H0are states for which the eﬀect never takes place. Examples

for states in H∞are bound states. A probability distribution for the time of the occurrence of the

event described by Pcan be deﬁned for states in Hﬁn. Namely, one deﬁnes the positive-operator

valued measure (POVM)

PP(I) := P(R)−1

2ZI

PtdtP(R)−1

2(6)

on Hﬁn for the distribution of arrival times. Here Iis a time interval (or a union thereof). The

ﬁrst moment of this measure, i.e.,

TP:= ZR

tPP(dt),(7)

is then an operator such that ⟨Ψ|TP|Ψ⟩is the expectation value (in the statistical sense) for the

time of occurrence of the event Pin the state |Ψ⟩∈Hﬁn. One may also relax the requirement

of Pbeing a projector and allow for Pbeing a positive operator.

We emphasize that for any Ψ∈ Hﬁn, (6) provides the full distribution of click times, not

just an expectation value. As we will see below, in the case of an inﬁnitesimally thin detector

and when restricting to positive momenta in the direction normal to the detector, one recovers

Kijowski’s distribution [18]. In many cases of practical interest, the state Ψis not fully known,

but only partial information is available, such as being peaked at some energy E. Then, the

formalism may still be predictive if we restrict to the expectation value TP, or rather diﬀerences

of such expectation values, i.e., time delays.

In the following, we will work on a subspace Hscat ⊂ Hﬁn of scattering states |Ψ⟩which are

fully characterized by a wave function ψ(E)depending only on the energy, i.e.,

|Ψ⟩=Zψ(E)|E⟩dE, (8)

with {|E⟩} providing a basis of eigenstates of energy E. For the matrix elements of Ptin this

basis, we have

⟨E′|Pt|E⟩=e−i(E−E′)tP(E′, E),(9)

with P(E′, E)the matrix elements of P. It follows that the matrix elements of P(R)are given

P(R)(E′, E)=2πδ(E−E′)P(E′, E),(10)

so that

PP(I)(E′, E) = 1

2πZI

e−i(E−E′)tcP(E′, E)dt, (11)

where we have introduced the integral kernel

cP(E′, E) := P(E′, E′)−1

2P(E′, E)P(E, E)−1

2,(12)

which fulﬁlls the normalization condition

cP(E, E) = 1.(13)

Let us apply this in a concrete example to verify that for a particular choice of the eﬀect P,

we recover Kijowski’s distribution [18]. We restrict to the one-dimensional case and, as Kijowski,

positive momenta. The generalized energy eigenstates |E⟩are then given by

⟨x|E⟩=rm

2πk eikx (14)

in position space, with

k=√2mE. (15)

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AngulartimedelayinquantummechanicalscatteringJochenZahnInstitutfürTheoretischePhysik,UniversitätLeipzigBrüderstr.16,04103Leipzig,Germanyjochen.zahn@itp.uni-leipzig.deJuly18,2023AbstractWeapplyBrunettiandFredenhagen’sconceptofthetimeofoccurrenceofaneventinquantummechanics[1]totheexampleofscatteringof...

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Angular time delay in quantum mechanical scattering Jochen Zahn Institut für Theoretische Physik Universität Leipzig

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