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Marginal picture of quantum dynamics related to intrinsic arrival times
Gabino Torres-Vega*Physics Department, Cinvestav, Apartado Postal 14-740, 07000 México City, Distrito Federal, Mexico
�Received 29 November 2006; published 5 September 2007�
We introduce a marginal picture of the evolution of quantum systems, in which the representation vectors arethe quantities that evolve and operators and wave packets remain static. The representation vectors can be seenas probe functions that are the evolution of a � function with initial support on q=X in coordinate space. Thispicture of the dynamics is suited for the determination of intrinsic arrival distributions for quantum systems,providing a clear physical meaning to the “time eigenstates” used in these calculations. We also analyzeGalapon et al.’s “confined time eigenstates” �Phys. Rev. Lett. 93, 180406 �2004�� from this point of view, andpropose alternative probe functions for confined systems without the need of a quantized time.
DOI: 10.1103/PhysRevA.76.032105 PACS number�s�: 03.65.Db
I. INTRODUCTION
Pauli �1–3� pointed out long ago that there is no timeoperator canonical conjugate to a semibounded Hamiltonian,and Allcock �4–6� argued against the precise quantum-mechanical description of the time of arrival concept. How-ever, there are several proposals for the calculation of intrin-sic or operational arrival-time distributions, and also thereare proposals for time operators �7–29� and conjugate pairs�30�, and there are even analyses and criticisms of Pauli’sassertion �29,31�. The existing literature on this subject isvast, and a few references are at the end of this paper, in-cluding a broad review by Muga and Leavens �32�.
A well-known “time eigenstate” for free motion isKijowski’s state, which, in momentum representation, isgiven by �24�
�p�T,�� = ���p�eip2T/2m�� �p�m�
, �1.1�
where ���p� is the Heaviside step function, and �= ±1 forright and left movers, respectively. The squared magnitude ofthis state has as classical counterpart the flux that arrives atq=0, at time T. As was noticed by Baute et al. �12�, Kijow-ski’s “time eigenstates for free motion” �19� can be seen asthe backward evolution of the initial state ���p���p� /m�,which corresponds to an initial probability density���p��p� /m�. The arrival-time distribution is the squaredmodulus of the inner product between this state and a giveninitial wave packet ���, for which one is interested in deter-mining the arrival time distribution,
��T,�� = ��T,�����2. �1.2�
Baute et al. have also generalized this to arbitrary potentialsand positions �12�.
Leavens �33� has argued that the use of the factor ��p� canlead to some paradoxical effects, but his arguments wererefuted in �34�.
Operator-normalized construction of a time operator mea-suring the occurrence time of an effect gives similar resultsas using Kijowski’s states or Positive Operator Valued �POV�measures �35–37� but with a normalization at the level ofoperators �10,11�. Then, we note that Kijowski’s type ofarrival-time eigenstates appear, quite frequently, indepen-dently of the procedure used to obtain an arrival-time distri-bution, intrinsic or operational �14,20,38�. Therefore, wewould like to have a clear physical meaning of these timeeigenstates.
In this paper, we reinterpret and further develop the aboveideas with the introduction of an approach for the analysis ofthe dynamics that is useful, for instance, for the determina-tion of arrival distributions for quantum nonrelativistic sys-tems. We do not use generalizations of standard quantummechanics, but instead we make use of probe functions thatare propagated backward in time with the evolution operator,as we do with any other state. These “probe functions”sample an initial probability density and then predict arrivaldistributions. They are well-localized functions in coordinateor momentum space and are similar to “presence” and Ki-jowski’s time eigenstates. The classical analysis, found inRef. �39�, clarifies what our quantum probe function, andKijowski’s and others, do: they pick up the part of the initialprobability density that will arrive at q=X at the arrival timeT, and also shows that some of the difficulties found in quan-tum systems are also found in the classical case.
The approach used in this work does not need modifica-tion of the Hamiltonian �19,24�, nor generalizations of quan-tum mechanics �35,36�, nor quantization rules and symme-trization of multivalued nonlinear functions �40�, nor thefinding of conjugates to the Hamiltonian �19�, nor the quan-tization of time �29�, but, instead, it makes use only of thebackward evolution of the basis vectors of the coordinate �q�or momentum �p� representations. This is a different pictureof quantum mechanics in which the representation vectorsevolve in time and probability densities and operators do not.The probe functions are solutions of the evolution equationand then they are suited for answering time-relatedquestions.
We will consider nonrelativistic, one-dimensional sys-tems. We will also make use of the Husimi transform�41–43� in order to make a comparison between classical and*[email protected]
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quantum quantities and then elucidate their physical mean-ing.
In Sec. II, we use the ideas developed in Ref. �39� forclassical densities in the quantum case and we introduce arepresentation picture of the dynamics. We study its proper-ties and relate it to the calculation of arrival quantities.
In Sec. III, we analyze the confined arrival-time distribu-tion of Galapon et al. �25–27,29�. We find that Galapon etal.’s states could be undersampling some of the quantumstates. Then we propose alternative probe function for thesesystems.
At the end there are some concluding remarks.
II. QUANTUM PROBE FUNCTIONS
We develop here a picture of quantum dynamics suited forarrival-time-related questions. We recognize, analyze, anduse a representation picture of the dynamics of quantum sys-tems in which the representation vectors are the ones thatevolve, and state vectors and operators are static. Later, thispicture is applied to the determination of some arrival distri-butions.
A function that is concentrated at q=X and that covers,equally weighted, the momentum values is e−ipX/�, in themomentum representation. Then a probe function for arrivalat q=X at time T is
�Q�p;T;X� = eiTH�p,i��/�p�/�e−ipX/� = �p�eiTH/��X� �2.1�
in momentum space. In the above definition, �X� is an eigen-
function of the coordinate operator Q for the particular pointq=X. This function is the backward propagation of �X�, andis appropriate for determining presence distributions. Forarrival-time distributions we need the factor ��p� /m�, butthis factor can be included as will be shown below. Earlier,researchers considered only free propagation or, in an indi-rect way, propagation in other potentials �13�. Recently,
Baute et al. also introduced the backward propagator eiTH/�
for arbitrary potentials �12�.Actually, this type of probe state has been in use since the
early days of quantum mechanics, and constitutes anotherpicture of quantum dynamics which has not been recognizedas such before. Recall that the relationship between coordi-nate and momentum representations of a wave packet ��� is
��q;t� = −�
�
dp eipq/���p;t� = −�
�
dp�q�p��p�e−itH/����t = 0��
= ��Q�t;q����t = 0�� . �2.2�
Then, the coordinate representation of quantum mechanicscan be seen as the sampling of the initial wave function���t=0�� for arrival at q at time t. Contrary to theSchrödinger or Heisenberg or interaction picture of quantummechanics, here the basis vectors are the time-dependentquantities, and wave packets and operators are static. This isa complementary picture of quantum mechanics. The onlydifference from the arrival-time probe states that we use isthat we sample for only a specific point q=X. This allows usto assign a time value to parts of the state vector by finding
the overlap between the probe function and the state vector,with the origin of time assigned to the initial probe function.
The same is true for the momentum representation
��p;t� = ��P�t;p����t = 0�� , �2.3�
where �q ��P�t ; p��=eitH�−i��/�q,q�/�eipq/� is the probe functionfor arrival with momentum p at time t.
In what follows we separate into positive and negativemomentum parts �right and left movers� and derive someproperties of the probe functions. In quantum mechanics, thedistinction between right and left movers is necessarily anapproximated concept, because the requirement q=X is notexactly compatible with the requirement p0 �or p0�, andwe have to take some of the results with caution �44�. Theprobe functions that we will use are
��O± �T;X�� =
0
�
dp eiTH/�O�P,Q�� ± p��±p�X�
eiTH/�O±�P,Q��X� , �2.4�
��O�T;X�� = eiTH/�O�P,Q��X� �2.5�
= ��O− �T;X�� + ��O
+ �T;X�� , �2.6�
where O�P , Q� is an observable of interest and O±�P , Q��0
�dp O�P , Q��± p��±p�. We will also use the notation Ip±
�0�dp�± p��±p�. We should keep in mind that for left mov-
ers the point p=0 is excluded. In what follows, we will con-sider only the vectors for arrival at q=X, ��O
± �T ;X��, and wecan replace �X� by �P� in order to get formulas for the otherprobe functions, for arrival with momentum P. When
O�P , Q�=1, we will omit the subscript O from the probefunction.
Since the position basis vectors are orthogonal, the probestates are also orthogonal with respect to X, with the same T,
��X� − X� = �X��X� = �X��Ip+e−iTH/�eiTH/�Ip�
+ �X�
+ �X��Ip−e−iTH/�eiTH/�Ip�
− �X�
= ��−�T;X����−�T;X�� + ��+�T;X����+�T;X���2.7�
=���T;X�����T;X�� , �2.8�
and
��−�T;X����+�T;X�� = 0. �2.9�
These functions behave like a clock variable since if weapply the evolution operator for a time t to the probe states,we obtain again the probe states but for a shorter arrival time,
e−itH/���O± �T;X�� = ��O
± �T − t;X�� , �2.10�
e−itH/���O�T;X�� = ��O�T − t;X�� . �2.11�
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The inner products between probe states for different Tand the same X, calculated in the energy representation, are
���T�;X����T;X�� = ���0;X����T − T�;X��
= ��
Sp�H�dEei�T−T��E/� �X�E,�� 2
�2.12�
=�X�ei�T−T��H/��X� �2.13�
and
��−�T�;X���+�T;X�� = �X�Ip�− ei�T−T��H/�Ip
+�X� , �2.14�
where and �Sp�H�dE indicates summation over the discreteand continuous parts of the spectrum, and � takes into ac-count the possible degeneracy of the energy eigenvalues�twofold for free motion�. Then the probe states, for differentT and same X, are not orthogonal, unless �X �E ,�� is con-stant with the same value for all values of E and �.
In terms of energy eigenstates, we find that
��O± �T;X�� =
Sp�H�dEeiTE/�IEO±�P,Q��X� , �2.15�
��O�T;X�� = Sp�H�
dEeiTE/�IEO�P,Q��X� , �2.16�
where IE���E ,���E ,��. Then, the energy components ofthe time probe functions are
��O± �E;X��
−�
�
dTe−iET/�
2����O
± �T;X�� = IEO±�P,Q��X� ,
�2.17�
��O�E;X�� = ��O− �E;X�� + ��O
+ �E;X�� = IEO�P,Q��X� .
�2.18�
A difference from the states of Refs. �19� and �15� is thefactor of the type �E,�
* �q=X� in the integrand, a factor whichindicates that this is a probe state for arrival at q=X. Withthese factors, we are considering only a narrow region in q,otherwise we would be taking into account the whole realline. These expressions can be useful for approximating theprobe states in specific applications.
The presence probe states do not form a complete set
when summed over T, unless �X �E ,�� and �E ,��Ip�± �X� have
the same constant values for all X, E, and �,
−�
�
dT��±�T;X�����±�T;X��
= 2����
Sp�H�dE�E,���E,��Ip�
± �X���X�Ip±�E,���E,�� ,
�2.19�
−�
�
dT���T;X������T;X��
= 2����
Sp�H�dE�E,���E,��X���X�E,���E,�� .
�2.20�
However, these states indeed are complete if the summationis carried out over X,
−�
�
dX��±�T;X����±�T;X�� = eiTH/�Ip±e−iTH/�, �2.21�
and
−�
�
dX���T;X�����T;X��
= −�
�
dX���−�T;X����−�T;X�� + ��+�T;X����+�T;X��� = I .
�2.22�
There are two ways of calculating an average with theprobe functions when the system is evolving. The first optionis written in a symmetrical form and involves a single type of
probe function which depends on �O�P , Q�,
���T��O�P,Q����T�� = −�
�
dX ���O�T;X���� 2.
�2.23�
A second option is asymmetric in the probe functions,
���T��O�P,Q����T�� = −�
�
dX����O�T;X�����T;X����
�2.24�
=−�
�
dX�����T;X����O�T;X���� .
�2.25�
Arrival densities for O�P , Q�, when the quantum systemis in a state ���, are given by
�O± ���T�;X� = ���O
± �T;X���� 2 �2.26�
=��±�T;X����*��O± �T;X���� , �2.27�
�O���T�;X� = ���O�T;X���� 2 �2.28�
=���T;X����*��O�T;X���� . �2.29�
If the state vector is normalized, the above definitions are the
“densities” of the quantity O�P , Q� “present� at the arrival
time. If O�P , Q� is the velocity, the integral over a timeinterval is the average number of arrivals �in a given direc-tion� in this time interval. In some cases, as for free-particle
MARGINAL PICTURE OF QUANTUM DYNAMICS RELATED… PHYSICAL REVIEW A 76, 032105 �2007�
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motion, the above densities can be normalized, and then be-come a probability density. Thus, it is possible to recover adistribution like Kijowski’s arrival-time distribution from
�2.26� when O�P , Q�= �P� /m�.
The densities for arrival of O�P , Q� in the interval Z,when the system is in the state ���, are given by
��T�,O± �Z;X� =
Z
���T�,O± �T;X�dT , �2.30�
with Z a Borel set for �−� ,��.
Some applications
We now will consider some examples of the use of therepresentation picture introduced above.
In Fig. 1 there is a plot of the squared magnitude of thepresence probe functions �±�q ;T ;X� and of their sum forarrival at X=−1, for free propagation and in coordinatespace. For instance, the probe function for right movers hassupport mainly on q�X, except for T=0, when the functionis like a Dirac � function, with a 1/q term added to it, at q=X and constant for p0.
Let us use the Husimi transform of wave packets in orderto get a phase-space representation of the free-evolutionpresence probe functions and thus have a better understand-ing of what these probe functions are. The Husimi transformis the projection of the given quantum state onto the coherentstate set �41–43�. Here we use the projection �in dimension-less units�
��p,q� −�
�
dq�e−�q� − q�2/2�2−ip�q�−q/2���q�� . �2.31�
This function now depends on �= �p ,q� and can be thoughtof as a phase-space function. The squared magnitudes of theHusimi transforms of the quantum presence probe functions,for free evolution, are shown in Fig. 2. They provide a neat
classical-like picture of the quantum functions, allowing usto make a comparison with the classical probe functions.These are densities with a small support around the supportof the classical probe functions of Ref. �39�, i.e., around thelines q=X−Tp /m.
From the Husimi transform, we can see why these statesare not orthogonal for the same X and different T. Similarlyto what happens with classical densities, and contrary towhat was expected earlier �4,45,46�, the probe functions arenot orthogonal over T because they actually overlap at p=0,and at other points when there is recurrence in the dynamics.For instance, a particle with zero momentum and located atq=X does not have a well-defined time of arrival because theparticle will be at q=X for all time, a fact which makes thispoint belong to the probe state for all time. But if we con-sider probe functions with different X and for the same T,they will not overlap; they will scan the phase space as wevary the value of X.
In Fig. 3, there is a snapshot of the probe function
0.0
0.5
1.0
−8 0 8−16q
|ν+(q
;T;X
)|2
FIG. 3. Squared magnitude of the unnormalized presence probefunction �q ��+�T ;X�� for a potential barrier of height 12.5 betweenq=−5 and q=−4. In this figure, T=2 and X=0, in dimensionlessunits.
0.0
0.5
1.0
0.0
0.5
1.0
(a)
(b) (c)
−5 0 5 −5 0 5−10 10 −10 10qq
|ν−(q
;T;X
)|2|ν(q;T
;X)|2
|ν+(q
;T;X
)|2FIG. 1. Squared magnitude of the unnormalized presence probe
functions for a free particle in coordinate space. Plots for �a� allmovers, �b� left movers, and �c� right movers. For this calculationwe have set m=�=1, X=−1, and T=1. Dimensionless units.
−10
−5 −5
−5
0
0
0
5
5
5
10
−5
−10−10 10 −10 0 5 10
(a)
(b) (c)
p
p
FIG. 2. Density plots of the squared magnitude of the Husimitransforms of the presence probe functions of Fig. 1. Plots for �a� allmovers, �b� left movers, and �c� right movers. Darker regions indi-cate that the density there is larger than in lighter regions. Dimen-sionless units.
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�q ��+�T ;X�� for a particle experiencing a potential barrier ofheight 12.5 from q=−5 to q=−4. For that calculation, due tothe finiteness of numerical calculations, we have limited themomentum of the probe function to the range �0,6�. Thissystem can be used for discussing arrival-time distributionsfor tunneling through a barrier �47–50�. There is a part whichis being reflected by the barrier and which comes from thenegative momentum region. This function will sample thepart of a wave packet that will arrive at q=0 at time T=2.
In their discussion of quantum backflow, Bracken andMelloy �51� have used the wave packet
��p� = � 18�35K
p��e−p�/K − e−p�/2K/6� , p� p0,
0 p� p0,�
�2.32�
where p�= p+ p0, with p0 a boost, and K is a positive con-stant with dimensions of momentum. We have added a boostin order to not have a significant amount of probability withmomentum around zero. With this state, Bracken and Melloyhave shown that a wave packet which is made up of onlypositive momentum components can have, when it propa-gates freely, a negative flux for some time. They even gavean upper bound of 0.04 for the magnitude of the backflow forany state in free motion. For comparison with the classicalquantity calculated in Ref. �39�, in Fig. 4 there is a plot of thepresence distribution for free evolution, for the initial wavepacket of Eq. �2.32�.
III. CONFINED ARRIVAL-TIME STATES
In order to circumvent the non-self-adjointness of arrival-time operators, Galapon et al. �25–27,29� have introducedarrival-time eigenfunctions ��q�, −l�q� l, for confined sys-tems which satisfy the boundary condition ��−l�=e−2i���l�,����.
For instance, for the symmetric case �=0, the even timeeigenfunctions are given by
�s±�q� = Bse
�irsq2J3/4,1/4
� �q2rs� +4Bse
�irs
�4rs�1/4 J1/4�rs� , �3.1�
where Bs is the normalization constant, J�,�� �x�
=x��J−��x�� iJ��x��, J��x� is the Bessel function of the firstkind, and rs are the roots of the equation J−3/4�x�
+ �2/3�J5/4�x�+J1/4�x� /x=0, with s=1,2 , . . .. The corre-sponding eigenvalues are n
±= ± ��l2 /4�rs�, with � the parti-cle’s mass.
The first four confined arrival-time eigenfunctions in mo-mentum space, for ��l�=0, can be seen in Fig. 5. They in-clude left and right movers, they mainly sample at the edgesof the system, at q= ± l, and they also give more weight tolarge momentum regions. However, these functions do notsample all the range of momentum values in the same way;there are regions which are poorly sampled and regionswhich are oversampled. Each of the time eigenstates in-volves many momentum eigenvalues, and the time eigenval-ues do not correspond to the times associated with the mo-mentum eigenvalues, �l2 /k�.
The common eigenfunctions of momentum and energyoperators, in coordinate representation, are
�k����q� =
1�2l
ei��+k��q/l, �3.2�
with eigenvalues pk,�= ��+k�� / l, Ek= pk,�2 /2�, k=0,
±1, ±2, . . ..Then, according to Eq. �2.15�, we can get probe functions
in coordinate space, for determining the arrival-time distri-bution, by summing up the energy eigenfunctions �3.2� times��p� /�:
���P�/��q;T;X� = ���P�/�− �q;T;X� + ���P�/�
+ �q;T;X� , �3.3�
where
���P�/�± �q;T;X� = �
k�−�/�,k−�/�
��� + k��2l��
� eiT�� + k��2/2�l2ei��+k���q−X�/l. �3.4�
For right �left� movers, the summation is carried out with k�−� /� �k−� /��. These functions will sample exactly themomentum values involved in the dynamics of the system
1.0
0.0
0.5
2 40T
|〈ν+(T
;X)|ψ
〉|2
FIG. 4. Plot of the unnormalized quantum, free-evolution, pres-ence distribution for the state Eq. �2.32�, with X=5, p0=1, and K=1. Dimensionless units.
−30 0 30 −30 0 30−60 60 −60 60
0.5
0.5
0.0
1.0
0.0
1.0
pp
|φ+ 1(p
)|2
|φ+ 2(p
)|2
|φ+ 3(p
)|2
|φ+ 4(p
)|2
FIG. 5. Squared magnitude of the unnormalized first four Gal-apon et al. time eigenfunctions for the symmetric case �=0 and��−l�=��l�=0, in momentum space. Dimensionless units.
MARGINAL PICTURE OF QUANTUM DYNAMICS RELATED… PHYSICAL REVIEW A 76, 032105 �2007�
032105-5
and do not need a discretization of time because they start abackward evolution aligned at q=X.
For numerical calculations, an approximation to the aboveprobe functions is to truncate the infinite sum. We havefound that with 30 terms we obtain a function with a thinenough support in q. In Fig. 6 we show plots of the approxi-mated initial probe states for right and left movers. If theparticular situation under study involves only the first 30 orfewer values of momentum, this approximated function hasthe properties needed for a probe function. Increasing thenumber of terms in the summation will increase the accuracyof the calculation.
Then the probe functions introduced in this paper are alsouseful when the spectrum is discrete without the need of aquantization of time.
Remarks
It is desirable to learn how to deal with the time variablein many physical theories. In this paper, we have introduceda marginal picture of quantum dynamics which is differentfrom the Schrödinger, Heisenberg, or interaction pictures,
and is intended to answer questions regarding the arrival ofdynamical quantities at some point q=X. In fact, this is apicture of quantum dynamics which has not been recognizedas such before; a picture in which the representation vectorsmove and the rest, namely, the state vectors and operators,remain static.
We can make contact with other works. The probe func-
tions ��O�T ;X��, with O�P , Q�=��P� /m, were mentioned byBaute et al. in Ref. �12�, the “crossing states,” as a generali-zation, for arbitrary potentials, of Kijowski’s time eigenstatesfor free motion. For X=0, �±p �X�=1, and for free motion
�E , ± � and �p± � coincide. In this case, with O�P , Q�=1, theprobe states become the arrival-time eigenstates �t , ± � ofDelgado and Muga �19�.
Our arrival probe functions differ from those of Skuli-mowski �15�,
�S�t�,�,X� =e−iXP
�2�e−iHt
Sp�H�dE�E,�� , �3.5�
and Delgado and Muga �19� in that the arrival point q=X is
considered inside the summation over eigenfunctions of Hwith the extra factor �X �E�*. This additional factor enables asound physical interpretation of the probe functions as back-ward time fronts with a classical analog �39�.
Probe functions represent the motion of a given system,taking a clock as a reference for how much the system haschanged. The classical picture for the probe functions is thatof the propagation of the line q=X in phase space. The back-ward propagation of that line, labeled by T, is taken as thesupport for probe functions that can sample an initial prob-ability density and predict, then, the amount of probabilitythat will arrive at X.
With the probe functions, it is possible to find out whatpart of a quantum final wave packet corresponds to a givenpart of the initial one. This is not like classical trajectories forsingle points in phase space, but it is close to that; we cantake a probe function concentrated around the line q=X andfind the amount of density that is coming from another re-gion of q at time T.
In this and in �39�, we have introduced pictures for clas-sical and quantum systems which allow us to treat classicaland quantum arrival in a similar way, but they are marginalstates since the momentum has been summed up. In �44� weuse this picture of evolution, without the summation over p,for the description of classical and quantal evolution in anenergy-time space.
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(a)
(c) (d)
(b)
0 50−50−100 100
1
2
0
3
0.0 0.5−1.0 1.0−0.5
0.5
1.0
0.00.0−0.5−1.0 0.5 1.0
0 50−50−100 100
pp
|ν− √|P
|/µ(q
;0;0
)|2
|ν+ √ |P|/µ
(q;0
;0)|2
|ν− √|P
|/µ(p
;0;0
)|2
|ν+ √ |P|/µ
(p;0
;0)|2
FIG. 6. Approximated unnormalized arrival probe functions Eq.�3.4� for Galapon’s et. al. confined particle, with �=T=X=0, incoordinate and momentum representations. We have consideredonly 30 terms of the infinite sums. �a� Left and �b� right movers incoordinate space. Momentum representation for �c� left and �d�right movers. Dimensionless units.
GABINO TORRES-VEGA PHYSICAL REVIEW A 76, 032105 �2007�
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