time in quantum mechanics

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Time in Quantum Mechanics

Curt A. Moyer

Department of Physics and Physical Oceanography, UNC Wilmington

Abstract

The failure of conventional quantum theory to recognize time as an observable and to admit

time operators is addressed. Instead of focusing on the existence of a time operator for a given

Hamiltonian, we emphasize the role of the Hamiltonian as the generator of translations in time

to construct time states. Taken together, these states constitute what we call a timeline, or

quantum history, that is adequate for the representation of any physical state of the system. Such

timelines appear to exist even for the semi-bounded and discrete Hamiltonian systems ruled out

by Paulis theorem. However, the step from a timeline to a valid time operator requires additional

assumptions that are not always met. Still, this approach illuminates the crucial issue surrounding

the construction of time operators, and establishes quantum histories as legitimate alternatives to

the familiar coordinate and momentum bases of standard quantum theory.

PACS numbers: 03.65.Ca, 03.65.Ta

Electronic address: [email protected]

1

arXiv:1305.55

25v1[quant-ph]2

3May2013
mailto:[email protected]:[email protected]


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I. INTRODUCTION

The treatment of time in quantum mechanics is one of the challenging open questions

in the foundations of quantum theory. On the one hand, time is the parameter entering

Schrodingers equation and measured by an external laboratory clock. But time also plays

the role of an observable in questions involving the occurrence of an event (e.g. when a nu-

cleon decays, or when a particle emerges from a potential barrier), and like every observable

should be represented in the theory by an operator whose properties are predictors of the

outcome of [event] time measurements made on physical systems. Yet no time operators

occur in ordinary quantum mechanics. At its core, this is the quantum time problem. As

further testimony to this conundrum, the uncertainty principle for time/energy is known to

have a different character than does the uncertainty principle for space/momentum.An important landmark in the historical development of the subject is an early theorem

due to Pauli [1]. Paulis argument essentially precludes the existence of a self-adjoint time

operator for systems where the spectrum of the Hamiltonian is bounded, semi-bounded,

or discrete, i.e., for most systems of physical interest. Pauli concluded that . . . the intro-

duction of an operator T [time operator] must fundamentally be abandoned. . . . Whilecounterexamples to Paulis theorem are known, his assertion remains largely unquestioned

and continues to be a major influence shaping much of the present work in this area. For a

comprehensive, up-to-date review of this and related topics, see [2],[3].

In this paper we advocate a different approach, one that emphasizes the statistical dis-

tribution of event times not the time operator as the primary construct. Essentially,

we follow the program that regards time as a POVM (positive, operator-valued measure)

observable [4]. Event time distributions are calculated in the usual way from wave functions

in the time basis. We show that this time basis exists even for the semi-bounded and discrete

Hamiltonian systems ruled out by Paulis theorem, and is adequate for the representation

of any physical state. However, the step from a time basis to a valid time operator requires

additional assumptions that are not always met. Still, this approach illuminates the cru-

cial issue surrounding the construction of time operators and, at the same time (no pun

intended), establishes the time basis as a legitimate alternative to the familiar coordinate

and momentum bases of standard quantum theory.

The plan of the paper is as follows: In Sec. II we introduce the time basis and establish

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the essential properties of its elements (the time states) for virtually any physical system.

Sec. III explores the relationship between time states and time operators, and establishes

existence criteria for the latter. In Secs. IV and V we show how some familiar results for

specific systems can be recovered from the general theory advanced here, and in Sec. VI we

obtain new results applicable to the free particle in a three-dimensional space. Results and

conclusions are reported in Sec. VII. Throughout we adopt natural units in which = 1, a

choice that leads to improved transparency by simplifing numerous expressions.

II. QUANTUM HISTORIES: A NOVEL BASIS SET

We introduce basis states in the Hilbert space | labeled by a real variable that we

will call system time. For a quantum system described by the Hamiltonian operator H,these states are defined by the requirement that H be the generator of translations amongthem. In particular, for any system time and every real number we require

| + = expiH | (1)

Since H is assumed to be Hermitian, the transformation from | to | + will be unitary(and therefore norm-preserving). Eq.(1) shows that if | is a member of this set then so

too is | + , implying that system time extends continuously from the remote past( = ) to the distant future ( = +). We refer to the set {| : } as atimeline or quantum history.

Eq.(1) is reminiscent of the propagation of quantum states, which evolve according to

| (t) = expiHt | (0) . It follows that the dynamical wave function in the time basis,

| (t) , obeys | (t) = t | (0) , (2)

a property known in the quantum-time literature as covariance. To appreciate its signif-icance, recall that | | (t) |2 is essentially the probability that a given [system] time will be associated with some measurement after a [laboratory] time t has passed; covariance

ensures that the same probability will be obtained for the initial state at the earlier [system]

time t. This is time-translation invariance, widely recognized as an essential feature thatmust be reproduced by any statistical distribution of time observables [5].

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We advance the conjecture that a quantum history exists for every system, with elements

(the time states) sufficiently numerous to span the Hilbert space of physical states. The

completeness of this basis is expressed in the abstract by the following resolution of the

identity:

1 =

| | d (3)

Specifically, we insist that for any two normalizable states | and | , we must be able towrite

| =

| | d (4)

The time states also are orthogonal, at least in a weak sense consistent with closure. More

precisely, if | in Eq.(4) can be replaced by a time state, we get

| =

| | d (5)

Eq.(5) expresses quantitatively the notion of weak orthogonality; it differs from standard

usage (strong orthogonality) by uniquely specifying the domain of integration and restrict-

ing | to be a normalizable state. But owing to the continuous nature of time, the timestates | are not normalizable, so Eq.(4) can be satisfied even when Eq.(5) is not. It fol-lows that weak orthogonality is a stronger condition than closure, and subject to separate

verification.Quantum histories are intimately related to spectral structure, and derivable from it.

Indeed, we regard as axiomatic the premise that the eigenstates of H, say | E, also spanthe space of physically realizable states to form the spectral basis {| E : E}. Since H isby assumption Hermitian, the elements of the spectral basis can always be made mutually

orthogonal (in the strong sense), and normalized so as to satisfy a closure rule akin to

Eq.(3). Now Eq.(1) dictates that the timelinespectral transformation is characterized by

functions

|

E

such that

+ |

E

= exp(iE)

|

E; equivalently (with

),

exp(iE) | E = cE (6)

This form is imposed by covariance. The constant on the right, while unspecified, is mani-

festly independent of . This enables us to write

| E

cEexp(iE) E| (7)

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The sum in this equation is symbolic, translating into an ordinary sum over any discrete levels

together with an integral over the continuum. A timeline exists if for every physical state

| the transform of Eq.(7) maps the set { E| : E} into a square-integrable function

|

on

< 0 for every j (the only vector orthogonal to all is the null element). Closureequivalence with the spectral basis further requires all coefficients cj to have unit magnitude;

since any phase can be absorbed into the stationary state | Ej , we will simply take cj = 1.The time basis {| p : p = 1, 2, . . . N } has discrete elements, yet supposedly is a contin-

uous variable. We resolve this apparent contradiction by noting that the argument leadingto this discrete time basis is unaffected if N is scaled by any integer and is reduced by

the same factor (leaving N = rev unchanged). Thus, every set of time states| (n)p : p = 1, 2, . . . n N : (n) = rev/nN, n = 1, 2, . . . (9)is complete in this N-dimensional space, but all except the primitive one (n = 1) is

overcomplete. Nonetheless, this observation allows us to write

Nj=1

| Ej Ej | = 1rev

0+rev=0

| (n)p (n)p | (n) 1rev 0+rev

0

| | d (10)

(Passage to the continuum limit presumes that has no granularity, even on the smallest

scale.) Scaling the time states by

1/rev then results in an indenumerable time basis that

enjoys closure equivalence with the spectral basis in this N-dimensional space. Explicitly,

the squared norm of any element | can be expressed as

|

=

N

j=1 | Ej | |2 =

0+rev

0 |

|

|2 d (11)

with

| = 1rev

Nj=1

exp(iEj) Ej | (12)

While it could be argued that N always has an upper bound in practice, it is typically

quite large and known only imprecisely. Convergence of these results as N thereforeis crucial to a viable theory of timelines. Now for every N no matter how large, Eq.(10)

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establishes closure equivalence of the time basis with the spectral basis, and leads directly

to the Plancherel identity [7] expressed by Eq.(11). But by definition, the norm of a nor-

malizable state remains finite even in the limit N , so the Plancherel identity ensuresthe existence of the square-integrable function

|

on [0, 0 + rev ] in this same limit.

We conclude that Eq.(12) maps the spectral components Ej | into square-integrablefunctions | in the time basis, as required for a timeline. This establishes convergence[in the norm] for the timeline wave function | .

Our final task is to write the transformation law of Eq.(12) in a form that is useful for

calculation in the large N limit. The way we proceed depends on how the recurrence time

varies with N. As more states are included in the model, all levels might remain isolated

no matter how numerous they become (rev saturates at a finite value); alternatively, some

levels may cluster to form a quasi-continuum (rev ), merging into a true continuumas N . Anticipating a mix of the two, we write the quasi-continuum contributions to | in terms of the characteristic energy E 2/rev associated with the recurrencetime rev :

1rev

continuum

(. . .) =12

continuum

exp(iEj)

Ej | E

E

The significance of the bracketed term can be appreciated by comparing discrete and con-

tinuum contributions to the squared norm of the state

|

:

| =

discrete

| Ej | |2 +

continuum

Ej | E2 E

The replacement Ej |

E Ej |

amounts to a renormalization of the quasi-continuum wave function known as energy

normalization [8] such that the [energy-] integrated density of the new function carries

the same weight as does an isolated state. Replacing sums over the quasi-continuum withintegrals becomes exact in the large N limit (with the inclusion of additional levels, rev and E 0). In this way, we arrive at a form of the transformation law that lends itselfto computation as N :

| = 1rev

discrete

exp(iEj) Ej | + 12

EmaxEmin

exp(iE) E| dE (13)

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Eq.(13) assumes that all discrete stationary states are normalized to unity ( Ej | Ek = jk),and all [quasi-]continuum elements are energy-normalized ( E| E = (E E)). Noticethat the continuum contribution references N only indirectly through the spectral bounds

Emin, Emax [9]. Furthermore, whenever a continuum is present (aperiodic systems, for which

rev ), it makes the dominant contribution to | for any fixed value of . To befair, there is one limitation: cannot be so large that exp (iE) varies appreciably over the

step size E = 2/rev ; this means the integral approximation to the quasi-continuum fails

for rev . In a similar vein, the discrete terms can never simply be dropped from Eq.(13),as they make the largest contribution to | in the asymptotic regime.

Although complete, the time basis typically includes non-orthogonal elements. In the

absence of a continuum, and with uniform level spacing for the discrete terms, it is not

difficult to show that the minimal time basis is composed of N mutually orthogonal states.

But otherwise the existence of even one orthogonal pair is not guaranteed. By contrast,

weak orthogonality is the rule in this N-dimensional space. This can be verified rigorously

by using closure of the [discrete] time basis in Eq.(9) to write

(n)q | =1

rev

0+rev=0

(n)q | (n)p (n)p | (n)

Scaling the time states by 1/rev and passing to the continuum limit then gives | =

0+rev0

| | d, (14)

with | again calculated from Eq.(13). Since N is not referenced explicitly here, weconclude from Eq.(14) that weak orthogonality persists in the limit as N at everyvalue where the timeline wave function | converges.

B. The Treatment of Degeneracy

Degenerate states require labels in addition to the energy to distinguish them. These

extra labels derive from the underlying symmetry that is the root of all degeneracy. Thus,

a central potential gives rise to a rotationally-invariant Hamiltonian and this, in turn, im-

plies that the Hamiltonian operator commutes with angular momentum (the generator of

rotations). In such cases, the energy label is supplemented with orbital and magnetic quan-

tum numbers specifying the particle angular momentum. The larger point is simply this:

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with the additional labels comes again an unambiguous identification of the spectral states,

and we write | Ej | Ej , , where is a collective label that symbolizes all additionalquantum numbers needed to identify the spectral basis element with a given energy. With

this simple modification, the arguments of the preceding section remain intact. Quantum

histories can be constructed as before, but now are indexed by the same good quantum

numbers that characterize the spectrum. That is, in the face of degeneracy we have not one

but multiple timelines, and we write | | , . Notice that any two time statesbelonging to distinct timelines will be orthogonal (in the strong sense)

, | , , , (15)

and of course all timelines must be included to span the entire Hilbert space, so that

Eq.(3) becomes1 =

| , , | d (16)

Oftentimes there is more than one way to select good quantum numbers. Following

along with our earlier example, states having different magnetic quantum numbers may be

combined to describe orbitals with highly directional characteristics that are important in

chemical bonding. The crucial point here is that the new states are related to the old by a

unitary transformation (a rotation) in the subspace spanned by the degenerate states. In

turn, the rotated stationary states gives rise to new time states, related to the old as

| (r) =

Ur| , , (17)

where Ur are elements of the same unitary matrix that characterizes the subspace rota-

tion (cf. Eq.(7)). Furthermore, unitarity ensures that the projector onto every degenerate

subspace is representation-independent:

r | (r)

(r)

|= , r UrU

r | , ,

|=,

UU

| , , | =

| , , | (18)

As we shall soon see, Eq.(18) has important ramifications for the statistics of [event] time

observables whenever degeneracy is present.

One final observation: degenerate or not, the timelines of Eq.(13) are not unique, inas-

much as they can be altered by an (energy-dependent) phase adjustment to the eigenstates

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of H. Consider the replacement | E exp(iE) | E. If E is proportional to E, sayE = 0E, then | | 0 , i.e., a linear (with energy) phase adjustment to the sta-tionary states shifts the origin of system time, underscoring the notion that only durations

in system time can have measureable consequences. Other, more complicated phase adjust-

ment schemes can be contemplated, with the time states always related by a suitable unitary

transformation. Some of these have clear physical significance, as later examples will show.

III. INTERPRETING TIME STATES, AND AN OPERATOR FOR LOCAL TIME

The time states of Sec. II can be used to formally construct a time operator; for non-

degenerate spectra,

T = 0+rev0

| | d (19)

(Degeneracy requires the replacement | | | , , | in this and subsequentexpressions.) At a minimum, the existence of T demands that matrix elements of Eq.(19)taken between any two normalizable states | and | be well-defined, i.e.,

|T | 0, and vanishes as || in the entire sector 0 arg() . Inturn, these properties of

|

in the complex plane ensure that

E

|

calculated from

Eq.(35) is truly zero for all negative values of E (follows from applying the residue calculus

to a contour of integration consisting of the real axis closed by an infinite semicircle in the

upper half-plane).

A. Free-Particle Timelines

We begin by taking the degenerate eigenfunctions to be plane waves, writing | E | k

with x | k = Ck exp(ikx). These are harmonic oscillations with wavenumber k and energyEk = k

2/2m. Orthogonality of these waves is expressed by

k | k = CkCk

dx exp(ikx ikx)

= C2k2 (k k) = C2k2 |k|

m(Ek Ek) ,

so that energy normalization in this case requires

Ck = m2 |k| (36)Plane waves running in opposite directions (k) give rise to distinct quantum histories,

which we distinguish by the direction of wave propagation: | | ,. Timeline elementsin this representation are described by the Schrodinger wave functions x | , (x),obtained by taking | = | x in Eq.(35):

(x) =12

0

exp(iEk) Ck exp(ikx) dEk

= 12

m

0

dk k exp ikx ik2 /2m (37)In this and subsequent expressions, the right (left) arrow is associated with the upper (lower)

sign. The integral of Eq.(37) is related to the parabolic cylinder function D(. . .); in partic-

ular, we have for m < 0 (e ( i) > 0) [18]

(x) =1

4

mz3/2 exp

x2z2/4D3/2 (ixz) z mi

(38)

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This form holds for |arg z| < /4, but since D is an entire function of its argument [19]the result can be analytically continued to all real values of (arg z = /4). Clearly, (x) = (x). For real and negative we take arg z = /4 in Eq.(38) to obtain

(x) =1

4m |z|3/2

exp ix2 |z|2 /4 + i3/8D3/2 (ix |z| exp (+i/4))=

(x)

, (39)

a relation that also is evident from the integral representation, Eq.(37). These results are

consistent with the pioneering 1974 work of Kijowski [5], who used an axiomatic approach

to construct a distribution of arrival times in the momentum representation; however, the

coordinate form (x) given by Eq.(38) did not appear in the literature until more than

twenty years later [20].

Another representation better suited to numerical computation relies on the degeneracy of

free-particle waves to construct histories from standing wave combinations of plane waves.

Since standing waves are parity eigenfunctions, parity not direction of travel is the

good quantum label in this scheme. The competing descriptions in terms of running waves

and standing waves are connected by a unitary transformation; as noted in Sec. II, this

same transformation also relates the timelines stemming from the two representations (cf.

Eq.(17)): | , | ,

= 12

1 i1 i

| , + | ,

(40)As it happens, standing waves are simply related to Bessel functions J of order = 1/2.Using exp(ikx) =

kx/2

J1/2(kx) iJ1/2(kx)

in Eq.(37), we find on comparing

with Eq.(40) that timeline elements in the standing-wave representation are described by

the coordinate-space forms x | , (x), where

(x) =

x

2m

0

dk k exp ik2 /2m J1/2(kx) x 0 (41)The sign label () specifies the parity of these waves and prescribes their extension to x < 0.

Once again the integrals in Eq.(41) can be evaluated in closed form. The odd-parity

timeline waves for > 0 and x 0 are given by [21]

(x) =

2

m(xz)3/2 exp

ix2z i/8 J3/4 x2z iJ1/4 x2z z m

4(42)

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Unlike Eq.(38), z in this expression is real and positive. Eq.(42) is essentially the result

reported in a recent paper by Galapon et. al. [22].

The odd-parity states by themselves constitute a complete history for an otherwise free

particle that is confined to the half-axis x > 0 (e.g., by an infinite potential wall at the

origin), but for a truly free particle we also need the even-parity states. The even-parity

timeline waves for > 0 and x 0 are [21]

+ (x) =

2

m(xz)3/2 exp

ix2z+ i/8

J1/4

x2z iJ3/4 x2z z m

4(43)

Eqs.(42) and (43) are valid for > 0; results for < 0 follow from (x) = [ (x)]

(cf.

Eq.(41)). Timeline waves of either parity are well-behaved for all finite values of x, but

diverge (as |x|1/2) for |x| .

The time states constructed from running waves admit an interesting physical interpreta-tion. For any > 0 the rightward-running timeline wave (x) diverges as x

1/2 for x ,but vanishes as |x|3/2 for x ; more precisely, the asymptotics of the parabolic cylinderfunction [23] show that for any > 0 (arg z = /4) and large |x|

(x) exp(imx2/2) x1/2 x > 0

|x|3/2 x < 0These features are reversed for the leftward-running wave (x). (Corresponding results

for < 0 follow directly from the relation

(x) = (x).) The changeover in behavioroccurs in a small neighborhood of x = 0. The width of this transition region narrows with

diminishing values of , approaching zero for = 0. The Schrodinger waveform (x) for

two [system] times straddling = 0 are shown as Figs. 1 and 2, using a color-for-phase

plotting style that captures both the modulus and phase of these complex-valued functions.

The functions at any [laboratory] time other than t = 0 are found by replacing | in theabove argument with the evolved state exp

i

Ht

| = | + t . Thus, for t = 0 the

function behavior shown in the figures is unaltered, but the origin of time is shifted so thatthe abrupt change in behavior around x = 0 occurs generally at the [system] time = t.Consequently, is designated an arrival time, inasmuch as it signals the [laboratory] timewhen the bulk of probability shifts from one side of the coordinate origin to the other [20].

(The minus sign can be understood by noting that as system time increases, the time to

arrival diminishes.) In summary, the construction of Eq.(38) leads in this case to time-

of-arrival states for leftward [] or rightward []-running waves, with specifying the

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FIG. 1: The timeline waveform (x) for = 0.005 constructed from rightward-running planewaves. The shading (coloring) represents varying phase values for this complex function. In units

where = m = 1, the plot extends from x = 1 to x = +1. Except for an overall phase factor, thisalso represents the conjugate of the waveform (x) at the system time = 0.005 (cf. Eq.(39)).

arrival time at the coordinate origin x = 0. This interpretation receives further support

from the recent work of Galapon [22], who showed that similar states in a confined space

(where they can be normalized) are such that the event of the centroid arriving at the origin

coincides with the uncertainty in position being minimal. Arguably this is the best we can

do in defining arrival times for entities subject to quantum uncertainty.Time-of-arrival states specific to an arbitrary coodinate point, say x = a, can be obtained

as spatial translates of those constructed here: | ,, a = exp(ipa) | , (p, the par-ticle momentum operator, is the generator of displacements). The associated Schrodinger

wave function is x | ,, a = x a | , = (x a). In keeping with our earlierobservation concerning the phase ambiguity of timelines, we note that spatial translates

also can be recovered from Eq.(37) by re-defining the phases of the stationary waves as

x

|k

exp(

ika)

x

|k

.

B. A Free-Particle Time Operator in One Dimension

A time operator for free particles can be constructed following the recipe of Sec. III.

The invariance expressed by Eq.(18) ensures that the same time operator results no matter

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FIG. 2: The timeline waveform (x) for = 0.005 constructed from rightward-running plane

waves. The shading (coloring) represents varying phase values for this complex function. In units

where = m = 1, the plot extends from x = 1 to x = +1. Except for an overall phase factor, thisalso represents the conjugate of the waveform (x) at the system time = 0.005 (cf. Eq.(39)).

which [degenerate] representation we choose for the computation. With parity as the good

quantum number, the free-particle time operator is composed from operators in the even-

and odd-parity subspaces: T = T+ T, where (cf. Eq.(27))T P

| , , | d (44)

Now the energy-normalized stationary waves of odd parity vanish at the lower spectral bound

as E1/4(cf. Eq.(36)), so the general theory of Sec. III implies that T is well-defined by theintegral above, the principal value notwithstanding. The coordinate-space matrix elements

of this operator are simply related to one of a class of integrals Il (r, r) studied in Appendix

A; using the result reported there, we find

x |T| x =

(x) (x) d = 12xxI0 (x, x) = im4 x< sgn(x x) (45)The case for T+ is more delicate, since the energy-normalized stationary waves of even

parity actually diverge at the lower spectral bound as E1/4 (cf. Eq.(36)). Nonetheless, Ap-

pendix A confirms that the principal value integral for the coordinate space matrix elements

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of T+ remains well-defined, and can be evaluated in closed form to give x |T+| x =

+ (x)

+ (x

) d =1

2xxI1 (x, x

) = im

4x> sgn(x x) (46)

Combining the even and odd-parity computations, we arrive at the provocatively simpleform

x |T1dfree| x = im4

(x + x)sgn(x x) (47)Eq.(47) agrees with the formula reported by Galapon et. al. [22] for a particle confined

to a section of the real line, in the limit where the domain size becomes infinite. Here we

arrive at the same result in an unbounded space using an alternative limiting process the

accessible states model.

VI. EXAMPLE: FREE PARTICLE IN THREE DIMENSIONS

In this case, the Hamiltonian H is the operator for kinetic energy in a three-dimensionalspace. The spectrum of H is semi-infinite (bounded from below by E = 0, but no upper limit)and composed of degenerate levels. This degeneracy breeds multiple timelines, conveniently

indexed by the same quantum numbers that label the spectral states. Again there is some

flexibility in labeling here depending upon what dynamical variables we opt to conserve

along with particle energy, but the general timeline wave | is constructed from itsspectral counterpart E| following the same prescription used in the one-dimensionalcase, Eq.(34).

A. Angular Momentum Timelines for a Free Particle

In the angular momentum representation, the stationary states are indexed by a continu-

ous wave number k (any non-negative value), an orbital quantum number l (a non-negativeinteger), and a magnetic quantum number ml (an integer between l and +l, not to be con-fused with particle mass): | E | klml . This stationary state has energy Ek = k2/2m.The associated Schrodinger waveforms are spherical waves r | klml = Ck jl(kr)Ymll (r),formed as a product of a spherical Bessel function jl with a spherical harmonic Y

mll . Ck is

a constant that for the construction of timelines is fixed by energy normalization. Not-

ing that the spherical harmonics are themselves normalized over the unit sphere, we apply

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the Bessel function closure rule [24] to evaluate the remaining portion of the normalization

integral:

klml | klml = CkCk

0 dr r2jl (kr)jl (k

r)

= C2k

2k2(k k) = C2k

2mk(Ek Ek)

Thus, energy normalization of these spherical waves requires

Ck =

2mk

(48)

The time states | lml in this representation have components in the coordinate basisgiven by

r

| lml

l(r)Y

mll (r) where

l(r), the radial piece of the timeline wave, is

calculated from (cf. Eq.(34)):

l(r) =12

0

exp(iEk) Ck jl (kr) dEk

=1

2mr

0

expik2 /2m kJl+1/2 (kr) dk (49)

The last line follows from the connection between spherical Bessel functions and the (cylin-

der) Bessel functions of the first kind. The closure rule obeyed by these time states

l(r)

l(r)

d =1

r2(r r)

can be confirmed from the integral representation of Eq.(49) using the closure rule for Bessel

functions [24].

The integral in Eq.(49) converges for all m 0 and any l 0. Defining 2 l 1/2,we find [21]

l(r) =

4r

mz3/2 exp

ir2z i 2 + 1

4

J+1

r2z iJ r2z z m

4(50)

For fixed , J(. . .) is a regular function of its argument throughout the complex plane cut

along the negative real axis. Thus, through the magic of analytic continuation, Eq.(50)

extends l(r) to the whole cut z-plane |arg(z)| < . Now for any real > 0, z is a positivenumber, say z = x. To recover results for < 0, z must approach the negative real axis

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from above (arg(z) for arg() ). Writing z = x exp(i) in Eq.(50) and usingJ(exp (i) x) = exp (i) J(x) [25] leads to the relation

l(r) =

l(r)

(51)

for any real value of , a result that also is evident from the integral form, Eq.(49).

The behavior of l(r) for small r and/or large follows directly from the power series

representation of the Bessel function [26]. Apart from numerical factors, we find from Eq.(50)

l(r) z3/2+

r5/2 r

l

l/2+5/4r2 |z| 1 (52)

and this result is valid in any sector of the cut z-plane. Similarly, the asymptotic series for

the Bessel function [27] furnishes a large-argument approximation to l(r), valid for any

l

0 and

|arg(z)

|< :

l(r) 1

r5/2

2

m

l (l + 1) + 1/4

4 i2z

exp

i2z i 2l + 1

4

+

2l + 1

4

r2 |z| 1

(53)

B. Uni-Directional Timelines for a Free Particle

Free particles also can be described by momentum eigenstates labeled by a wave vectork . These momentum states have energy Ek = k

2/2m, and so must be expressible as a

superposition of angular momentum states with the same energy:

| k =l=0

lml=l

Umll

k | klml (54)Here k is the unit vector specifying the orientation of the wave vector with modulus k.The transformation from the angular momentum representation to the linear one should be

unitary to preserve the energy normalization required for the construction of timelines. To

identify the transformation coefficients Umll k, we note first that the Schrodinger wave-forms associated with | k are plane waves multiplied by a suitable normalizing constantC

(uni)k :

r | k = C(uni)k exp

ik r

(55)

Next, we appeal to the spherical wave decomposition of a plane wave [28]

exp

ik r

= 4

l=0

iljl(kr)l

ml=l

Ymll (r) [Ymll (k)]

27


28/39

to write the coordinate-space projection of Eq.(54):

4C(uni)k

l=0

iljl(kr)l

ml=l

Ymll (r) [Ymll (k)]

=l=0

lml=l

Umll

kCk jl(kr)Ymll (r)This will be satisfied if for every l 0 and |ml| l we have

4C(uni)k i

l [Ymll (k)] = Umll

kCkFor l and ml both zero this last relation reduces to

4C

(uni)k = U

00

kCk, leaving

4U00

k il [Ymll (k)] = Umll k. Setting 4U00 k = 1 then leads toUmll

k = il [Ymll (k)] (56)that describes the desired unitary transformation [29]:

l=0

lml=l

Umllk2 Umll k1 =

l=0

lml=l

[Ymll (2)] Ymll (1) = (1 2)

It follows that the energy-normalized plane waves are described by the normalizing factor

C(uni)k =

Ck4

U00

k = 14

2mk

(57)

Uni-directional time states are formed from plane waves all moving in the same direction,

but with differing energy. Accordingly, we adopt the unit vector

k as an additional label

for such time states, writing | | ,k . These uni-directional time states can be relatedto the angular momentum time states of the preceding section. Combining Eqs.(34), (54),

and (56), we find that the uni-directional timeline wave in the coordinate basis, r | ,k

k(

r ), can be computed from the spherical-wave expansion

k(

r ) =l=0

ill(r)l

ml=l

Ymll (r) [Ymll (k)]

, (58)

where l(r) is the radial timeline wave of Eq.(50).

Alternatively, we might try to calculate

k(r ) directly by taking | = | r in Eq.(34).With the help of Eqs.(55) and (57), we obtain in this way

k(

r ) = 12

0

exp(iEk) r | k dEk

=1

42

m

0

expik2 /2m + i kk r k3/2 dk (59)

28


29/39

Eq.(59) shows that the dependence of k(

r ) on k and on r occurs only through thecombination k r , which is nothing more than the projection of the coordinate vector ronto the direction of plane wave propagation. (Indeed,

k(

r ) itself is a plane wave albeitnot a harmonic one with the surfaces of constant wave amplitude oriented perpendicular

to k.) In terms of, then, there is a universal timeline applicable to any direction in space,as befits the expected isotropy of a free-particle environment. This universal timeline has

elements that we denote simply as (), and are given by

() =1

42

m

0

expik2 /2m + i k k3/2 dk (60)

Unlike a similar integral encountered in the one-dimensional case, Eq.(60) fails to converge

for real values of . But the integral does define a function that is analytic throughout the

lower half plane m < 0, and can be analytically continued onto the real axis. Fore ( i) > 0 we have [18]

() =3

16

3mz5/2 exp

2z2/4D5/2 (iz) z mi

, (61)

where D5/2 (. . .) is another of the parabolic cylinder functions. Eq.(61) limits to the sector

< arg < 0 (e ( i) > 0), but the mapping from z to allows analytic continuation

to the whole -plane cut along the negative real axis, arg < . The complexvariable z then is mapped into the sector 3/4 < arg z /4. And because D is anentire function of its argument [19], Eq.(61) defines a single-valued function throughout this

domain. Comparing Eq.(61) for > 0 (arg z = /4) and < 0 (arg z = /4), we discoverfor all real values of and

() = [()] (62)

For > 0 (arg z = /4), the asymptotics of the parabolic cylinder function [23] imply

() exp(im2/2) 3/2 > 0

||5/2 < 0

Thus, () diverges as 3/2 for and vanishes as ||5/2 for . Analogous

results for < 0 follow from Eq.(62). The behavior is reminiscent of the timeline functions

constructed from running waves in one dimension. Indeed, it appears that in () we again

have time-of-arrival functions, with denoting the arrival time at the coordinate origin for

29


30/39

FIG. 3: The universal timeline waveform () for the system time = 0.005. The shading(coloring) represents varying phase values for this complex function. In units where = m = 1,

the plot extends from = 1 to = +1.

FIG. 4: The universal timeline waveform () for the system time = +0.005. The shading

(coloring) represents varying phase values for this complex function. In units where = m = 1,

the plot extends from = 1 to = +1.

waves moving in the direction ofk, and k r . This interpretation is supported by theillustrations in Figs. 3 and 4 showing () for system times just prior to, and immediately

following, arrival at the coordinate origin.

30


31/39

C. The Three-Dimensional Free-Particle Time Operator

Lastly, we investigate the time operator for this example. We exercise the freedom allowed

by the degeneracy of free particle states to work in the angular momentum representation.

From Eq.(48) we find that the stationary spherical waves r | klml = Ck jl(kr)Ymll (r)vanish at the lower spectral edge (limk0r | klml = 0), so that a free-particle time oper-ator in three space dimensions does exist by the theory of Sec. III. The matrix elements ofT in the coordinate basis are given by (cf. Eq.(19))

r1 |T3dfree| r2 = l,ml

P

r1 | lml lml | r2 d

= l,mlYmll (1) [Ymll (2)] P

l

(r1) l(r2) dThe principal value integrals are studied in Appendix A, where their existence is rigorously

established and a closed-form expression given for their evaluation:

P

l(r1)

l(r2)

d = im

2sgn(r1 r2) 1

r>

r

lCollecting the above results, we obtain

r1 |T3dfree| r2 = im2

sgn(r1 r2) 1r>l,ml

Ymll (1) [Ymll (2)]

rl

The remaining sums also can be evaluated in closed form. Combining the generating

function for the Legendre polynomials [30] with the addition theorem for spherical harmonics

[31], we obtain for any |t| < 11

1 2t cos + t2 =

l=0

Pl(cos )tl

= 4

l=0

lml=l

Ymll (1) [Ymll (2)] t

l

2l + 1 ,

from which it follows that

2

l=0

lml=l

Ymll (1) [Ymll (2)]

tl = t1/2

t

t1/2

1 2t cos + t2

=1 t2

2 (1 2t cos + t2)3/2

31


32/39

Finally, identifying t with r gives

r1 |T3dfree| r2 = im2

sgn(r1 r2) 14

r2> r2 2r cos + r2


33/39

not admit a well-defined mean, or variance. Time operators when they exist are system

specific, useful for calculating moments of the [event] time distribution in those instances

where said moments can be shown to converge. Interestingly, we find that time operators

for periodic systems are never canonical to the Hamiltonian, but canonical time operators

can and do arise in [aperiodic] systems with a vanishing point spectrum (no isolated levels).

As examples of these general principles, we have examined several systems (particle in

free-fall, free particle in one dimension) for which results have been reported previously in

the literature. Our objective has been to illustrate how these diverse results follow from

the unified approach developed here. We also have gone beyond the familiar and applied

that same approach to the free particle in three dimensions. To the best of our knowledge,

results for the latter have never before appeared. Most importantly, they confirm that

the notion of an arrival time first encountered in the one-dimensional case extends to

three dimensions, complete with an accompanying canonical time operator. Possibilities for

future investigations abound. For instance, how to generate correct arrival-time statistics

for a particle scattering from even the simplest one-dimensional barrier remains a subject of

controversy [32]; we expect that discussion and numerous others to be informed by the

results presented here.

Appendix A: Time Operators and The Integrals Il (r1, r2)

In this Appendix we investigate the principal value integrals that arise in the construction

of a canonical time operator for free particles:

Il (r1, r2) P

l(r1)

l(r2)

d limR

+RR

l(r1)

l(r2)

d (A1)

Here l(r) is the timeline wavefunction in the lth angular momentum subspace, given by

Eq.(50). Inspection of Eqs.(42) and (43) shows that the l = 1 and l = 0 integrals alsoappear in the context of the time operator for a free particle in one dimension. Our objec-

tive here is to establish the existence of these integrals, and obtain closed-form expressions

suitable for their evaluation.

The relation l(r) =

l(r)

can be used to show that Il (r1, r2) is purely imaginary,

as well as antisymmetric under the interchange r1 r2, properties that can be used to

33


34/39

reduce the integral to the half-axis 0:

Il (r1, r2) = 2i m

lim

R

R0

l(r1)

l(r2)

d

(A2)

Substituting from Eq.(50), this becomes (recall 2 l 1/2)

Il (r1, r2) = im

r1r2

2(I1 + I2)

where

I1 limsR0+

sR

sin

s

r21 r22

J+1

s r21

J+1

s r22

+ J

s r21

J

s r22

ds

I2 limsR0+

sR

cos

s

r21 r22

J+1

s r21

J

s r22 J s r21 J+1 s r22 ds

The small-argument behavior of J ensures that both integrals exist in the indicated limits

provided > 1: accordingly, explicit reference to the limits will be omitted from subse-quent expressions.

Our next goal is to relate I2 to I1. To that end we define the related (and simpler)

integrals I1,2 byI1 (r1, r2)

0

sin

s

r21 r22

J

s r21

J

s r22

ds

I2 (r1, r2) 0

cos

s

r21 r22

J+1

s r21

J

s r22

ds

Then I1

I+11 (r1, r2) +

I1 (r1, r2) and I2

I2 (r1, r2)

I2 (r2, r1). Integrating

I2 once by

parts (the out-integrated part vanishes for >

1) and using the Bessel recursion relation

[25]

J1 (z) J+1 (z) = 2dJ (z)dz

34


35/39

results in

I2 (r1, r2) = 1r21 r220

sin

s

r21 r22

s

J+1

s r21

J

s r22

ds

= r2

2

2 (r21 r22)

0

sin s r21 r22 J+1 s r21 J+1 s r22 J1 s r22 ds+

r212 (r21 r22)

0

sin

s

r21 r22

J

s r22

J+2

s r21 J s r21 ds

In terms of yet a third integral I3 defined as

I3 (r1, r2)

0

sin

s

r21 r22

J+1

s r21

J1

s r22

ds

=2

r22

0

sin

s

r21 r22

J+1

s r21

J

s r22 ds

s I+11 (r1, r2)

=2

r21

0

sin

s

r21 r22

J

s r21

J1

s r22 ds

s I11 (r1, r2)

we can write the result for I2 compactly asI2 (r1, r2) =

1

2 (r21

r22) r21I

1 (r1, r2) + r

22I

+11 (r1, r2) r22I

3 (r1, r2) + r

21I

+13 (r1, r2)

=1

(r21 r22)

r21I1 (r1, r2) + r22I+11 (r1, r2) + 0

sin

s

r21 r22

J+1

s r21

J

s r22 ds

s

Then

I2 =1

(r21 r22)r21I1 (r1, r2) + r22I+11 (r1, r2) + r22I1 (r1, r2) r21I+11 (r1, r2)

+1

(r21 r22)

0

sin

s

r21 r22

J+1

s r21

J

s r22

J+1

s r22

J

s r21

ds

s

= I1 + 1(r21 r22)

0

sin

s

r21 r22

J+1

s r21

J

s r22 J+1 s r22 J s r21 dss

and so

Il (r1, r2) = im

r1r2

2 (r21 r22)

0

sin

s

r21 r22

J+1

s r21

J

s r22 J+1 s r22 J s r21 dss

(A3)

35


36/39

Finally, from integral tables [33] we have for a, b > 0 and > 1:0

J+1 (ax) J (bx)sin(cx)dx

x= 0 for 0 < c < b a

= a1

b

c for 0 < c < a bThis result is applied separately to each integral in Eq.(A3). In terms of the smaller (r) of its two arguments, the final form for Il (r1, r2) can be written most compactly

as

Il (r1, r2) = im

2sgn(r1 r2) 1

r>

r

l(A4)

Appendix B: Canonical Property of the Free-Particle Time Operator in Three

Dimensions

In this Appendix we establish that the time operator of Eq.( 63) is canonical to the free-

particle Hamiltonian H = p2/2m. To that end we examine the coordinate-space matrixelements of the commutator

r1 |T3dfree, H | r2 = 1

2mr1 |T3dfreep2 p2T3dfree| r2

=1

2m 21 22

r1 |

T3dfree| r2 (B1)

For evaluating the Laplacians in this expression, we apply the vector calculus identity [ 34]

A B

=A

B

+

B

A

+

A

B +

B

A (B2)

with the identificationsA =

(r1 r2 )|r1 r2 |3

;B = r1 + r2

Noting thatA is essentially the electrostatic field of a point charge, we have

1

A = 4 (r1

r2 ) =

2

A

1 A = 0 = 2 AB also is curl-free, so the first two terms in Eq.(B2) are zero. Further, with r1,2 =(x1,2, y1,2, z1,2), the simplicity of

B allows us to write

A 1

Bx =

Ax

x1+ Ay

y1+ Az

z1

(x1 + x2)

= Ax, etc.

36


37/39

and

x1

B 1

Ax

=

x1

(x1 + x2)

Axx1

+ (y1 + y2)Axy1

+ (z1 + z2)Axz1

=

B

1

Ax

x1+

Ax

x1, etc.

Then

21

A B

= 1 1

A B

= 21 A

+

B 1

1 A

(B3)

Replacing 1 with 2 in this last expression generates an equally valid result, but since Adepends only on r1 r2 , we obtain

22

A B

= 21 A

+

B 1

1 A

(B4)

and finally,

r1 |T3dfree, H | r2 = i 1

16

21 22 A B= i

1

4

1 A

= i (r1 r2 ) (B5)

We conclude thatT3dfree, H = i, i.e., that T3dfree is canonical to the free-particle Hamil-

tonian

H.

[1] W. Pauli in Handbuch der Physik (Springer, Berlin, 1933), Vol. 24 p. 83.

[2] Time in Quantum Mechanics -Vol. 1 (Lec. Notes Phys. 734), 2nd ed. edited by G. Muga, R.S.

Mayato, and I. Egusquiza (Springer, Berlin Heidelberg 2008).

[3] Time in Quantum Mechanics -Vol. 2 (Lec. Notes Phys. 789) edited by G. Muga, A.

Ruschhaupt, and A. del Campo (Springer, Berlin Heidelberg 2009).

[4] M.D. Srinivas and R. Vijayalakshmi, Pramana 16, 173 (1981).

[5] J. Kijowski, Rep. Math. Phys. 6, 362 (1974).

[6] D.T. Pegg, Phys. Rev. A 58, 4307 (1998).

[7] See for example, Peter J. Olver, Introduction to Partial Differential Equations, (University of

Minnesota, 2010), p 97-8.

[8] E. Merzbacher, Quantum Mechanics, 2nd ed. (John Wiley & Sons, New York, 1970), p 86.

37


38/39

[9] We expect that Emax grows without bound as N increases, implying that there is no natural

upper limit to the energy spectrum. While this may seem evident on its face, there exist model

spectra for which it is not true (e.g., the discrete spectrum of atomic hydrogen).

[10] C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York,

1976), pp. 5380.

[11] For an in-depth discussion of this point see G.C. Hegerfeldt, D. Seidel, J.G. Muga, and B.

Navarro, Phys. Rev. A 70, 012110 (2004), and references therein.

[12] R. Giannitrapani, Int. J. Theor. Phys. 36, 1575 (1997).

[13] A. Erdelyi, Asymptotic Expansions (Dover Publications Inc., New York, 1956), p. 47.

[14] Peter J. Olver, Introduction to Partial Differential Equations, (University of Minnesota, 2010),

p 301.

[15] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon Press, Oxford,

1965), p. 269.

[16] D.E. Aspnes, Physical Review 147, 554 (1966).

[17] Handbook of Mathematical Functions edited by M. Abramowitz and I. A. Stegun, (Dover, New

York, 1965), p. 447.

[18] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic

Press, New York,1965), p. 337.

[19] Higher Transcendental Functions Vol. II edited by A. Erdelyi, (McGraw-Hill, New York,

1953), p.117.

[20] J.G. Muga, C.R. Leavens, and J.P. Palao, Phys. Rev. A 58, 4336 (1998).


Press, New York,1965), p. 757-8.

[22] E. A. Galapon, F. Delgado, J. G. Muga, and I. Egusquiza, Phys. Rev. A72, 042107 (2005).

Apart from an overall phase, (x) is the complex conjugate of the function labeled oddt (q)

in this reference. The difference is inconsequential, since only oddt | 2 is specified in theapproach taken by these authors. In the same vein, the time operator introduced here differs

from the time-of-arrival (TOA) operator in this reference by an overall phase ().


1953), p.122-3.

[24] G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists, 5th ed. (Harcourt Aca-

38


39/39

demic Press, San Diego, 2001), p. 691. The closure equation for spherical Bessel functions

follows directly from their definition and appears explicitly on p. 735 of this same reference.


1953), p.12.


York, 1965), p. 360.


York, 1965), p. 364.

[28] The spherical wave expansion of a plane wave can be found in any advanced text on scattering.

See, for example, R.G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Dover,

New York, 2002), p. 38.

[29] Unitarity here stems from closure of the spherical harmonics on the unit sphere; see G.B.

Arfken and H.J. Weber, Mathematical Methods for Physicists, 5th ed. (Harcourt Academic

Press, San Diego, 2001), p. 801.


demic Press, San Diego, 2001), p. 740.


demic Press, San Diego, 2001), p. 796.

[32] A.D. Baute, I.L. Egusquiza, and J.G. Muga, Phys. Rev. A 64, 012501 (2001).


Press, New York,1965), p. 749. The extension to c = |a b| required in our application isobtained as the limit of the tabulated results for c < |a b|.

[34] Such identities are readily verifed by writing the component relations in rectangular coordi-

nates, and appear in various resources. See for example, the inside front cover of D. J. Griffiths,

Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, Upper Saddle River, 1999).

time in quantum mechanics

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