Muga G. Time in Quantum Mechanics

Подождите немного. Документ загружается.

112 J. Mu

noz et al.

5.5.1 The Harmonic Oscillator and Systems with Bound States

If we were to try the direct application of deﬁnition (5.1) to the case of the harmonic

oscillator we would soon run into trouble due to divergent integrals. A natural option

is to restrict the total time to one period of the system, as follows:



T/2

−T/2

dτ e

Hτ/

(.x)e

−i

Hτ/

, (5.47)

where the Hamiltonian is .p

/2m + mω

/2 and T = 2π/ω. We can thus write

the dwell-time operator, restricted to one period, as

= T

∞



n=0



(.x)







. (5.48)

The eigenvalues, T



(.x)





, can be understood as the period times the propor-

tion of that period spent by the particle in the stationary state in the region D.This

suggests an alternative useful quantity for systems whose Hamiltonian has a purely

point spectrum. Deﬁne

.τ

= lim

T →∞



T/2

−T/2

dτ e

Hτ/

(.x)e

−i

Hτ/

. (5.49)

Let the Hamiltonian be given by

H =



n,α

,α



,α

, (5.50)

with α a degeneracy index. Then it is easy to compute

.τ



n,α,β

,α



,α

(.x)

,β



,β

, (5.51)

that is to say, the operator that gives us the fraction of time spent in region D is

diagonal in the energy basis, up to degeneracy in energy.

5.5.2 Times of Residence

The construction above suggests an extension to the time of residence, which is an

analog of the dwell time valid for systems in which the concept of a region in space

is not applicable. The question we now purport to answer is the following one: how

much time has a system spent in a state



? Or, alternatively, from time 0 to T , what

is the proportion of time the system has spent in that state? Let then the projector

5 Dwell-Time Distributions in Quantum Mechanics 113

associated with the state



be denoted by P and the unitary evolution operator by

U(t). We deﬁne the time-of-residence operator as

.τ

(0; T) =



dtU

†

(t)PU(t) . (5.52)

For example [45], consider a two-level system with Hamiltonian

H =





0 Ω

Ω 0



(5.53)

and the state







. The time of residence gets written as

.τ

(0; T) =

⎛

⎝

2Ω

sin

(

ΩT

)

−i

2Ω

(

1 − cos

(

ΩT

))

2Ω

(

1 − cos

(

ΩT

))

−

2Ω

sin

(

ΩT

)

⎞

⎠

. (5.54)

This entails that the measurement of this quantity would inevitably lead to one of

the following two (eigen)values:

sin

(

ΩT/2

)

. (5.55)

The fact that only two values can be obtained in each realization of the experiment,

and not a continuous distribution of time of presence in the state







, has

been attributed to a predictive character of the measurement involved, as opposed to

an observation that extends over time through the coupling with a weakly interacting

clocking system [45].

As a matter of fact, the average time spent by a particle prepared at instant 0 in

a generic state ranges from τ

−

to τ

for the eigenstate of P. Represent a generic

pure or mixed state ρ by a vector in or on the Bloch sphere, ρ = (1 +r ·σ )/2; then

the average time of residence in state





over the time interval [0, T ]isT/2 +

(1 − cos

(

ΩT

)

+sin

(

ΩT

)

/2Ω, which, as stated, ranges over [τ

−

,τ

It should be noticed that the operator of time of residence from instant 0 to instant

T is generically not stationary; for the example at hand we have

H,.τ

(0; T)

i



1 − cos

(

ΩT

)

i sin

(

ΩT

)

−i sin

(

ΩT

)

−1 + cos

(

ΩT

)



. (5.56)

Notice that whenever T = 2nπ/Ω, with n a natural number, the corresponding time

of residence does indeed commute with the Hamiltonian; this is in keeping with the

expressions of Sect. 5.5.1.

114 J. Mu

noz et al.

5.5.3 Multiparticle Systems

Deﬁnition (5.1) admits a straightforward extension to systems of many particles,

using the formalism of second quantization, in which we perform the substitution

(.x) →.n



dx .a

†

, (5.57)

where .a

is the annihilation operator at point x. For free particles, that is to say, for

a Hamiltonian of the form

H =



∞

−∞





∞

−∞



†

, (5.58)

with .a

the operator that annihilates a particle with wavenumber k, we can compute

the dwell-time operator as



∞

−∞

|k|



†

ikL

sin

(

)

†

−k



. (5.59)

It is also feasible to write a dwell-time density operator, which for the free case

reads

(t) =



∞

−∞



(

t −t

(k)

)

+δ

(

t −t

−

(k)

)

†

−ikL

(

t −t

(k)

)

−δ

(

t −t

−

(k)

)

†

−k



. (5.60)

It is easy to see that ψ|

(t)|ψ reduces to the dwell-time density for the one-

particle marginal wavefunction. The real multiparticle aspect of this construction

will only be accessible through dwell-time–dwell-time correlation functions, for

which a suitable operational interpretation needs to be built.

In addition, we note that for multiparticle systems, the alternative question might

be posed as what is the time during which a given number of particles n can be

found in the region D. This naturally leads to the introduction of the dwell-time

operator for n particles:

(n) =



∞

−∞

†

(t)δ

U(t)

2π



∞

−∞



2π

dθ

†

(t)e

iθ(.n

−n)

U(t) , (5.61)

where .n

is the density operator restricted to the region D, deﬁned in Eq. (5.57).

Letting .ρ be the density matrix describing the state of the system, it is convenient to

introduce the characteristic function

5 Dwell-Time Distributions in Quantum Mechanics 115

F(θ; t) = Tr[.ρ

†

(t)e

iθ

U(t)] , (5.62)

whose Fourier transform is the atom number distribution in the region D [21]:

(n, t) =

2π



2π

−inθ

F(θ; t)dθ, (5.63)

with n ∈ N. The meaning of P

(n, t) is precisely the probability for n particles to

be found in the spatial domain D at time t.

Knowledge of the atom number distribution can be used to compute the average

dwell time for different number of particles, namely



(n)=



∞

−∞

dtTr[.ρ(t)δ

] =



∞

−∞

dtP

(n, t) . (5.64)

5.6 Relation to Flux–Flux Correlation Functions

This section follows [35] and examines a link between the dwell time and ﬂux–ﬂux

correlation functions (ffcf) that have been considered mostly in chemical physics to

deﬁne reaction rates for microcanonical or canonical ensembles [24]. The motiva-

tion for this exercise is to relate the dwell-time distribution, and not just the average

value, to other observables.

5.6.1 Stationary Flux–Flux Correlation Function

Pollak and Miller [38] have shown a connection between the average stationary

dwell time and the ﬁrst moment of an ffcf. They deﬁne a quantum microcanonical

ffcf C

(τ,k) = Tr{Re

(τ,k)} by means of the operator

(τ,k) = 2π[

J(x

,τ)

J(x

, 0) +

J(x

,τ)

J(x

, 0)

−

J(x

,τ)

J(x

, 0) −

J(x

,τ)

J(x

, 0)]δ(E −

H) , (5.65)

where

J(x, t) = e

Ht/

[.pδ(.x −x)+δ(.x −x).p]e

−i

Ht/

is the quantum mechanical

ﬂux operator in the Heisenberg picture, and .p and .x are the momentum and position

operators.

This deﬁnition is motivated from classical mechanics: Eq. (5.65) counts ﬂux cor-

relations of particles entering D through x

) and leaving it through x

) a time

τ later. Moreover, particles may be reﬂected and may leave the region D through

its entrance point. This is described by the last two terms, where the minus sign

compensates for the change of sign of a back-moving ﬂux. Note that these negative

terms lead to a self-correlation contribution that diverges for τ → 0.

116 J. Mu

noz et al.

We shall ﬁrst derive the average correlation time and show its equivalence with

the average dwell time. We shall only consider positive incident momenta and deﬁne

by substituting δ(E −

H)byδ

(E −

H):= δ(E −

H)Λ

, where Λ

is the

projector onto the subspace of eigenstates of H with positive momentum incidence.

By means of the continuity equation

−

J(x, t) =

.ρ(x, t) , (5.66)

where.ρ(x, t) = e

Ht/

δ(.x−x)e

−i

Ht/

is the (Heisenberg) density operator,

(τ,k)

can be written as

(τ,k) =−2π 



dτ

.χ

(τ )



.χ

(t)



t=0

(E −

H) . (5.67)

By a partial integration and using the Heisenberg equation of motion, the ﬁrst

moment of the Pollak–Miller correlation function is given by





∞

dττ

(τ,k)



= Tr



2π



∞

dτ.χ

(τ )

i

[.χ

(0),

H]δ

(E −



(5.68)

Boundary terms of the form lim

τ →∞

.χ

(τ ),γ = 0, 1, 2, are omitted here and

in the following. The contribution of these terms should vanish when an integration

over stationary wavefunctions is performed to account for the wavepacket dynamics,

as it is done explicitly in the next section. For potential scattering the probability

density decays generically as τ

−3

, which assures a ﬁnite dwell-time average, but for

free motion it decays as τ

−1

[32], making τ

inﬁnite, unless the momentum wave

function vanishes at k = 0 sufﬁciently fast as k tends to 0 [11], as we have discussed

before.

Writing the commutator explicitly and using the cyclic property of the trace give





∞

dττ

(τ,k)



= Tr



2π



∞

dτ



−

dτ

.χ

(τ )



.χ

(0)δ

(E −



, (5.69)

and integration over τ yields the ﬁnal result:





∞

dττ

(τ,k)



= 2π Tr



.χ

(0)δ

(E −



= T

. (5.70)

Expressing the trace in the basis |φ

 gives back the stationary dwell time of

Eq. (5.6), i.e., the diagonal element of the on-the-energy-shell dwell-time operator,

5 Dwell-Time Distributions in Quantum Mechanics 117

The calculation of the average in [38] is different in some respects: (a) The coor-

dinates x

and x

are taken to minus and plus inﬁnity, but it can be carried out for

ﬁnite values modifying Eq. (8) of [38] accordingly; (b) Formally there are no explicit

boundary terms at inﬁnity but a regularization is required in Eq. (16) of [38], which

is justiﬁed for wave packets; (c) δ(E −

H) is used instead of δ

(E −

H). This

simply provides an additional contribution for negative momenta parallel to the one

obtained here for positive momenta; (d) In our derivation the average correlation

time is found to be real directly, even though

(τ,k) is not self-adjoint, whereas

in [38] the real part is taken. (The discussion of the imaginary time average in [38]

is based on a modiﬁed version of Eq. (5.65).)

Next, we will show that the second moment of the Pollak–Miller ffcf equals the

second moment of T. This was not observed in Ref. [38]. Proceeding in a similar

way as above, we start with

I = Tr





∞

dττ

(τ,k)



. (5.71)

Integrating by parts twice, neglecting the term at inﬁnity, using Heisenberg’s equa-

tion of motion and the fact that φ

is an eigenstate of

H, the real part is

I + I

∗

2πm

k



∞

dτ φ

|[.χ

(τ ).χ

(0) +.χ

(0).χ

(τ )]|φ

 . (5.72)

Introducing resolutions of the identity,

ReI =



2πm

k



∞

dτ



∞

−∞







i(E−E



)τ/

∗

(x)φ



(x)φ

∗



)φ



)



+c.c., (5.73)

where c.c means complex conjugate. Making the changes τ →−τ and x, x



→



, x in the c.c-term, it takes the same form as the ﬁrst one, but with the time inte-

gral from −∞ to 0. Adding the two terms, the τ -integral provides an energy delta

function that can be separated into two deltas which select k



=±k to arrive at





∞

dττ

(τ,k)



4π







dx |φ

(x)|





dx φ

∗

(x)φ

−k

(x)





= (T

)

. (5.74)

In other words, the relation between dwell times and ﬂux–ﬂux correlation functions

goes beyond average values and C

(τ,k) includes quantum features of the dwell

time: note that the ﬁrst summand in Eq. (5.74) is nothing but (T

)

, whereas the

second summand is positive, which allows for a nonzero on-the-energy-shell dwell-

time variance (T

)

− (T

)

. We insist that the stationary state considered has

118 J. Mu

noz et al.

positive momentum, φ

(x), k > 0, but this second term implies the degenerate

partner φ

−k

(x) as well, and is generically nonzero.

We shall see in the next section with a more general approach that these connec-

tions do not hold for higher moments.

5.6.2 Time-Dependent Flux–Flux Correlation Function

A time-dependent version of the above ﬂux–ﬂux correlation function can be deﬁned

in terms of the operator

C(τ ) =



∞

−∞

J(x

, t +τ )

J(x

, t) +

J(x

, t +τ )

J(x

, t)

−

J(x

, t +τ )

J(x

, t) −

J(x

, t +τ )

J(x

, t)

, (5.75)

which leads to the ﬂux–ﬂux correlation function

C(τ ) =Re

C(τ )

, (5.76)

where the real part is taken to symmetrize the nonself-adjoint operator

C(τ )as

before.

As in the stationary case, Eq. (5.76) counts ﬂux correlations of particles entering

D through x

or x

at a time t and leaving it either through x

or x

a time τ later.

Moreover we integrate over the entrance time t. It can be shown that the ﬁrst moment

of the classical version of Eq. (5.75), where

J is replaced by the classical dynamical

variable of the ﬂux, gives the average of the classical dwell time.

As in Eq. (5.67), we may rewrite

C(τ )as

C(τ ) =−



∞

−∞

dτ

.χ

(.x, t + τ)

.χ

(.x, t) . (5.77)

From here we note that the ffcf C(τ ) is not normalized:



∞

dτ

C(τ ) =



∞

−∞

dt .χ

(t)

.χ

(t) = 0 , (5.78)

as a result of the self-correlation.

Next we derive the average of the time-dependent correlation function. With a

partial integration one ﬁnds



∞

dττ

C(τ ) =



∞

−∞



∞

dτ .χ

(.x, t + τ)

.χ

(.x, t) .

A second partial integration with respect to t replacing d/dt by d/dτ and integrating

over τ gives

5 Dwell-Time Distributions in Quantum Mechanics 119



∞

dττ

C(τ ) =



∞

−∞

dt .χ

(.x, t) =



∞

−∞

dt .χ

(.x, t) =

, (5.79)

where .χ

= .χ

has been used. Equation (5.79) generalizes the result of Pollak and

Miller to time-dependent dwell times.

A similar calculation can be performed for the second moment of C(τ ). After

three partial integrations with vanishing boundary contributions to get rid of the

factor τ

one obtains



∞

dττ

C(τ ) = 2



∞

−∞



∞

dτ .χ

(.x, t + τ).χ

(.x, t) .

Making the substitutions t + τ → t and τ →−τ in the complex conjugated term,

we ﬁnd



∞

dττ

C(τ ) =

. (5.80)

5.6.3 Example: Free Motion

For a stationary ﬂux of freely moving particles with energy E

, k > 0, described

by φ

(x) =x|k=(2π)

−1/2

ikx

, the ﬁrst three moments of the ideal dwell-time

distribution on the energy shell are given by

k

, (5.81)

)



1 +

sin

(kL)

, (5.82)

)



1 + 3

sin

(kL)

. (5.83)

As proved above, the ﬁrst two moments agree with the corresponding moments of

the Pollak–Miller ffcf, but for the third moment we obtain instead





∞

dττ

(τ,k)







1 −

3[1 + cos

(kL)]

3sin(2kL)



(5.84)

In Fig. 5.5 the ﬁrst three moments are compared. The agreement between (T

)

and Eq. (5.84) is very good for large values of k, but they clearly differ for small k.

Nevertheless, the agreement of the ﬁrst two moments suggests a similar behavior of

Π(τ ) and C(τ ).

120 J. Mu

noz et al.

Fig. 5.5 (Color online) Comparison of the ﬁrst three moments: T

,(T

)

and (T

)

,withthe

corresponding moments of the ﬂux–ﬂux correlation function, for a free-motion stationary state

with ﬁxed k.(

)

is also shown.  = m = 1andL = 2

To calculate Π

(τ ) for a wavefunction

ψ(k):=k|ψwith only positive momen-

tum components we use Eq. (5.11) and the explicit forms of the eigenvectors,

Eq. (5.14) and eigenvalues t

, Eq. (5.15):

(τ ) =



γ =±

ψ(k

(τ ))|



(τ ))|

, (5.85)

where the j-sum is over the solutions k

(τ ) of the equation F

(k) ≡ t

(k) −τ = 0

and the derivative is with respect to k.

We use the following wavefunction [27]:

ψ(k) = N(1 − e

−αk

−(k−k

)

/[4(Δk)

]

−ikx

Θ(k) , (5.86)

where N is the normalization constant and Θ(k) the step function. For the free

ﬂux–ﬂux correlation function we write

C(τ ) = Re



∞



∞



∗

(k)

ψ(k



)k|

C(τ )|k



 , (5.87)

and C



(τ ) =k|

C(τ )|k



 in the free case



(τ ) =

2πk

δ(k − k



)

dτ

2g(kτ/m) − g(kτ/m − L) − g(kτ/m + L)

(5.88)

5 Dwell-Time Distributions in Quantum Mechanics 121

where

g(x) =−2e

imx

/(2τ )



iπτ



1/2

+iπ x erﬁ



2τ

. (5.89)

The result is shown in Fig. 5.6. The ffcf shows a hump around the mean dwell time

but it oscillates for small τ and diverges for τ → 0. As discussed above, this is

due to the self-correlation contribution of wavepackets which are at x

or x

at the

times t and t + τ without changing the direction of motion in between. A similar

feature has been observed in a traversal-time distribution derived by means of a path

integral approach [14].

0 10203040

−0.6

−0.4

−0.2

0.2

0.4

0.6

Dwell Time Distribution

C(τ)

π(τ)

Π(τ)

Fig. 5.6 (Color online) Comparison of dwell-time distribution Π(τ )(dashed line) and ﬂux–ﬂux

correlation function C(τ )(solid line) for the freely moving wave packet (5.86). Furthermore, the

alternative free-motion dwell-time distribution π(τ ), Eq. (5.27), is plotted (circles). We set

 =

m = 1and|x

|large enough to avoid overlap of the initial state with the space region D = [0, 45].

= 2, Δk = 0.4, and α = 0.5

In contrast, Π

(τ ) behaves regularly for τ → 0, but shows peaks in the region of

the hump. This is because the denominator of Eq. (5.85) becomes zero if the slope

of the eigenvalues t

(k) is zero, which occurs at every crossing of t

(k) and t

−

(k).

The distribution π

(τ ), see Eq. (5.27), is also computed: it agrees with C(τ )in

the region near the average dwell time and it tends to 0 for τ → 0. However, it does

not show the resonance peaks of Π

(τ ).

In the absence of a direct dwell-time measurement, the physical signiﬁcance of

and

depends on their relation to other observables. The present results indicate

that the second moment of the ﬂux–ﬂux correlation function is related to the former

and not to the later, providing indirect support for the physical relevance of the

dwell-time resonance peaks, but other observables could behave differently.

Muga G. Time in Quantum Mechanics - Vol. 2

Подождите немного. Документ загружается.