More about Tunnelling Times, the Dwell Time and the “Hartman Effect”

(1)

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More about Tunnelling Times, the Dwell Time and the

“Hartman Effect”

Vladislav Olkhovsky, Erasmo Recami, Fabio Raciti, Aleksandr Zaichenko

To cite this version:

Vladislav Olkhovsky, Erasmo Recami, Fabio Raciti, Aleksandr Zaichenko. More about Tunnelling

Times, the Dwell Time and the “Hartman Effect”. Journal de Physique I, EDP Sciences, 1995, 5 (10),

pp.1351-1365. �10.1051/jp1:1995202�. �jpa-00247141�

(2)

J. Phys. I France 5

(1995)

1351-1365 OCTOBER 1995, PAGE 1351

Classification Physics Abstracts

73.40Gk 03.80+r 03.65Bz

More about TunnelEng Times, trie Dwell Time and trie "Hartman

Elfiect" (*)

Vladislav S.

Olkhovsky (~),

Erasmo Recami

(2),

Fabio Raciti

(3)

and Aleksandr K.

Zaichenko

(~)

(~) Institute for Nuclear Research, Ukrainian Academy of Sciences, Kiev, Ukraine and I.N.F.N., Sezione di Catania, 57 Corsitalia, Catania, Italy

(~) ^Facoltà ^di Ingegneria, Università Statale di Bergamo, Dalmine

(BG),

Italy; I.N.F.N., Sezione di Milano, Milan, Italy; Dept. ^of Apphed Mathematics, State University at Campinas, Camp- inas, S-P-, Brazil

(~) Dipartim. di Fisica, Università di Catania, Catama, Italy

(~) Institute for Nuclear Research, Ukrainian Academy of Sciences, Kiev, Ukraine

(Received

18 April 1995, received in final form 29 May 1995, accepted 20 June

1995)

Abstract. In _a recent review paper

[Phys.

Reports 214

(1992)

_339] _we proposed, within

conventional quantum mechanics, _new definitions for the sub-borner tunnelling and reflection times. Aims of the present paper are: 1) presenting and analysing the results of various numerical

calculations

(based

on our

equations)

_on the penetration and retum times _< _rp~n >, < _rRet >, during tunnelling inside _a rectangular potential barrer, for _vanous penetration depths _xi;

ii)

putting fortin and discussmg suitable definitions, ^besides of the _mean values, also of the _uanances

(or dispersions)

DTr and DrR for the time durations of transmission and reflection processes;

iii) mentioning, _moreover, that _our definition _< _rr _> for the average transmission time results to constitute _an improuement of trie ordinary dwell-time f~~ formula:

iv)

commenting, at last,

on the basis of _our _new numencal results, _upon _some recent criticism by C.R. Leavens. We stress

that _our numencal evaluations confim that _our approach implied, and implies, the existence

of the lfartman ejfect: _an effect that in these days

(due

to the theoretical connections between tunnefling and evanescent-wave

propagation)

is receivmg -at Cologne, Berkeley, Florence and

Vienna- indirect, but quite interesting, expenmental venfications. Eventully, _we briefly analyze

some other definitions of tunnelling times.

1. Introduction

In _our review article

[ii

_we put forth _an

analysis

of the main theoretical definitions of the sub-barrer

tunnelling

and reflection times, and

proposed

_new definitions for such durations

(*)Work

partially supported by INFN, MURST and CNR

(Italy),

by CNPq

(Brazil),

and by the I.N.R.

(Kiev, Ukraine)

and S-K-S-T- of Ukraine

(project

_no.

2.3/68).

(3)

which _seem to be self-consistent within con~entionai quantum nlechanics

(~).

In

particular,

the

"prediction" by

_our

theory [ii

of the

reality

of the Hartman

ejfect

_[2] in

tunnelling

_processes has

recently

^received -due to the

analogy

_[3] between

tunnelling

electrons

and evanescent waves- quite

interesting,

even if

indirect, experimental

verifications _at

Cologne

[4],

Berkeley

_[5], Florence [6] and ^Vienna [6].

Main a1nls of the present paper are:

presenting

and

analysing

the results of several numerical calculations of the

penetration

and _return times inside _a

rectangular potential

barrier

during tunnelling (Sect. 3);

^and

proposing

_new suitable formulae for the distribution variances of the transmission and reflection times

(Sect. 2).

^The ^results ^of _our ^numerical evaluations _seem _to confirm that _our

approach

is

physically acceptable,

and that it _moreover

implied,

and

implies,

the existence of the so-called "Hartman

ejfect

^" _even for

non-quasi-nlonochromatic packets.

The present ^research

field,

however, is

developing

_so

rapidly

and in such _a controversial manner, that it may be convenient to add here -before ail- _some brief _comnlents about _a few _papers

appeared during

the last three _years:

1)

First,

let _us mention that _we had overlooked _a _new _expression for the dwell-lime

Y~~

derived

by Jaworsky

and Wardlaw

[7,8]

Y~~ (xi,

_~i;

k)

₌

/~

dt _t

J(xf, t) /~

_dt _t

_J(xi,

t)) /~

_dt _J;n

_(xi,

t))

~

il

-m -m -m

which is indeed

equivalent

_[7] to our

equation (16)

of reference

[ii (all

notations

being

defined

therein):

ce xi ce -1

Y~~ (~;,

_xf;

k)

₌

~Îoe

dt

Î,

dz

p(~, t) _~Îoe

dt In

lx;, t) (2)

This

equivalence

reduces the

difference,

between _our definition _< _rT _> of _the _average transmission lime -under _ouf

assumptions-

and

quantity Y~~,

ta the difference between

the average made

by using

the

positive-definite probability density

dtJ+(x,t)/ fc~dtJ+(x,t)

and the _average made

by using

the

ordinary "probability density"

dt

J(x,t)/ fÎ~

dt

J(x,t). Generally speaking,

the last

expression

is not

always

positive

definite,

_as it _was

explained

at page 350 of reference

[ii,

and hence does non possess any direct

physical meaning.

ii)

^In ^reference _[9] an attempt _was made to

analyze

the evolution of the _wave

packet

_mean position <

x(t)

>

("center

of

gravity" ), averaged

over

pdx, during

ils

tunnelling through

a

potential

barrier. Let _us here observe that the conclusion to be found

therein,

about the absence of _a causal relation between the incident _space centroid and its transmitted

equivalent,

holds

oniy

when the contribution

coming

^front the barrier

region

to the _space

integral

is

negligible.

iii)

Let _us also add that in reference

[loi

it _was

analyzed

the distribution of the transmission time _rT in _a rather

sophisticated

_way, which is very similar to the dwell-time

approach,

however with _an

artificial, abrupt switching

_on of the initial _wave

packet.

We _are

going

to propose, on the contrary, ^and ⁱⁿ

analogy

with _our

equations (30)

to

(31)

in reference [11, (~) ^Let us take advantage of the present opportumty ^for pointing out that _a mispnnt entered _our equation

(10)

in reference [1], whose last term ka ought to be elimmated. Moreover, due to _an editorial _error, _m the footnote at page 32 of _Dur reference [16] ^the dependence of G _on Ak disappeared,

whilst in that _paper _we had assumed

G(k Î)

_% C

exp[-(k

k)~

/(Ak)~]

(4)

N°10 TUNNELLING TIMES AND THE "HARTMAN EFFECT" 1353

the

following

expressions, as

physically adequate

definitions for the ~ariances

(or disper- sions)

DrT and DrR of the transmission and reflection time

[see

Sect.

2], respectively:

DTr

₊

Dt+ lxf)

+

Dt+ (xi

13)

and

DrR

+ Dt-

ix;

+ Dt-

(xi (4)

where

ii

dl t~

J+jx, tj ii

dt t

J+jx, tj

^~

~~~ i~i

_~

jr~

dt

J+ j~, t) jz~

_dt

_J+ _j~, _t)

^i~~

Equations (3)

to

(5)

_are based _on the formalism

expounded

in reference

[iii,

_as well

as on our definitions for

J+(~,t)

in reference

iii.

Of course, we are

supposing

that the

integrations

over

J+(~f)dt, J+(x;)dt

and

J-(x;)dt

_are

independent

of _one another. We shall devote Section

2, below,

to these

problems,

_1-e-, to the

problem

of

suitably defining

mean values and variances of

durations,

for various transmission and reflection _processes

during tunnelling.

iv) Below,

in Section

4,

_we shall

briefly re-analyse

_some other definitions of

tunnelling

durations.

Before

going

_on, let _us recall that several _reasons

"justify"

the existence of different _ap-

proaches

to the definition of

tunnelling

times:

a)

the

problem

of

defining tunnelling

durations is

dosely

connected with that of

defining

_a time operator, 1-e-, of

introducing

time _as _a

(non-

selfadjoint)

_quantum mechanical

observable,

and

subsequently

of

adopting

_a

general

definition for collision durations in quantum ^mechanics. Such

preliminary problems

did receive _some clar-

ification in recent times

(see,

for

example,

reference

[ii

and citations [8] and [22]

therein); b)

the motion of _a

particle tunnelling

inside _a

potential

barrier _is _a

purely

quantum

phenomenon,

devoid of _any

classical,

intuitive

limit; c)

the various theoretical

approaches

_may dioEer in the choice of the

boundary

conditions _or in the

modelling

of the

experimental

situations.

2. Mean Values and Variances for Various Penetration and Return Times

during

Tunnelling

In _our previous _papers, _we

proposed

for the transmission and

reflection

times _some formulae which

imply

_-as functions of the penetration

depth- integrations

_over time of

J+ ix, t)

^and

J-

ix, t), respectively.

Let _us recall that the total flux

J(x,t)

inside _a barrier consists of _two components,

J+

and J-, associated with motion

along

the

positive

and the

negative

x-direction,

respectively.

Work in similar directions did

recently

_appear in reference [12].

Let _us refer ourselves -here- to

tunnelling

and reflection _processes of _a

partide by

_a

potential barrier, confining

ourselves to _one space dimension.

Namely,

let _us

study

the evolution of _a _wave

packet l#(x,t), starting

^from the initial state

l#in(x,t);

and follow the notation introduced in reference

[ii.

In the _case of uni-directional motions it is

already

known [13]

that the flux

density J(x,t)

e

Re[(ià/m) l#(x,t) àl#*(x,t)/ôx]

_can be

actually interpreted

as the

probability

that the

partide (wave packet)

_passes

through position

_x

during

_a unitary time-interval centered at t, _as it

easily follows from

the

continuity eq~ation

and

from

the

fact

that

quantity p(x,t)

e

(l#(x,t)(~

is the

probabiiity density

for _our

"partide"

to be

located,

at time t, ^inside a unitary space-interval centered at _x.

Thus,

in order to determine the _mean

instant at which _a

moving

wave

packet l#(x, t)

_passes

through

position _x, we have to take the

average of the time variable t with respect to the

weight w(x, t)

%

J(x, t) / ff~~ J(x, t)

dt.

(5)

However,

if the motion direction _can _~ary, then

quantity w(x, t)

is _no

longer positive definite,

and _moreover is not

bounded,

because of the

variability

of the

J(x, t) sign.

In such _a _case, _one

can introduce the two

weights:

w+

ix, t)

m

J+ ix, t) /°' _J+ _{ix, tj} ^tj

^~~

_j6j

w~jx,tj

_m

J~jx,tj /°' J~jx,tj tj

^~~

jij

where

J+(x, t)

and J-

lx, t)

_represent the positive and

negative

parts of

J(x,t), respectively,

which _are

bounded, positive-definite functions,

normalized to 1. Let _us show

that,

from the

ordinary probabilistic interpretation

of

p(x, t)

and from the well-known

continuity equation

apjx, tj aJjx, tj

at ⁺ ax ⁼ ^° 18)

it follows aiso in this

(more generai)

_case that quantities _w+ ^and _w-,

represented by

_equa-

tions

(6), iii,

_can be

regarded

_as the

probabilities

that _our

"particle"

_passes

through position

x

during

a unit time-interval centered _at t

(in

the _case of forward and backward motion,

respectively).

Actually,

for those time mtervals for which J

=

J+

_or J

= J-, one can rewrite equation

(8)

as follows:

~~ÎÎ~

~~

~~ÎÎÎ~

^~~ ^~~~~

~~ÎÎ~

~~ ~~

ÎÎ~

_~~

~~~~

respectively.

Relations

(9a)

and

(9b)

_can be considered _as formal definitions of

ôp> lût

and

ôp< lût.

Let _us _now

integrate equations (9a)

and

(9b)

_over time from _-cc _to t; _we obtain:

p>

ix, t)

=

j~ _~~(~'~~~

^dt'

^j10a)

p<

ix, t)

_m

/~ ôJ-j~'~')

^dt'

^jiob)

with the initial conditions _p>

(x, -ccl

=

p<(x, -cc)

= 0. Then, let _us introduce the quantities

N> ix,

_cc;

t)

_e _p>

ix', t)

dz'

=

J+ ix, t')

dt' _> ₀

il la)

Î~ oe

N< j-cc,

_x;

t)

_e

/~ ^p<jxi, ^t)

^dz' _m

_/~ J~jx, t')

^dt' > o

jiib)

which have the

meaning

of

probabilities

for _our

"partide"

to be located at time _t _on the

semi-axis

ix, cc)

_or

(-cc, x) respectively,

_as functions of the flux densities

J+ lx, t)

_or J-

lx, t),

provided

that the normalization condition

fÎ~ p(x, t)dx

= 1 is fulfilled. The r-h-s-'s of _equa- tions

Il la)

and

(1lb)

have been obtained

by integrating

the r.h.s.'s of equations

(10a)

and

(lob)

and

adopting

the

boundary

conditions

J+(-cc,t)

=

J-1-cc, t)

= 0. Now,

by dioEerenciating

equations

(lia)

and

(llb)

with respect to t, _one obtains:

~ijj ^"'

=

J+jx, t)

> 0

j12a)

(6)

~~( ^"'

= J-

lx, t)

_> 0.

(12b) Finaiiy,

from

equations (lia), (llb), (12a)

and

(12b),

_one _can infer that:

~°~' ~Î(Î~~ÎÎÎÎ~

^~~~~~

~°-l~,t)

"

~))~j£))/~ l13b)

, ,

which

justify

the abovementioned

probabilistic

interpretation of _w+

ix, t)

and _w-

ix, t).

Let _us notice,

incidentally,

that _our

approach

does not _assume any ad hoc

postulate, contrarily

to what believed

by

the author of reference [14].

At this

point,

_we _can

eventually

^define ^the mean ~aiue of the time at which _our

"parti-

de" passes

through

position _x,

travelling

ⁱⁿ ^the

positive

_or

negative

direction of the _x axis,

respectively,

_as:

ff~~

^t

^J+

(X,

t)

dt

~ ~~~~~ ^~

ff~~ ^J+

(X,

t)

dt ~~~~~

ii tJ~jx, t)

dt

<

~-i~i

> ~

jz~ _{J-ix, tj}

_dt

^i~~~i

and,

_moreover, the ~ariances of the distributions of these times _as:

ii

^t~

J+ ix, t)dt

~~~

_i~~ _~

jz~ _{J+j~, tjdt} ^l~

^~~^i~~

^~i

~ ^i~~~~

-i

+

Îi~ _J~jÎ, ^)Ît

1<

~-i~i >i~ i~~~i

in accordance with the

proposal presented

in references

il,15].

Thus,

_we have _a formahsm for

defining

_mean values, variances

(and

other central

moments)

related to the duration distributions of all

possible

_processes for _a

partide, tunnelling through

a

potential

barrier located in the interval

(0,a) along

the _~ axis; and not

only

for

tunnelling,

but also for all

possible

kinds of

collisions,

with

arbitrary energies

and

potentials.

For instance,

we have that

<

r~j~~,x~j

_> _a _<

t+jx~j

_> _<

t+j~jj

_>

j16j

with _-cc < ~j < 0 and _a < ~f < cc; and therefore

(as

anticipated in equation

(3))

that

~TT (~i, ~f)

~

~~+(~f)

~

~~+ (~i

for transmissions from region

i-cc, 0)

to region

(a, cc)

which _we called

[ii regions

I and III,

respectively. Analogously,

for the pure

(complete) tunnelling

_process _one has:

<

~Tuni°, ~)

> ~ <

t+i~)

_> _<

t+i°)

>

i~~)

and

DrTuni0, aj

a

Dt+ jaj

+

Dt+ j0j il 8j

while _one has

(7)

<

rpen(0,~f)

> + <

t+(~f)

> <

t+(0)

>

(19)

and

Drpen

(0, ~f)

₊

Dt+ (xf)

+

Dt+ (0) (20)

[with

0 _< _xf _<

ai

^for penetration mside the barrier

region (which

_we called

region II).

Moreover:

<

r~e~(~,x)

> e <

t~(x)

> <

t+(x)

_>

(21)

Dr~e~

(x, x)

e Dt~

(xi

₊

Dt+ ix) j22)

[with

0 < _x <

ai

^for "return

processes"

inside the barrier. At

last,

for

reflections

in

region I,

we have that:

<

r~jx;, xi)

_> ₊ _< t~

(xi)

> <

t+jx~)

_>

j23) [with

_-cc < xi <

ai,

and

(as

anticipated in

equation (4))

that D

rR(xj, xi)

₊ D t-

(xi)

+ D t+

(xi).

Let _us stress that _our definitions hold within the framework of conventional quantum me-

chanics,

without the introduction _{of any} _new

postulates,

and with the

single

measure

expressed by weights (13a), (13b)

for all time averages

(both

in the initial and in the final

conditions).

According

to our

definition,

the

tunnelling phase

time

(or, rather,

the transmission

duration),

defined

by

the stationary

phase

approximation for

quasi-monochromatic partides,

is

meaningful oniy

in the limit _xi

- cc when

J+ ix, t)

is the flux

density

of the initial

packet

Jjn of incoming

waves

(in

absence

of reflected

_wa~es,

therefore).

Analogously,

the dwell

time,

which _can be

represented (cf. Eqs. (l)

and

(2) by

the

expression

[7,

8,16]

Y~~(xi> xf)

₌

/~

_t

_{J(xf, t)}

_dt

/~

_t

J(xi> t)

dt

/~ _{Jin(xi, t)}

_dt

,

-m -m -m

with _-cc < ~j < 0, and _~f _> _a, _is not

acceptable, generally speaking.

In

fact,

the

weight

in the time averages is

meaningful, positive

definite and normalized to 1

oniy

in the _rare _cases when _~j _- _-cc and Jjn ₌ JIII

ii-e-,

when the barrier _is

transparent).

3. Penetration and Return Process

Durations,

inside _a

Rectangular Barrier,

for

Tunnelling

Gaussian Wave Packets: Numerical Results

We put ^forth here the results of _our calculations of _mean durations for various

penetration (and return)

_processes, inside _a

rectangular barrier,

for

tunnelling gaussian

_wave

packets;

_one

of _our aims

being

to

mvestigate

the

tunneiiing speeds.

In _our

calculations,

the initial _wave

packet

is

l#jn(x, t)

₌

/ ^~ _G(k _-1) _exp[ikx _iEt/à]

_dk

₍₂₄₎

with

G(k ii

_e

Cexp[-(k -1)~ /(2 Ak)~] (25)

ezactly

_as in reference [8]; ^and ^with ^E =

à~k~/2m; quantity

C

being

the normalization constant, and _m the electron _mass. Our

procedure

of

integration

_was descnbed in reference [16].

(8)

N°10 TUNNELLING TIMES AND THE '~HARTMAN EFFECT" 1357

1o

io

0 2 3 5

X(À)

Fig. 1. Behaviour of the _average penetration time <

rp~n(0,x)

>

(expressed

in

seconds)

_as _a

function of the penetration depth _xi % x

(expressed

in

Angstroms)

through _a rectangular barrier with width _a

= 5 À, for Ak

= 0.02 À~~

(dashed fine)

and Ak

= 0.01 À~~

(continuous fine),

respectively.

The other parameters _are listed in footnote

(1).

It is worthwhile _to notice that _< _rpen _> rapidly

increases for the first, few initial àngstroms _{(+~ 2.5}

À),

_tending afterwards to _a saturation value. This

seems to confirm the existence of the so-called "Hartman eifect" [1,

2,15].

Let _us express the

penetration depth

in

Angstroms,

and the penetration ^time m seconds.

In

Figure

1 _we show the

plots corresponding

to a = 5

À,

for Ak

= 0.02 and 0.01

À~~,

respectively.

The penetration time < rpen >

always

tends _to _a saturation value.

In

Figure

2 _we

show,

for the _case Ak

= 0.01

À~~,

the

plot corresponding

to a = 10

À.

_It

is

interesting

that _< _rpen _> is

practically

the _same

(for

the _same

Ak)

for _a

= 5 and _a

= 10

À,

a result that

confirm,

let _us repeat, the existence

[ii

^{of the} so-called Hartman

ejfect

[2]. ^Let us

add

that,

when varying the parameter Ak between 0.005 and 0.15

À~l

_and

letting

_a to _assume values _even

larger

than 10

À, analogous

results have been

always

gotten. Similar calculations have been

performed (with

_quite reasonable

results)

also for various

energies

^É in the range 1

to10 eV

(~).

In

Figures

3, 4 and 5 _we show the behaviour of the _mean

penetration

and return durations _as function of the penetration

depth (with

_xi

= 0 and 0 < xf e x <

a),

for barriers with

height

Vo " 10 eV and width _a

= 5

Àor

₁₀

À.

_In

Figure

3 _we present ^the

plots

of _<

rpen(0, xi

>

correspol~ding

to different values of the _mean kinetic _energy: É

= 2.5 eV, 5 eV and 7.5 eV

(plots

1, 2 and 3,

respectively)

with Ak ₌ 0.02

À;

and E ₌ 5 eV with Ak ₌ 0.04

À~~ (plot 4), always

with _a

= 5

À.

_In

Figure

4 we show the

plots

of _<

rpen(0, xi

_>,

corresponding

to (~) ^For ^the ^interested reader, let _us recall that, when integrating _over dl, _we used the interval

-10~~~

s to

+10~~~

s

(symmetrical

with respect to t ₌

o),

_very much larger than the tempo- ral _wave packet extension.

[Recall

that tl~e extension _in time of _a _wave packets _is of the order of

1/(ô6k)

=

(Akfi$)~~

ci 10~~~ si. Our "centroid" has been always _ÉD

" 0j _~o ₌ 0. For clanty's

sake, let _us underline again that in _our approach the initial _wave packet 4fjn(~,

t)

is net regarded _as prepared at _a certain instant of time, but it is expected to flow through _any

(initial)

_point _~, dunng the infinite time interval

j-oc, +oc),

_even if with _a jimte time-centroid to. The value of such centrera to _is essentially defined by the phase of the weight amplitude

G(k k),

and in _our _case _is equal to 0

when

G(k

k) is real.

(9)

0 2 6 8 10

X(À)

Fig. 2. Trie _same plot _as _m Figure 1, for Ak

= 0.01

À~~,

_except that _Dow trie barrer width is

a = 10 À. _Let

us observe that the numencal values of the

(total)

tunnelling time < _rr > practically

does not change when _passing from _a

= 5 À _to

a = 10

À,

again _in agreement with the charactenstic

features 1 of the Hartman eifect. Figures 1 and 2 do improve

(and correct)

the corresponding _ores,

preliminarily presented by _us _in reference [16].

q

Î

Îc

~

Î)~~

0 0 2.0 3.0 1-O 5.0

(À)

Fig. 3. Behaviour of <

rPen(0,~)

_>

(expressed

in

seconds)

_as _a function of _~

(expressed

in

Angstroms),

for tunnelling through _a rectangular barrer with width _a

= 5 À _and _for _diiferent _values _of É and of Ak.

(curve

1): Ak

= 0.02 À~~ _and _E

= 2.5 eV;

(curve

2): Ak

= 0.02 À~~ _and É

= S-o eV;

(curve

_3): Ak

= 0.02 À~~ _and _E

= 7.5 eV;

(curve

4): Ak

= 0.04 À~~ _and _E

= s-ù eV.

(10)

2

0 2 6 B 10

X(À)

rp~n(0,

~) _>

(in seconds)

_as _a function of _~

(in

Angstroms) for É

= 5 eV

and diiferent values of _a and Ak.

(curve

1): _a

= 5 À _and _Ak

= 0.02 À~~;

(curve

2): _a

= 5 À and

Ak = 0.04 À~~;

(curve

3): _a =10 À _and _Ak

= o.02 À~~;

(curve

4): _a ₌ 10 À _and _Ak

= 0.04 À~~.

a = 5

À,

with Ak

= 0.02

À~~

_and _0.04

À~~ (plots

1 and

2, respectively);

and to _a ₌ 10

À,

with Ak

= 0.02

À~~

_and 0.04

À~~ (plots

3 and 4,

respectively),

the _mean kinetic _energy E

being

5

eV,

1-e-, one half of Vo.

In

Figure

5 the

plots

_are shown of _<

rRet(x, xi

_>. The _curves 1, 2 and 3

correspond

to

É

= 2.5

eV,

5 eV and 7.5

eV, respectively,

for Ak

= 0.02

À~~

_and

a = 5

À;

the _curves 4,

5 and 6

correspond

to E ₌ 2.5

eV,

5 eV and 7.5 eV,

respectively,

for Ak

= 0.04

À~~

_and

a = 5

À;

while the _curves

7,

8 and 9

correspond

to Ak

= 0.02

À~~

_and _0.04

À~~, respectively,

for

É

= 5 eV and _a

= 10

À.

Also from trie _new

Figures

3-5 one can see tl~at:

1)

at variance with reference [8], no

plot

considered

by

_us for tl~e _mean

penetration

duration _< _rpen

(0, xi

> of _our _wave

packets

presents any interval with

negative values,

_nor with _a

decreasing

<

rpen(0, x)

> for

increasing

_x;

and,

moreover, that

2)

the _mean

tunnelling

duration <

rTun(0, a)

> does _not

depend

_on the barrier width _a

("Hartman effect");

and

finally

that

3) quantity

<

rTun(0, a)

> decreases when the

energy increases.

Furthermore,

it is noticeable that also from

Figures

3-5 _we observe:

4)

_a

rapid

increase for the value of the electron

penetration

time _m the initial part ^of ^the ^barrier

region (near

_x

=

0);

and

5)

_a

tendency

of _<

rpen(0, x)

> to a saturation value _m the final part of the

barrier,

_near _x ₌ _a.

Feature 2,

firstly

observed for

quasi-monocl~romatic partides,

_[2] does

evidently

_agree with the

predictions

made _m reference

[ii

for

arbitrary

_wave

packets.

Feature 3 is also _m agreement with previous evaluations

performed

for

quasi-monochromatic

partiales and

presented,

for instance, in references

[1,2,15].

Features 4 and 5 _can be

apparently explained by

interference

(11)

8

0 2 6 8 10

x

(À)

rRet(~,~)

>

(in

seconds) _as _a function of _~

(in

Angstroms) for diiferent values of _a, É _and _Ak.

(curve

1): a = 5 À, E ₌ 2.5 eV and Ak ₌ o.02 À~~;

(curve

2): _a ₌ 5

À,

É

= à-o eV and Ak

= 0.02 À~~;

(curve

3): _a ₌ 5

À,

E

= 7.5 eV and Ak

= 0.02 À~~;

(curve 4):

a = 5

À,

E

= 2.5 eV and Ak

= o.02 À~~;

(curve

5): _a

= 5

À,

É

= S-o eV and Ak

= 0.02

À~~; (curve

6): _a ₌ 5

À,

E

= 7.5 eV and Ak

= 0.02 À~~;

(curve

7): _a

= 5

À,

E ₌ S-o eV and Ak

= 0.02 À~~;

(curve

8): _a

= 5

À,

E

= S-o eV and Ak

= 0.02 À~~.

between those initial

penetrating

and

returning

_waves inside the

barrier,

whose

superposition yields

tl~e

resulting

fluxes

J+

and J-. In

particular,

if in tl~e initial part ^{of tl~e} ^barrier ^tl~e

retuming-wave packet

is

comparatively large,

it does

essentially extinguisl~

the

leading edge

of the

incoming-wave packet. By

contrast, if for

growing

_x the returning-wave

packet quickly vanishes,

then the contribution of the

leading edge

of the

incoming-wave

packet to the _mean

penetration duration <

rpen(0,x)

> does

initially (quickly)

_grow, while in the final barrier

region

its increase does

rapidly

slow down.

Furthermore,

the

larger

is the barrier width _a, the

larger

is the part ^{of the} ^back

edge

^{of the}

mcoming-wave

packet

which is

extinguished by

interference with the

returning-wave packet.

Quantitatively,

these

phenomena

will be studied elsewhere.

Finally,

_m connection with the

plots

of <

rRet(x,x)

_> _as _a function of _x,

presented

in

Figure

5, let _us observe that:

i)

the

mean reflection duration <

rR(0, 0)

> % <

rRet(0, 0)

> does not

depend

_on the barrier width a;

ii)

in

correspondence

with the barrier _region betwen 0 and

approximately

0.6 _a, the value of <

rRet(0,x)

_> is almost constant; while

iii)

its value increases with _x

only

_m the barrier

region

near x = a

(even

if it should be

pointed

out that _our calculations _near _x ₌ _a _are not so

good,

due to the _very small values assumed

by fÎ~

J-

ix, t)dt therein).

Let _us notice that point 1), ^also observed

firstly

for

quasi-monochromatic partides,

_[2] is _as well in accordance with the results obtained in reference _[1] for

arbitrary

_wave

packets. Moreover,

also points

ii)

and

iii)

_can be

explained by

interference

phenomena

mside the barrier:

if,

_near _x

= a, the

initial

returning-wave packet

is almost

totally quenched by

the initial mcommg-wave

packet,

(12)

then

only

_a

negligibly

small

piece

of its back

edge (consisting

^of the components with the smallest

velocities)

does remain. With

decreasing

_x

ix

_-

0),

the

unquenched

part ^of ^the

returning-wave packet

_seems to become _more and _more

large (containing

_more and _more

rapid components),

thus

making

the difference _<

rRet(0,x)

> <

rpen(0,x)

> almost constant.

And the interference between

inconling

and reflected _waves at

points

_x < 0 does

effectively

constitute _a

retarding phenomenon

_[so that _< _t-

ix

=

0)

> is

larger

^than <

rR(x

=

0)],

which

can

explain

tl~e

larger

values of _<

rR(x

= 0,~

=

0)

> in comparison with <

rTun(x

= 0,

~ =

a)

>.

Therefore _our

evaluations,

in ail the _cases considered

above,

_appear to confirm _our

previous analysis

at page ³⁵² of reference [1], ^and our conclusions therein

concerning

in

particular

the

validity

^of ^the Hartman effect also for

non-quasi-monochromatic

_wave

packets.

Even _more, since the interference between

incoming

and reflected _waves before the barrier

(or

between

penetrating

and

retuming

_waves, inside the

barrier,

_near the entrance

wall)

does

just

increase the

tunnelling

time _as well _as the transmission

times,

_we _can expect that _our non-relativistic

formulae for _<

rTun(0, a)

> and _<

rT(xj

< 0, _xf >

a)

_> will

always

forward

positive

values

(~).

At this

point,

it is necessary -however- _to observe the

following.

Even if _our non-relativistic equations _are not

expected (as

_we have

just seen)

_to

yield negative

times, nevertheless _one

ought

to bear in mind that

(whenever

it is met _an

object, tJ, travelling

at

Superluminal speed) negative

contributions should be

expected

to the

tunnelling

times: and this

ought

not to be

regarded

_as

unphysical.

In

fact,

whenever _an

"object"

tJ _o~ercomes the infinite

speed

_[18] with respect to a certain

observer,

it will afterwards _appear to the _same observer _as its

"anti-object"

à travelling

in the

opposite

_space ^direction

[18].

^For instance, when

passing

from the lab _to

a frame F _moving in tl~e _same direction _as the

partides

_or waves entenng ^tl~e ^barrier

region,

the

objects

tJ

penetrating through

the final part of the barrer

(with

almost infinite

speeds,

like in

Figs. 1-5)

will _appear in the frame F _as

anti-objects ^à crossing

that

portion

^of ^the barrier in the

opposite space-direction

_[18]. In tl~e _new frame

F, therefore,

such

anti-objects ^à

would

yield

_a

negati~e

contribution to the

tunnelling

time: which could _even

result,

in

total,

to be

negative

_[2]. For any

clarifications,

_see references

[18]. So,

_we have _no

objections

a

priori against

the fact that Leavens _can

find,

in certain _cases,

negative

values [8,

iii:

_e-g, when

applying

our formulae to _wave

packets

with suitable initial conditions. What _we _want to stress here is that the _appearance of

negative

times

(it being predicted by Relativity itself,

_[18] when in presence of

anything travelling

faster than

c)

is not a valid _reason to rule out _a theoretical

approach.

At

last,

let _us

-incidentally-

recall and _mention the

following

fact. Some

prel1nlinary

calculations of

penetration

times

(inside

_a

rectangular barrier)

for

tunnelling gaussian

wave

packets had been

presented by

us in 1994 in reference [16]. ^Later on

-looking

for _any

possible explanations

for the

disagreement

between the results in reference [8] ^and ⁱⁿ our reference

[16]-

_we

discovered, however,

that _an

exponential

factor _was

missing

in _a term of _one of trie fundamental formulae _on which the numerical

computations (performed by

_our _group in

Kiev)

were based: _a mistake that could not be

detected,

of _course,

by

_our careful checks about the

(~)A diiferent daim by Delgado, Brouard and Muga _[17] does not _seem to be relevant to _our calcu-

lations, _smce it is based _once _more, hke reference [8], ^not on our but _on diiferent _wave packets

(and

over-barrer components are also retained in reference [17], at _vanance with us). Moreover, _m their

classical example, they overlook the fact that the

mean entrance time < t+(0) > gets contribution

mainly by the rapid components of the _wave packet; they forget, _m fact, that the slow components

are

(almost)

totally reflected by the initial wall, _causing _a quantum-mechanical reshaping that _con- tributes to the initial "time decrease" discussed by _us already in the last few paragraphs of _page 352

in reference [1]. ^All ^such phenomena reduce the value of <

t+(0)

_>, and _we expect ^it to be

(m

_our

non-relativistic

treatment)

less than _<

t+(a)

_>.

(13)

computing

_process.

Therefore,

trie _new results of _ours

appearing

in

Figures

1-2 should

replace Figures

^1-3 of reference [16]. ^One may observe

tl~at, by using

the _same parameters _as

(or

parameters _very _near

to)

the _ones

adopted by

Leavens for his

Figures

3 and 4 in reference [8],

our new, corrected

Figures

1 and 2 result _to be _more similar to Leavens' than the uncorrected

ones

(and

this is of _course _a welcome step towards the solution of the

problem).

One _can

verify

_once _more,

however,

that _our

theory

_appears to

yield for

those parameters

non-negative

results for _<

rpen(xf)

_>,

contrarily

to _a daim in reference [8].

Actually,

_our

previous general

conclusions have _not been

apparently

affected

by

the mentioned mistake. In

particular,

tl~e value of _<

rpen(xf)

_> increases with

increasing

_xf, ^and ^tends to saturation for _xf _- _a. We

acknowledge, however,

that the difference in the

adopted integration

_ranges

[-cc

to +cc for _us, and 0 to +cc for

Leavens]

does Dot

play

_an

important role, contranly

to _our

previous belief,

_[16]

in

explaining

the

remaining discrepancy

between _our results and Leavens'. Such _a

discrepancy might perl~aps depend

_on the fact tl~at the functions to be

integrated

do fluctuate

heavily

_(~)

(anyway,

_we did

carefully

check that _our _own

elementary integration

step ⁱⁿ ^the

integration

over dk _was small

enough

in order to guarantee ^the

stability

of the numerical

result,

and, m

particular,

of their sign, ^for

strongly oscillating

functions in the

integrand).

More

probably,

the

persisting disagreement

can be

merely

due _to the fact _-as

recently

daimed also

by Delgado

et ai.

Ii?i

that

dijferent

initial conditions for the _wave

packets

_were

actually

chosen in reference [8] ^and ⁱⁿ reference

iii. Anyway,

_our

approach

_seems to get support, at least in

some

particular

_cases, also

by

_a recent article

by

Brouard et

ai.,

which

"generalizes"

_-even ^if

starting

from _a

totally

different

point

of view- _some of _our results

[12,15].

Let _us take

advantage

of the present opportunity for

answering

other criticisms

appeared

in reference [8], ^where ^it ^has ^been furthermore commented about _our _way of

performing

actual averages over the

physical

t1nle. We cannot agree with those comments: let _us

re-emphasize

in fact

that,

within conventional quantum

mechanics,

the time

t(x)

at which _our

partide (wave packet)

_passes

through

the

position

x is

"statistically

distributed" with tl~e

probability

densities

dtJ+ ix, t) / fÎ~ dtJ+ ix, t),

_as _we

explained

at page 350 of reference

iii.

Tl~is distribution meets tl~e

requirements

of tl~e

time-energy uncertainty

relation.

We also answered in Section Leavens' comments about _our

analysis iii

of tl~e dwell-time approacl~es [19].

The last

object

of the criticisnl in reference _[8] refers to the

impossibility,

_m _our

approach,

of

distinguishing

between "to be transmitted" and "to be reflected" _wave

packets

at the

leading edge

of tl~e barrier.

Actually,

_we do

distinguish

between

tl~em; only,

_we cannot -~f _course-

separate

tl~em,

due to tl~e obvious superposition

(and interference)

of both _wave functions in

p(x,t),

in

J(x,t)

and _even in

J+ ix, t).

This _is known to be _an unavoidable consequence of the superposition

principle,

valid for _wa~e

functions

_m conventional quantum ^mechanics.

That last

objection, tl~erefore,

sl~ould be addressed _to quantum

mecl~anics,

ratl~er then to _us.

Nevertheless,

Leavens' aim of

comparing

the definitions

proposed by

us for the

tunnelling

times not

only

witl~ conventional, but also witl~ non-standard quantum ^mecl~anics

might

be

regarded

a priori as

stimulating

and

possibly

wortl~ of furtl~er

investigation.

4. Further Remarks

In connection witl~ tl~e

question

of

"causality"

for relativistic

tunnelling partides,

let _us stress tl~at tl~e Hartman-Fletcl~er

pl~enomenon (very

small

tunnelling durations),

witl~ tl~e _conse-

(~)We

can only _say that _we succeeded in reproducing results of the type put forth in reference [8]

by using larger _steps; whilst the "non-causal" results disappeared _-m the considered _case- when

adopting small enough integration steps.