Change time of identical particles tunneling through the rectangular barrier at their exchange interaction
Work is devoted to studying the influence of exchange effects on a time of simultaneous crossing by identical particles through the rectangular quantum barrier. It is shown, that such effects essentially influence on the tunneling parameters. A change of identical particles tunneling time is first c...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2014 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Цитувати: | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction / L.S.Martsenyuk // Вопросы атомной науки и техники. — 2014. — № 5. — С. 35-38. — Бібліогр.: 5 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859666669734985728 |
|---|---|
| author | Martsenyuk, L.S. |
| author_facet | Martsenyuk, L.S. |
| citation_txt | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction / L.S.Martsenyuk // Вопросы атомной науки и техники. — 2014. — № 5. — С. 35-38. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Work is devoted to studying the influence of exchange effects on a time of simultaneous crossing by identical particles through the rectangular quantum barrier. It is shown, that such effects essentially influence on the tunneling parameters. A change of identical particles tunneling time is first computation taking into account their exchange interaction in the field of rectangular quantum barrier.
Работа посвящена изучению влияния обменных эффектов на время одновременного туннелирования тождественных частиц через прямоугольный квантовый барьер. Показано, что такие эффекты существенно влияют на параметры туннелирования. Впервые рассчитано изменение времени туннелирования тождественных частиц с учетом их обменного взаимодействия в поле прямоугольного квантового барьера.
Робота присвячена вивченню впливу обмiнних ефектiв на час синхронного туннелювання тотожних частинок через прямокутний квантовий бар'єр. Показано, що такi ефекти iстотно впливають на параметри туннелювання. Вперше розрахована змiна часу туннелювання тотожних частинок при урахуваннi Їх обмiнної взаємодiї в полi прямокутного квантового бар'єру.
|
| first_indexed | 2025-11-30T11:27:43Z |
| format | Article |
| fulltext |
CHANGE TIME OF IDENTICAL PARTICLES TUNNELING
THROUGH THE RECTANGULAR BARRIER AT THEIR
EXCHANGE INTERACTION
L.S.Martsenyuk∗
Institute for Nuclear Researches NAS Ukraine, 03680, Prospect Nauky 47, Kiev, Ukraine
(Received June 22, 2013)
Work is devoted to studying the influence of exchange effects on a time of simultaneous crossing by identical par-
ticles through the rectangular quantum barrier. It is shown, that such effects essentially influence on the tunneling
parameters. A change of identical particles tunneling time is first computation taking into account their exchange
interaction in the field of rectangular quantum barrier.
PACS: 03.65.-w, 03.65.Xp, 03.65.Nk
1. INTRODUCTION
In given work research of influence of identical parti-
cles exchange interaction on time of their simultane-
ous tunneling through the rectangular quantum bar-
rier is conducted.
Tunneling is one of the most essential quantum-
mechanical phenomenas, with which face the devel-
opers of modern devices nanotechnology, quantum
optics, physicists of the condensed matter and etc.
Research of tunneling interesting as in theoretical as-
pect, by virtue of some unusual features of signal
passing through area, ”forbidden” with positions of
the laws classical physicists (in them impulses of par-
ticles accept imaginary values)), so and in practical.
The practical interest is stimulated by perspective
of development on base of these phenomena new de-
vices, which, possible, will possess the feature, not
having analogue amongst material of surrounding us
nature.
Attention to problem of the tunneling especially
increased in connection with opening of the Hartman
effect (1962.). This effect is direct consequence of
tunneling Vigner’s time definition [1].
As is well known, there are a few definition of tun-
neling time. Vigner’s time (or phase time) is defined
as time of the intersection by maximum of the wave
package the region of quantum potential barrier:
τϕ = h̄
∂ϕ
∂E
,
where: ϕ – is a phase of a wave transmitted through a
barrier, E – energy of particle, incident on a barrier.
Hartman T.E., using the traditional approach in
definition of tunneling parameters, has established
that Vigner’s time of tunneling through a rectangular
barrier is determined by the following formula (Hart-
man’s formula):
τϕ =
h̄√
E(U0 − E)
,
where: U0 – is height of a barrier.
A few important consequences follow from this
formula [1].
1. Phase time of tunneling does not depend ex-
plicit on weight of a particle, but only from energy.
2. Time of tunneling does not depend on width
of a barrier and, hence, at barrier wide enough and
at big energies, speed of a particle can exceed speed
of light.
The last statement conflicts with the generally ac-
cepted position about impossibility of exceeding of
light velocity by particles.
There are some experimental results testifying to
possibility of display of Hartman’s effect [1]. There-
fore there is a question, whether has place such effect
at scattering of identical particles on each other?
The similar task frequently arises in experiments
on scattering in nuclear physics. The simplified
model of such process is simultaneous crossing by
identical particles the area of a rectangular barrier at
their movement in opposite directions. For Coulomb
potential such task has been solved by Mott N.F.
(Mott’s formula) [2].
At research of processes of identical particles tun-
neling in view of their exchange interaction it is con-
venient to consider tunneling as a limiting case of
elastic scattering.
Particles can have spin (protons, electrons, etc.),
or have not spin, as in case of scattering of alpha
particles on alpha particles.
Scattering of identical particles has the features
[3] which are finding out their fundamental difference
from not of identical particles.
∗Corresponding author E-mail address: prolisok77@yandex.ua
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.35-38.
35
This difference is found out from the circuit re-
sulted on Fig.1.
Fig.1. Two possible variants of identical particles
scattering in system of mass center which are not
distinct experimentally [3]
As particles are not distinct from each other, and
we cannot establish, which from particles gets in one
of counters, at calculation of scattering section it is
necessary to summarize amplitudes of probabilities
of both processes specified on Fig.1. It means that
can be revealed interference constituent, the contri-
bution of which to the section of dispersion will de-
pend on the spin state and from what particles dis-
perse: bosons or fermions
For two identical spinless Bose particles in system
of center of mass the effective scattering section of one
particle in direction of spatial angle dΩ = sin θdθdφ
looks like [1]:
σ(p← p0) = |f(p← p0) + f(−p← p0)|2 . (1)
Because events, proper to the hit of particle 1 or
particles 2 in some one of counters experimentally
not distinguished (they are adequate to amplitudes
of probabilities f of events, presented by each from
components in a formula (1)), for spherically sym-
metric interactions have:
σ = |f(E, θ) + f(E, π − θ)|2 =
= |f(E, θ)|2 + |f(E, π − θ)|2 +
+2f(E, θ)f∗(E, π − θ) . (2)
For two not identical particles we would have expres-
sion:
σ = |f(E, θ)|2 + |f(E, π − θ)|2 .
Thus, for identical particles in section of scatter-
ing there is the additional component responsible for
quant mechanical effect of interference, observable ex-
perimentally in case of scattering only for identical
particles. The Mott’s formula [2] describing scatter-
ing of identical particles in a field of Coulomb po-
tential, also contains additional component, arising
by virtue of exchange interaction of particles. For
two identical fermions with spin 1/2, scattering in
the field of potential, not dependent from spins, wave
functions of two-partial system look like:
Ψ = Ψ(x1, x2)χ ,
where: χ is spinor, responding to singlet state with a
spin s = 1, m = +1, 0,−1 or triplet state with a spin
s = 0.
In order that a complete wave function was anti-
symmetric, its coordinate part must be antisymmet-
ric for the triplet state and symmetric in the singlet
state.Thus, two identical fermions with a spin in the
singlet state are examined at processes of scattering
similar two identical spinless Bose particles [3].
For triplet state of polarized fermions a coordinate
function must be antisymmetric.
At scattering of non-polarized beams of fermions
have [3]:
σnon−polarized =
1
4
σsinglet +
3
4
σtriplet =
= |f(E, θ)|2 + |f(E, π − θ)|2 +
−Ref(E, θ)f∗(E, π − θ) .
Thus, for non-polarized fermions as well as for
triplet state, we have antisymmetric coordinate wave
function. As follows from [2], the wave function de-
scribing collision of two particles in system of the cen-
ter of mass can be representing by the following ex-
pression:
u(r⃗)r→∞ → eikz + r−1f(θ, φ)eik⃗r⃗ , (3)
where: r, θ, – spherical coordinates of a vector r⃗. For
identical particles, taking into account necessity sym-
metrization of wave function of scattering, asymp-
totic expressions for symmetric and antisymmetric
wave functions should be written down as follows:
Ψ = (eikz±e−ikz)+[f(θ, φ)± f(π − θ, φ+ π)] r−1eik⃗r⃗ ,
(4)
where ”+” corresponds to symmetric function, ”-” -
to antisymmetric function. As follows from (2) at ac-
count of identity of particles at their scattering in field
of a quantum barrier, to section of scattering of two
particles it is necessary to bring in the addition de-
termined third element in formula (2). Just the same
difference in the formula of scattering and stipulates
a change time of tunneling for identical particles by
virtue of their exchange interaction.
2. AN ESTIMATIONS OF THE TIME OF
IDENTICAL PARTICLES INTERACTION
IN FIELD OF A RECTANGULAR
POTENTIAL BARRIER
The circuit of two particles tunneling in one-
dimensional variant is represented on Fig.2.
Fig.2. The circuit of two identical particles inter-
action in field of a rectangular potential barrier
36
If there is a scattering potential, asymptotic form of
wave function looks like [4]:
rΨ =
∑
l
Pl(cos θ)gl(r) ∼=
∑
l
AlPl(cos θ) sin(kr+∆l) ,
(5)
where: ∆l = δl − lπ/2. From the scattering the-
ory with use of a method of partial waves [4] follows,
that cross-section of scattering not identical particles
is described by the formula:
σ = |f(θ)|2 =
1
k2
∣∣∣∣∣∑
l
(2l + 1)
2
Pl(cos θ)(e
2iδl − 1)
∣∣∣∣∣
2
.
(6)
This formula shows a dependence of cross-section on
a phase δl. At l = 0 angular dependence is absent,
and we have the following expression:
σ =
1
k2
(e2iδl − 1)2
4
. (7)
Taking into account this expression, we shall have:
Re(fbf
∗
c ) =
1
4k2
(e2iδ−1)(e−2iδ−1) = 1
2k2
(1−cos 2δ) .
(8)
Thus, the function describing scattering of two iden-
tical particles, will have a next view:
Ψ = (eikz ± e−ikz) + r−1eikr
√
1
2k2(1− cos 2δ)
=
= (eikz ± e−ikz) + r−1eikr
1
k
√
2
sin δ . (9)
For small values δ (as it is underlined in [4], at small
values k the phase also is small): sin δ ∼= δ ∼= eδ − 1.
Then we have:
Ψ = (eikz ± e−ikz) + r−1eikr
1
k
√
2
(eδ − 1) . (10)
Thus, for scattering of the identical particles de-
scribed by symmetric wave function, additional shift
of a phase appeared approximately equal δ (while
usually - for not identical particles, its value 2δ [4]).
The additive to time of tunneling in view of ex-
change interaction of the identical particles described
by symmetric wave function, according to [4] makes:
τϕ =
1
νg
d
dk
δ =
m
h̄k
d
dk
δ . (11)
Expressions for δ are resulted in the educational lit-
erature for various forms of potential. Having substi-
tuted these expressions in (11) it is possible to define
the interesting us size of the additive addition to time
of tunneling.
In [2, 5] it is shown, that for a barrier with poten-
tial
V (r) =
{
U, r ≤ a,
0, r > a.
the equation, describing behavior of one particle in-
side a barrier, looks like:(
d2
dr2
+K2
)
R0l(r) = 0 , R0l(0) = 0 , (12)
were:
K2 = k2 −K2
0 , K2
0 = 2mU/h̄2 .
Outside of a barrier the decision looks like:
R0l(r) = C sin(kr + δ0) . (13)
Inside of a barrier:
R0l = Cl sin Kr , if k ≥ K0 ,
R0l = Cl sin Qr , if k ≤ K0 , (14)
were: Q =
√
K2
0 − k2.
If energy of incident particles are small,Q ≈
K0 and from a condition of equality of logarithmic
derivative functions (13) and (14) it is received (at
ka≪ 1):
δ0 = arctg(KD)− ka , (15)
were:
D =
th(Qa)
Q
≈ th(K0a)
K0
.
From the equation (15) we find:
dδ0
dk
=
D
k2D2 + 1
− a .
Then ∆τϕ (change of particles tunneling time be-
cause of their exchange interaction), according to (11)
will be expressed by the formula:
∆τϕ =
m
hk(E)
(
D
k2D2 + 1
− a
)
. (16)
For calculations we shall take:
D =
th(K0a)
K0
; k2 = 2mE/h̄2 ;
a = 10−10 cm; U = 10−11 erg; E = 2 · 10−12 erg.
Having substituted these values in (16) it is possi-
ble to receive dependences of impulses incident simul-
taneously on a barrier of identical particles from en-
ergy for particles with different masses and to choose
the proper region (where ka≪ 1) for which the con-
ducted calculations will be correct [2].
On Fig.3 calculation dependences of a delay time
|∆τϕ| on energy of identical particles are presented.
Calculations are lead for particles with mass of elec-
trons (a), protons (b) and α-particles (c). As is obvi-
ous from the resulted dependences, exchange interac-
tion reduces absolute value of tunneling time and de-
pendence |∆τϕ| on mass is found out. Taking into ac-
count the requirement of symmetrization, mentioned
above, we shall note, that at scattering of α-particles
|∆τϕ| negatively, and of fermion’s particles - it is nec-
essary to take into account a spin state, since particles
can be in singlet or triplet state.
37
Fig.3. Dependences of delay time |∆τϕ| on energy identical particles with mass of electrons (a), protons (b),
and α-particles (c) at a condition ka≪ 1
Fig.4 illustrates dependence |∆τϕ| on width
of barrier for particles with mass of proton.
Fig.4. Change time of tunneling at depending on
width of a potential barrier to particles with mass of
a proton
As is obvious from figures 3-4, on change of tun-
neling time influence both mass of particles, and
width of a potential quantum barrier. Thus, the con-
ducted calculations within the framework of applied
approaching do not confirm existence of Hartman’s
effect in case of account of identical particles interac-
tion.
3. CONCLUSIONS
In work the influence of exchange effects on time of
simultaneous tunneling identical particles through a
rectangular quantum barrier is investigated. It is
shown, that such effects essentially influence on a pa-
rameters of tunneling. A change of tunneling time
of identical particles is first expected taking into ac-
count their exchange interaction in a field of rectan-
gular quantum barrier. It is discovered, that addi-
tional tunneling time (it can be negative or positive
depending on a spin state of particles and, whether
scattering bosons or fermions) depends on mass of
particles and width of barrier. It means that Hart-
man’s effect does not take place in this case. The
received results can be exploited in practical calcula-
tions and at designing the devices using processes of
quantum tunnel transitions.
References
1. A.B. Shvartsburg. Tunneling of electromagnetic
waves - paradoxes and perspectives // Successes
of physical sciences. 2007, v. 177, N1. p. 43-57 (in
Russian).
2. A.S.Davidov. Quantum mechanics. M: ”Science”,
1973, 703 p. (in Russian).
3. D.Taylor. Scattering theory. M: ”F.L.”, 1975, 567 p.
(in Russian).
4. D.Bohm. Quantum theory.M: ”Science”, 1965, 727 p.
(in Russian).
5. L. Shiff. Quantum mechanics. M: ”F.L.”, 1959, 473 p.
(in Russian).
ÈÇÌÅÍÅÍÈÅ ÂÐÅÌÅÍÈ ÒÓÍÍÅËÈÐÎÂÀÍÈß ÒÎÆÄÅÑÒÂÅÍÍÛÕ ×ÀÑÒÈÖ
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Ë.Ñ.Ìàðöåíþê
Ðàáîòà ïîñâÿùåíà èçó÷åíèþ âëèÿíèÿ îáìåííûõ ýôôåêòîâ íà âðåìÿ îäíîâðåìåííîãî òóííåëèðîâàíèÿ
òîæäåñòâåííûõ ÷àñòèö ÷åðåç ïðÿìîóãîëüíûé êâàíòîâûé áàðüåð. Ïîêàçàíî, ÷òî òàêèå ýôôåêòû ñóùå-
ñòâåííî âëèÿþò íà ïàðàìåòðû òóííåëèðîâàíèÿ. Âïåðâûå ðàññ÷èòàíî èçìåíåíèå âðåìåíè òóííåëèðîâà-
íèÿ òîæäåñòâåííûõ ÷àñòèö ñ ó÷åòîì èõ îáìåííîãî âçàèìîäåéñòâèÿ â ïîëå ïðÿìîóãîëüíîãî êâàíòîâîãî
áàðüåðà.
ÇÌIÍÀ ×ÀÑÓ ÒÓÍÍÅËÞÂÀÍÍß ÒÎÒÎÆÍÈÕ ×ÀÑÒÈÍÎÊ ×ÅÐÅÇ
ÏÐßÌÎÊÓÒÍÈÉ ÁÀÐ'�Ð ÏÐÈ �Õ ÎÁÌIÍÍIÉ ÂÇÀ�ÌÎÄI�
Ë.Ñ.Ìàðöåíþê
Ðîáîòà ïðèñâÿ÷åíà âèâ÷åííþ âïëèâó îáìiííèõ åôåêòiâ íà ÷àñ ñèíõðîííîãî òóííåëþâàííÿ òîòîæíèõ
÷àñòèíîê ÷åðåç ïðÿìîêóòíèé êâàíòîâèé áàð'¹ð. Ïîêàçàíî, ùî òàêi åôåêòè iñòîòíî âïëèâàþòü íà ïàðà-
ìåòðè òóííåëþâàííÿ. Âïåðøå ðîçðàõîâàíà çìiíà ÷àñó òóííåëþâàííÿ òîòîæíèõ ÷àñòèíîê ïðè óðàõó-
âàííi ¨õ îáìiííî¨ âçà¹ìîäi¨ â ïîëi ïðÿìîêóòíîãî êâàíòîâîãî áàð'¹ðó.
38
|
| id | nasplib_isofts_kiev_ua-123456789-80504 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T11:27:43Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Martsenyuk, L.S. 2015-04-18T15:47:32Z 2015-04-18T15:47:32Z 2014 Change time of identical particles tunneling through the rectangular barrier at their exchange interaction / L.S.Martsenyuk // Вопросы атомной науки и техники. — 2014. — № 5. — С. 35-38. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 03.65.-w, 03.65.Xp, 03.65.Nk https://nasplib.isofts.kiev.ua/handle/123456789/80504 Work is devoted to studying the influence of exchange effects on a time of simultaneous crossing by identical particles through the rectangular quantum barrier. It is shown, that such effects essentially influence on the tunneling parameters. A change of identical particles tunneling time is first computation taking into account their exchange interaction in the field of rectangular quantum barrier. Работа посвящена изучению влияния обменных эффектов на время одновременного туннелирования тождественных частиц через прямоугольный квантовый барьер. Показано, что такие эффекты существенно влияют на параметры туннелирования. Впервые рассчитано изменение времени туннелирования тождественных частиц с учетом их обменного взаимодействия в поле прямоугольного квантового барьера. Робота присвячена вивченню впливу обмiнних ефектiв на час синхронного туннелювання тотожних частинок через прямокутний квантовий бар'єр. Показано, що такi ефекти iстотно впливають на параметри туннелювання. Вперше розрахована змiна часу туннелювання тотожних частинок при урахуваннi Їх обмiнної взаємодiї в полi прямокутного квантового бар'єру. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Ядерная физика и элементарные частицы Change time of identical particles tunneling through the rectangular barrier at their exchange interaction Изменение времени туннелирования тождественных частиц через прямоугольный барьер при их обменном взаимодействии Змiна часу туннелювання тотожних частинок через прямокутний бар'єр при їх обмiннiй взаємодiї Article published earlier |
| spellingShingle | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction Martsenyuk, L.S. Ядерная физика и элементарные частицы |
| title | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction |
| title_alt | Изменение времени туннелирования тождественных частиц через прямоугольный барьер при их обменном взаимодействии Змiна часу туннелювання тотожних частинок через прямокутний бар'єр при їх обмiннiй взаємодiї |
| title_full | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction |
| title_fullStr | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction |
| title_full_unstemmed | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction |
| title_short | Change time of identical particles tunneling through the rectangular barrier at their exchange interaction |
| title_sort | change time of identical particles tunneling through the rectangular barrier at their exchange interaction |
| topic | Ядерная физика и элементарные частицы |
| topic_facet | Ядерная физика и элементарные частицы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80504 |
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