The time of simultaneous tunneling of identical particles through the rectangular quantum barrier
Work is devoted to studying the influence of exchange processes on a time of simultaneous crossing by identical particles of a rectangular quantum barrier. It is shown, that such processes essentially influence on the parameters of tunneling. The size of addition to time of identical particles tunne...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2010
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | The time of simultaneous tunneling of identical particles through the rectangular quantum barrier / L.S. Martseniuk, V.S. Olhovsky // Вопросы атомной науки и техники. — 2011. — № 3. — С. 19-21. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1110682017-01-09T03:02:23Z The time of simultaneous tunneling of identical particles through the rectangular quantum barrier Martseniuk, L.S. Olhovsky, V.S. Ядерная физика и элементарные частицы Work is devoted to studying the influence of exchange processes on a time of simultaneous crossing by identical particles of a rectangular quantum barrier. It is shown, that such processes essentially influence on the parameters of tunneling. The size of addition to time of identical particles tunneling, arising up because of their exchange interaction in a field of a rectangular quantum barrier is first counted. Робота присвячена теоретичному дослідженню впливу обмінної взаємодії тотожних частинок на часові характеристики їх синхронного тунелювання через прямокутний потенційний бар'єр. Показано, що, оскільки включення обмінної енергії змінює фазові характеристики хвильових функцій, що описують процес тунелювання, то обмінні процеси впливають на часові характеристики цього процесу. Вперше було розраховано значення додаткового часу тунелювання тотожних частинок з урахуванням їх обмінної взаємодії. Работа посвящена теоретическому исследованию влияния обменного взаимодействия тождественных частиц на временные характеристики их одновременного туннелирования через прямоугольный потенциальный барьер. Показано, что, поскольку включение обменной энергии изменяет фазовые характеристики волновых функций, описывающих процесс туннелирования, то обменные процессы влияют на временные характеристики этого процесса. Впервые было рассчитано значение дополнительного времени туннелирования тождественных частиц с учетом их обменного взаимодействия. 2010 Article The time of simultaneous tunneling of identical particles through the rectangular quantum barrier / L.S. Martseniuk, V.S. Olhovsky // Вопросы атомной науки и техники. — 2011. — № 3. — С. 19-21. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 03.65.-w, 03.65.Xp, 03.65.Nk http://dspace.nbuv.gov.ua/handle/123456789/111068 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Martseniuk, L.S. Olhovsky, V.S. The time of simultaneous tunneling of identical particles through the rectangular quantum barrier Вопросы атомной науки и техники |
| description |
Work is devoted to studying the influence of exchange processes on a time of simultaneous crossing by identical particles of a rectangular quantum barrier. It is shown, that such processes essentially influence on the parameters of tunneling. The size of addition to time of identical particles tunneling, arising up because of their exchange interaction in a field of a rectangular quantum barrier is first counted. |
| format |
Article |
| author |
Martseniuk, L.S. Olhovsky, V.S. |
| author_facet |
Martseniuk, L.S. Olhovsky, V.S. |
| author_sort |
Martseniuk, L.S. |
| title |
The time of simultaneous tunneling of identical particles through the rectangular quantum barrier |
| title_short |
The time of simultaneous tunneling of identical particles through the rectangular quantum barrier |
| title_full |
The time of simultaneous tunneling of identical particles through the rectangular quantum barrier |
| title_fullStr |
The time of simultaneous tunneling of identical particles through the rectangular quantum barrier |
| title_full_unstemmed |
The time of simultaneous tunneling of identical particles through the rectangular quantum barrier |
| title_sort |
time of simultaneous tunneling of identical particles through the rectangular quantum barrier |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2010 |
| topic_facet |
Ядерная физика и элементарные частицы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111068 |
| citation_txt |
The time of simultaneous tunneling of identical particles through the rectangular quantum barrier / L.S. Martseniuk, V.S. Olhovsky // Вопросы атомной науки и техники. — 2011. — № 3. — С. 19-21. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-08T01:34:29Z |
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| fulltext |
THE TIME OF SIMULTANEOUS TUNNELING OF
IDENTICAL PARTICLES THROUGH THE RECTANGULAR
QUANTUM BARRIER
L.S. Martseniuk∗, V.S. Olhovsky
Institute for Nuclear Research, National Academy of Sciences of Ukraine, 03680, Kiev, Ukraine
(Received April 6, 2010)
Work is devoted to studying the influence of exchange processes on a time of simultaneous crossing by identical
particles of a rectangular quantum barrier. It is shown, that such processes essentially influence on the parameters of
tunneling. The size of addition to time of identical particles tunneling, arising up because of their exchange interaction
in a field of a rectangular quantum barrier is first counted.
PACS: 03.65.-w, 03.65.Xp, 03.65.Nk
1. INTRODUCTION
At research of processes of two identical particles dis-
persion on each other it is necessary to take into ac-
count their exchange interaction which essentially in-
fluences on the parameters of dispersion. Such ac-
count for coulomb potential has been lead earlier [1-
3]. The Mott’s formula is describing the dispersion of
identical particles in a field of coulomb potential, it
contains additional element, arising up owing to effect
of exchange interaction. Parameters of the tunneling
process which can be considered as a limiting case
of dispersion at a zero corner of dispersion also can
depend on characteristics of exchange processes be-
tween identical particles. In the given work research
of influence of exchange interaction of identical parti-
cles on a time of their simultaneous tunneling through
a rectangular quantum barrier is lead. As follows
from [1], a wave function describing the collision of
two particles in the system of center of masses can be
presented by the following expression:
u(r)r→∞ → eih̄z + r−1 f(θ, ϕ)eikr , (1)
where r, θ, ϕ - are spherical coordinates of vector r.
For identical particles, taking into account the neces-
sity of symmetrisation of wave function of dispersion,
asymptotical expressions for symmetric and antisym-
metric functions of waves it is necessary to write down
as follows:
Ψ = (eikz ± e−ikz) +
[f(θ, ϕ)± f(π − θ, ϕ + π)]r−1eikr , (2)
where ”+” corresponds to symmetric function, ”-”
to antisymmetric function. The effective section of
dispersion, according to [1], will be described by:
σ(θ, ϕ) = |f(θ, ϕ)|2 + |f(π − θ, ϕ + π)|2 ±
2Re[f(θ, ϕ)f∗(π − θ, ϕ + π)] . (3)
As follows from this expression, at the account of
identity of particles at their dispersion in the field of
quantum barrier, to the section of dispersion for two
particles it is necessary to bring in addition, deter-
minate by the third element in a formula (3). Just
the same difference in the formula of dispersion and
determines a change time for identical particles by
virtue of their co-operation.
2. ESTIMATION THE TIME OF
IDENTICAL PARTICLES TUNNELING
THROUGH THE RECTANGULAR
QUANTUM BARRIER
The chart of tunneling of two identical particles in
one dimension variant is represented on the Figure
The circuit of interaction of two identical particles
in a field of rectangular potential barrier
In a theory of dispersion are shown that if present a
dispersive potential, the asymptotical form of wave
function looks like [2]:
∗Corresponding author E-mail address: prolisok77@yandex.ru
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N3.
Series: Nuclear Physics Investigations (55), p.19-21.
19
rΨ =
∑
l
Pl(cos θ) gl(r) ∼=
∑
l
AlPl(cos(θ) sin(kr + ∆) , (4)
where∆l = δl− lπ
2 It ensues from the theory of disper-
sion with the use of method of partsial waves [2], that
expression for transversal dispersion of not identical
particles looks like
σ = |f(θ)|2 =
1
k2
|
∑
l
(2l + 1)
2
Pl(cos θ)(e2iδl − 1)|2 . (5)
This formula shows dependence of transversal section
on a phase δ. At l = 0 angular dependence is absent,
and have the following expression:
σ =
1
k2
[e2iδl − 1]2
4
. (6)
Taking into account this expression will have:
Re(fbf
∗
c ) =
1
4k2
(e2iδ − 1)(e−2iδ − 1) =
1
2k2
(1− cos 2δ) . (7)
Thus, function, describing dispersion of two identical
particles will have the following kind:
Ψ = (eikz ± e−ikz) + r−1eikr
√
1
2k2
(1− cos 2δ) =
(eikz ± e−ikz) + r−1eikr 1
k
√
2
sin δ . (8)
For small values δ (as specified in [2] at small values
δ a phase is also small): sin δ ∼= δ ∼= eδ − 1
Ψ = (eikz ± e−ikz) + r−1eikr 1
k
√
2
(eδ − 1) . (9)
Thus, for dispersion of symmetric identical particles
the additional change of phase appeared approxi-
mately equal δ (while in ordinary case - for not iden-
tical particles, his value 2δ , [2]. Addition to time of
tunneling taking into account exchange interaction of
identical particles, described a symmetric wave func-
tion, in accordance with [2, 5] makes:
τf =
1
νg
d
dk
δ =
m
h̄k
d
dk
δ . (10)
Expressions for δ are led in educational literature for
various forms of potential. Having substituted these
expressions in (10) it is possible to define the interest-
ing us size of time of tunneling. In [2] such expressions
are brought for tan δ
tan δ =
k
a − tan ka
1 + k
a tan ka
∼=
k
a − ka
1 + k
aka
, (11)
where:a = k1 cot k1a,k2
1 = (E − U)2m
h̄2 for quantum
hole and k2
1 = (U − E) 2m
h̄2 -for a barrier in high U ;
k2 = 2m
h̄2 E. From expression for k and k1 follow, that
dk1
dk = k
k1
. Then:
dσ
dk
=
1
1 + (
k
a−ka
1+ k
a ka
)2
×
(1 + k2
α a)(α−k dα
dk
α2 − a)− ( k
α − ka)[(α−k dα
dk
α2 )ka + ka
α ]
(1 + k2a
α )2
,
(12)
where:
dα
dk
=
dk1
dk
(cot k1a− k
a
sin2 k1a
) =
k
k1
sin 2k1a + 2k1a
2 sin2 k1a
. (13)
Then to addition of tunneling time get next expres-
sion:
∆τf =
1
νg
d
dk
δ =
m
h̄k
(α + k2a)(α− α2a− k dα
dk )− (k − αak)(2αak − k dα
dk )
α3[(1 + k
αka)2 + ( k
α − ka)2]
.
(14)
3. CONCLUSIONS
Additional time of tunnelling for identical particles,
simultaneously crossing through a quantum poten-
tial barrier (a potential hole) in opposite directions is
designed, taking into account their exchange interac-
tion. Tunnelling was considered as a limiting case of
dispersion at a corner of dispersion aspiring to zero.
It is shown, that the exchange interaction changes
the time of tunnelling. Such analysis of the temporal
descriptions of simultaneous tunnelling processes is
conducted first.
References
1. L. Schiff. Quantum mechanics. M: ”F.L.”, 1959,
473 p. (in Russian).
2. D. Bohm. Quantum theory. M: ”Science”, 1965,
727 p. (in Russian).
3. L.D. Landau and E.M. Lifshitz. Quantum me-
chanics, course of Theoretical Physics, v. 3. M:
”Nauka”, 1989, p.768 [in Russian; eng. variant:
Oxford, Uk, Pergamon, 1982].
4. V.G. Levich, J.A. Vdovin, V.A. Miamlin. Course
of theoretical physics. v. 2. M: Science, 1971,
p.936 (in Russian).
5. G. Privitera, G. Salesi, V.S. Olkhovsky, E. Re-
cami. Tunneling times: An elementary introduc-
tion // Rivista del Nuovo Cimento. 1991, 26 (4),
p.55.
20
ВРЕМЯ СИНХРОННОГО ТУННЕЛИРОВАНИЯ ТОЖДЕСТВЕННЫХ ЧАСТИЦ
ЧЕРЕЗ ПРЯМОУГОЛЬНЫЙ КВАНТОВЫЙ БАРЬЕР
Л.С. Марценюк, В.С. Ольховский
Работа посвящена теоретическому исследованию влияния обменного взаимодействия тождествен-
ных частиц на временные характеристики их одновременного туннелирования через прямоугольный
потенциальный барьер. Показано, что, поскольку включение обменной энергии изменяет фазовые ха-
рактеристики волновых функций, описывающих процесс туннелирования, то обменные процессы влия-
ют на временные характеристики этого процесса. Впервые было рассчитано значение дополнительного
времени туннелирования тождественных частиц с учетом их обменного взаимодействия.
ЧАС СИНХРОННОГО ТУНЕЛЮВАННЯ ТОТОЖНИХ ЧАСТИНОК ЧЕРЕЗ
ПРЯМОКУТНИЙ КВАНТОВИЙ БАР’ЄР
Л.С. Марценюк, В.С. Ольховський
Робота присвячена теоретичному дослiдженню впливу обмiнної взаємодiї тотожних частинок на
часовi характеристики їх синхронного тунелювання через прямокутний потенцiйний бар’єр. Показано,
що, оскiльки включення обмiнної енергiї змiнює фазовi характеристики хвильових функцiй, що опи-
сують процес тунелювання, то обмiннi процеси впливають на часовi характеристики цього процесу.
Вперше було розраховано значення додаткового часу тунелювання тотожних частинок з урахуванням
їх обмiнної взаємодiї.
21
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