Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles
Work is devoted to theoretical research of influence of exchange interaction of identical particles on processes of simultaneous their tunneling through a rectangular potential barrier. Expression for exchange integral has been received and the analysis of dependence on parameters of identical parti...
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| Zitieren: | Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles / L.S. Martseniuk, S.P. Maydanyuk, V.S. Olhovsky // Вопросы атомной науки и техники. — 2009. — № 3. — С. 51-53. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1112742017-01-10T03:02:20Z Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles Martseniuk, L.S. Maydanyuk, S.P. Olhovsky, V.S. Ядерная физика и элементарные частицы Work is devoted to theoretical research of influence of exchange interaction of identical particles on processes of simultaneous their tunneling through a rectangular potential barrier. Expression for exchange integral has been received and the analysis of dependence on parameters of identical particles and characteristics of a quantum barrier is lead. It is shown, that as including of exchange energy changes phase characteristics of the wave functions describing process of tunneling, exchange processes influence on the time characteristics of tunneling. Робота присвячена теоретичному дослідженню впливу обмінної взаємодії тотожних часток на процеси одночасного туннелювання їх через прямокутний потенційний бар'єр. Був отриманий вираз для обмінного інтеграла V й проведений аналіз залежності V від параметрів тотожних часток і характеристик квантового бар'єра. Показано, що, оскільки включення обмінної енергії змінює фазові характеристики хвильових функцій, що описують процес туннелювання, то обмінні процеси впливають на часові характеристики туннелювання. Работа посвящена теоретическому исследованию влияния обменного взаимодействия тождественных частиц на процессы одновременного туннелирования их через прямоугольный потенциальный барьер. Было получено выражение для обменного интеграла V и проведен анализ зависимости V от параметров тождественных частиц и характеристик квантового барьера. Показано, что, поскольку включение обменной энергии изменяет фазовые характеристики волновых функций, описывающих процесс туннелирования, то обменные процессы влияют на временные характеристики туннелирования. 2009 Article Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles / L.S. Martseniuk, S.P. Maydanyuk, V.S. Olhovsky // Вопросы атомной науки и техники. — 2009. — № 3. — С. 51-53. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 03.65.-w, 03.65.Xp, 03.65.Nk http://dspace.nbuv.gov.ua/handle/123456789/111274 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. Maydanyuk, S.P. Olhovsky, V.S. Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles Вопросы атомной науки и техники |
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Work is devoted to theoretical research of influence of exchange interaction of identical particles on processes of simultaneous their tunneling through a rectangular potential barrier. Expression for exchange integral has been received and the analysis of dependence on parameters of identical particles and characteristics of a quantum barrier is lead. It is shown, that as including of exchange energy changes phase characteristics of the wave functions describing process of tunneling, exchange processes influence on the time characteristics of tunneling. |
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Article |
| author |
Martseniuk, L.S. Maydanyuk, S.P. Olhovsky, V.S. |
| author_facet |
Martseniuk, L.S. Maydanyuk, S.P. Olhovsky, V.S. |
| author_sort |
Martseniuk, L.S. |
| title |
Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles |
| title_short |
Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles |
| title_full |
Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles |
| title_fullStr |
Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles |
| title_full_unstemmed |
Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles |
| title_sort |
role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2009 |
| topic_facet |
Ядерная физика и элементарные частицы |
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http://dspace.nbuv.gov.ua/handle/123456789/111274 |
| citation_txt |
Role of exchange interaction of identical particles of the discrete spectrum at simultaneous their tunneling through the potentia barrier dividing these particles / L.S. Martseniuk, S.P. Maydanyuk, V.S. Olhovsky // Вопросы атомной науки и техники. — 2009. — № 3. — С. 51-53. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| fulltext |
ROLE OF EXCHANGE INTERACTION OF IDENTICAL
PARTICLES OF THE DISCRETE SPECTRUM AT
SIMULTANEOUS THEIR TUNNELING THROUGH THE
POTENTIA BARRIER DIVIDING THESE PARTICLES
L.S. Martseniuk∗, S.P. Maydanyuk†, V.S. Olhovsky
Institute for Nuclear Research, National Academy of Sciences of Ukraine, 03680, Kiev, Ukraine
(Received March 19, 2009)
Work is devoted to theoretical research of influence of exchange interaction of identical particles on processes of
simultaneous their tunneling through a rectangular potential barrier. Expression for exchange integral has been
received and the analysis of dependence on parameters of identical particles and characteristics of a quantum barrier
is lead. It is shown, that as including of exchange energy changes phase characteristics of the wave functions describing
process of tunneling, exchange processes influence on the time characteristics of tunneling.
PACS: 03.65.-w, 03.65.Xp, 03.65.Nk
1. INTRODUCTION
As is known [1, 2], processes of interaction between
quantum objects in some cases can be approximated
precisely enough, having presented potential of inter-
action as a rectangular quantum barrier that allows
to simplify mathematical calculations considerably.
Such approximation has allowed solving a num-
ber of the important problems of quantum mechanics,
and also welling to describe behavior of some quan-
tum objects [2]. Therefore attractive is the use of it
and for the decision of other tasks which yet it is not
enough investigated.
To the number questions which to the present
tense in general was not explored, a question on influ-
ence of exchange interaction on the processes of par-
ticles tunneling in the field of rectangular potential
barrier belongs to theoretical development of which
the present work is devoted.
2. PASSAGE OF PARTICLES THROUGH
A RECTANGULAR QUANTUM
BARRIER: THE GENERAL
CONSIDERATION
In the beginning, let us consider a variant when on a
rectangular potential barrier in width 2a and height
U particles with a pulse p from left to right fall. Equa-
tion of Schrödinger for particles looks like:
HΨ(x, t) = − p2
2m
∆Ψ + UΨ = EΨ ,
or
∆Ψ +
2m
h̄2 [E − U ] = 0.
Time dependence of own functions is described by
expression:
∆Ψ +
2m
h̄2 [E − U ] = 0.
Therefore for various areas decisions can be functions:
ΨT = AT (x) exp ip1r/h̄ (1)
at r > a,
Ψ = B exp p2r/h̄ + C exp−p2r/h̄ (2)
at r ∈ [−a, a],
Ψ = D exp ip1r/h̄ + E exp−ip1r/h̄ (3)
at r < −a, where
p1 =
√
2mE, p2 =
√
2m (U − E). (4)
Into a barrier to one particle correlations take place
[2]:
B =
A
2
{
1 +
ip1
p2
}
exp
[
(ip1 − p2)
2a
h̄
]
, (5)
C =
A
2
{
1− ip1
p2
}
exp
[
(ip1 + p2)
2a
h̄
]
. (6)
Provided that these functions and their first deriva-
tives are continuous on border of a barrier, it is possi-
ble to get the exact decision of a task which is resulted
in [1, 2] and other the sources.
∗Corresponding author E-mail address: prolisok77@yandex.ru
†E-mail address: maidan@kinr.kiev.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2009, N3. .
Series: Nuclear Physics Investigations (51), p.51-53
51
3. EXCHANGE INTERACTION OF
IDENTICAL PARTICLES IN A FIELD OF
A RECTANGULAR QUANTUM BARRIER
Processes of collision are considered in system of
the center of weights, for which speed of two particles
before collision and after are opposite. Interaction of
particles takes place in the field of a rectangular po-
tential barrier (see figure).
Let us consider two cases. The 1-st case: let us
assume that the barrier is opaque to tunneling. In
this case At in a formula (1) equals a zero. We will
assume that a barrier becomes opaque to tunneling at
some value of potential corresponding U ′. We have:
Ψb,c = Ψb(r1)Ψc(r2), where Ψb(r1) is a solution of
equation of: HΨb(r1) = EbΨb(r1), answering a value
Eb, and, similarly, for Ψc(r2), corresponding to a
value Ec. Energy of the system: E0 = Ec + Eb. Be-
cause at a change placed of particles nothing changes,
Ψc,b = Ψb(r2)Ψc(r1) will correspond to the same en-
ergy E0, i.e. we have double degeneration of power
level of the system.
-a a
R
0
r1 r2
cb
∆U U
U‘
The circuit of interaction of two identical particles
in a field of rectangular potential barrier
The 2-nd case: a barrier is transparent for tunnel-
ing. This case is realized, when the additional con-
stant field is enclosed to area of a barrier in opposite
to first direction, creating additional potential |∆U |.
Then the size of potential of barrier will fall down on
a size |∆U |, and have:
U = U ′ − |∆U |.
Let us consider additional potential |∆U | as small
indignation in system of two particles, originally be-
tween itself not interactive because of opacity of a
barrier (a case 1).
The additional probability of particles to pass
through a barrier leads to removal of degeneration.
At simultaneous approaching of particles b and
c to the borders of a barrier there can be the ef-
fects connected to exchange interaction of particles.
Such effects will take place only for identical particles
and their size is determined by “overlapping” of wave
functions, i.e. they are realized when wave functions
of particles will be distinct from zero in the same area
of space.
Taking to account that at interaction of identi-
cal particles symmetric or antisymmetric states are
realized only [1, 2], we shall have for symmetric or
antisymmetric wave function of expression:
Ψ1,2 = Ψ± = 1√
2
(Ψ1 ±Ψ2) =
= 1√
2
[Ψb(r1)Ψc(r2)±Ψc(r1)Ψb(r2)] ,
(7)
Size |Ψ±|2 - expresses probability of finding of the
system in the symmetric or antisymmetric state de-
pending on what particles are bosons or fermions.
If particles were not identical, the system would
be described a non symmetric function and we would
have:
|Ψ1,2|2 = |Ψb(r1)Ψc(r2)|2.
For identical particles there is an additional element:
|Ψb(r1)Ψc(r2)|∗Ψc(r1)Ψb(r2), and, consequently, ad-
ditional energy, proper to energy of interaction of
identical particles, appears. This energy, taking to
account that a potential barrier has a form, indicated
on figure, it is possible to calculate, in accordance
with [1] on a formula:
V = |∆U |
∫
r
Ψ∗b(r1)Ψ∗c(r2)Ψb(r2)Ψc(r1). (8)
From equalization (2) evidently, that a size V is
determined by “overlapping” the wave functions Ψb
and Ψc.
For values r < −a and r > a the value of inte-
gral is identically equally to zero, because for these
areas U = 0, therefore will conduct integration for
−a ≤ r ≤ a.
Into a barrier for one particle the ratio (2), (5)
and (6) are fair.
At p2a/h̄ À 1 (i.e. provided that parameters an
exhibitor strongly change from one border of a barrier
to another) C À B [2].
Then it is possible to do such approaching:
Ψ∗b(r1) = C∗ exp(
p2r1
h̄
) ,
Ψ∗c(r2) = C∗ exp(−p2r2
h̄
) ,
Ψb(r2) = B exp(
p2r2
h̄
) + C exp(−p2r2
h̄
) ,
Ψc(r1) = B exp(−p2r1
h̄
) + C exp(
p2r1
h̄
) .
In accordance with (8) have:
52
V = |∆U |
∫ ∫
C∗ exp
p2r1
h̄
· C∗ exp−p2r1
h̄
·
{
B∗ exp
p2r1
h̄
+ C∗ exp−p2r1
h̄
}
×
×
{
B exp−p2r1
h̄
+ C exp
p2r1
h̄
}
dr1 dr2 = |∆U | ·
{[
aBC∗
]2 −
[
CC∗
h̄
2 p2
exp−2p2a
h̄
]2
}
.
(9)
From (9) follows, that V = 0 at
|B| =
∣∣∣ Ch̄
2p2a
exp−2p2a
h̄
∣∣∣,
i.e. exchange interaction at such parameters of tun-
neling is absent.
More exact values it is possible to get for the
proper parameters of tunneling, if to present the wave
function of a moving particle as a wave package [3].
During [3] time of tunneling it is determined as:
τPh
T = h̄
∂ arg AT
∂E
. (10)
As to exchange energy V there corresponds a certain
value ω′ (V = h̄ω′) and, accordingly, ΨT passes in
Ψ′T = A′T exp( i
h̄
√
2m(E ± V ) that, hence, is received
also change of value of time of tunneling.
4. CONCLUSION
Within the framework of approximation of poten-
tial of interaction of particles as a rectangular poten-
tial barrier expression for exchange integral of identi-
cal particles, simultaneously tunneling through a bar-
rier towards each other is found.
It is shown, that the account of exchange in-
teraction changes time of tunneling. Such analysis
of processes of simultaneous tunneling is conducted
first.
References
1. L.D. Landau and E.M. Lifshitz. Quantum me-
chanics, course of Theoretical Physics. Vol. 3.
Moskva: ”Nauka” 1989, p. 768 (in Russian; Eng-
lish variant: Oxford, Uk, Pergamon, 1982).
2. D. Bohm. Quantum theory. Moskva: ”Science”,
1965, 727p. (in Russian)
3. G. Privitera, G. Salesi, V.S. Olkhovsky, E. Re-
cami. Tunneling times: An elementary introduc-
tion // Rivista del Nuovo Cimento. 1991, v.26(4),
p.55.
4. V.G. Levich, J.A. Vdovin, V.A. Miamlin. Course
of theoretical physics. Vol.2. Moskva: ”Science”,
1971, p. 936 (in Russian).
РОЛЬ ОБМЕННОГО ВЗАИМОДЕЙСТВИЯ ТОЖДЕСТВЕННЫХ ЧАСТИЦ
ДИСКРЕТНОГО СПЕКТРА ПРИ ОДНОВРЕМЕННОМ ТУННЕЛИРОВАНИИ ИХ
ЧЕРЕЗ ПОТЕНЦИАЛЬНЫЙ БАРЬЕР, РАЗДЕЛЯЮЩИЙ ЭТИ ЧАСТИЦЫ
Л.С. Марценюк, С.П. Майданюк, В.С. Ольховский
Работа посвящена теоретическому исследованию влияния обменного взаимодействия тождествен-
ных частиц на процессы одновременного туннелирования их через прямоугольный потенциальный
барьер. Было получено выражение для обменного интеграла V и проведен анализ зависимости V от
параметров тождественных частиц и характеристик квантового барьера. Показано, что, поскольку
включение обменной энергии изменяет фазовые характеристики волновых функций, описывающих
процесс туннелирования, то обменные процессы влияют на временные характеристики туннелирова-
ния.
РОЛЬ ОБМIННОЇ ВЗАЄМОДIЇ ТОТОЖНИХ ЧАСТОК ДИСКРЕТНОГО СПЕКТРА
ПРИ ОДНОЧАСНОМУ ТУННЕЛЮВАННI ЇХ ЧЕРЕЗ ПОТЕНЦIЙНИЙ БАР’ЄР, ЩО
РОЗДIЛЯЄ ЦI ЧАСТИНКИ
Л.С. Марценюк, С.П. Майданюк, В.С. Ольховський
Робота присвячена теоретичному дослiдженню впливу обмiнної взаємодiї тотожних часток на про-
цеси одночасного туннелювання їх через прямокутний потенцiйний бар’єр. Був отриманий вираз для
обмiнного iнтеграла V й проведений аналiз залежностi V вiд параметрiв тотожних часток i характе-
ристик квантового бар’єра. Показано, що, оскiльки включення обмiнної енергiї змiнює фазовi харак-
теристики хвильових функцiй, що описують процес туннелювання, то обмiннi процеси впливають на
часовi характеристики туннелювання.
53
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