One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ
A condition, at which inverse power one-dimensional potential becomes reflectionless during propagation through it of a plane wave, is obtained on the basis of SUSY QM methods. A scattering of a particle on spherically symmetric potential is analysed with taking into account of the reflectionless...
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irk-123456789-805162015-04-19T03:02:22Z One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ Maydanyuk, S.P. Ядерная физика и элементарные частицы A condition, at which inverse power one-dimensional potential becomes reflectionless during propagation through it of a plane wave, is obtained on the basis of SUSY QM methods. A scattering of a particle on spherically symmetric potential is analysed with taking into account of the reflectionless possibility. На основі методів SUSY QM отримано умову, при якій обернено степеневий потенціал стає абсолютно прозорим при проходженні крізь його плоскої хвилі. Представлено аналіз розсіювання частинки на сферично- симетричному потенціалі з врахуванням можливості абсолютної прозорості. С помощью методов SUSY QM получено условие, при котором обратно степенной потенциал становится абсолютно прозрачным при прохождении через него плоской волны. Представлен анализ рассеяния частицы на сферически-симметричном потенциале с учетом возможности абсолютной прозрачности. 2004 Article One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ / S.P. Maydanyuk // Вопросы атомной науки и техники. — 2004. — № 5. — С. 22-25. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 03.65.-w, 03.65.Db, 03.65.Nk, 03.65.Xp, 24.30.-v http://dspace.nbuv.gov.ua/handle/123456789/80516 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 Maydanyuk, S.P. One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ Вопросы атомной науки и техники |
| description |
A condition, at which inverse power one-dimensional potential becomes reflectionless during propagation through it of a plane wave, is
obtained on the basis of SUSY QM methods. A scattering of a particle on spherically symmetric potential is analysed with taking into account of the reflectionless possibility. |
| format |
Article |
| author |
Maydanyuk, S.P. |
| author_facet |
Maydanyuk, S.P. |
| author_sort |
Maydanyuk, S.P. |
| title |
One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ |
| title_short |
One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ |
| title_full |
One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ |
| title_fullStr |
One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ |
| title_full_unstemmed |
One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ |
| title_sort |
one-dimensional inverse power reflectionless potentials v(x) = const│x-x₀ │⁻ⁿ |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Ядерная физика и элементарные частицы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80516 |
| citation_txt |
One-dimensional inverse power reflectionless potentials V(x) = const│x-x₀ │⁻ⁿ / S.P. Maydanyuk // Вопросы атомной науки и техники. — 2004. — № 5. — С. 22-25. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT maydanyuksp onedimensionalinversepowerreflectionlesspotentialsvxconstxx0n |
| first_indexed |
2025-07-06T04:31:40Z |
| last_indexed |
2025-07-06T04:31:40Z |
| _version_ |
1836870586089865216 |
| fulltext |
ONE-DIMENSIONAL INVERSE POWER REFLECTIONLESS
POTENTIALS nxxconstxV −−⋅= 0)(
S.P. Maydanyuk
National Academy of Sciences of Ukraine, Institute for Nuclear Research, Kiev, Ukraine
e-mail: maidan@kinr.kiev.ua
A condition, at which inverse power one-dimensional potential
nxxxV 0)( −= α (α = const, x0 = const,
[;] + ∞∞−∈x , n is a natural number) becomes reflectionless during propagation through it of a plane wave, is
obtained on the basis of SUSY QM methods. A scattering of a particle on spherically symmetric potential
nrrrV 0)( −±= α is analysed with taking into account of the reflectionless possibility.
PACS: 03.65.-w, 03.65.Db, 03.65.Nk, 03.65.Xp, 24.30.-v
1. INTRODUCTION
Methods of supersymmetric quantum mechanics
(SUSY QM) allow finding quantum systems (both in
the region of continuous energy spectrum and discrete
one), which potentials have a penetrability coefficient of
particles through them equal to one. One can name such
quantum systems (and their potentials) as reflectionless
[1].
A resonant tunneling phenomenon and, especially,
papers, directed to study of its demonstration in
concrete physical problems (for example, see Ref. [2]),
have been caused an increased interest. The
penetrability coefficient of the barrier during the
resonant tunneling becomes large to the maximum. But
the reflectionless potentials are interested in that they
have the penetrability coefficient, practically equal to
one in a whole region of the energy spectrum, whereas
the resonant tunneling exists at selected energy levels
only. A number of papers devoted to study of properties
of the reflectionless quantum systems have been
increasing each year. Here, note the bright reviews [3,
4], where both the methods for detailed study of
properties of one- and multichannel reflectionless
quantum systems, and enough simple approaches for
their qualitative understanding are presented. All these
methods have found their application in scattering
theory (both in direct problem and in inverse one).
Note, that SUSY QM methods for study of the
properties of the systems in the continuous energy
spectrum are developed less than in the discrete one.
Besides, majority of the obtained reflectionless
potentials are expressed with use of series in enough
complicated form, and any found reflectionless potential
with a simple analytical form can be useful by its
clearness in qualitative analysis of the quantum systems
properties. In this paper we analyse the one-dimensional
and spherically symmetric quantum systems in the
region of the continuous energy spectrum, which
potentials have an inverse power dependence on a space
coordinate, and we obtain conditions, when these
systems (and potentials) become reflectionless.
2. INTERDEPENDENCE BETWEEN
SPECTRAL CHARACTERISTICS OF
POTENTIALS-PARTNERS
In the beginning we consider an one-dimensional
case of a motion of a particle with mass m inside a
potential field V(x). Let's introduce the following
operators A and A+:
),(
2
),(
2
xW
dx
d
m
A
xW
dx
d
m
A
+−=
+=
+
(1)
where W(x) is a function, given on the whole space
region x. We suppose that this function is continuous on
the whole region of its definition except for some
possible points of discontinuity. On the basis of
operators A and A+ one can construct two Hamiltonians
for a motion of this particle inside two different fields
V1 (x) and V2 (x):
),(
2
),(
2
22
22
2
12
22
1
xV
dx
d
m
AAH
xV
dx
d
m
AAH
+−==
+−==
+
+
(2)
where potentials V1 (x) and V2 (x) are defined as follows:
.)(
2
)()(
,)(
2
)()(
2
2
2
1
dx
xdW
m
xWxV
dx
xdW
m
xWxV
+=
−=
(3)
In development of SUSY QM theory the function
W(x) is named as superpotential, whereas the potentials
V1 (x) and V2 (x) are named as supersymmetric
potentials-partners [5]. Composition of Hamiltonians of
two quantum systems on the basis of the operators A
and A+ establish interdependence between spectral
characteristics (spectra of energy, wave functions) of
22 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 22-25.
mailto:maidan@kinr.kiev.ua
these systems. One can see a reason of such
interdependence in that two different potentials V1 (x)
and V2 (x) express through the same function W(x).
If the energy spectra of these systems are discrete,
then one can write:
,
,
)2()2()2()2(
2
)1()1()1()1(
1
nnnn
nnnn
EAAH
EAAH
ϕϕϕ
ϕϕϕ
==
==
+
+
(4)
where )1(
nE and )2(
nE are the energy levels with
number n (n is a natural number) for two systems with
potentials V1 (x) and V2 (x), )1(
nϕ and )2(
nϕ are wave
functions corresponding to these levels. We obtain:
).()(
),()(
)2()2()2()2(
1
)1()1()1()1(
2
nnnn
nnnn
AEAAAAH
AEAAAAH
ϕϕϕ
ϕϕϕ
++++
+
==
==
(5)
We displace V1 (x) by such a way that 0)1(
0 =E (it
has no influence into levels distribution inside energy
spectra and into a form of wave functions). Analysing
Eq. (5), one can obtain the following interdependencies
between the energy spectra and the wave functions [5]:
.)(
,)(
,0
,
)2(2/1)2()1(
1
)1(
1
2/1)1(
1
)2(
)1(
0
)1(
1
)2(
nnn
nnn
nn
AE
AE
E
EE
ϕϕ
ϕϕ
+−
+
+
−
+
+
=
=
=
=
(6)
Here, a normalisation condition for wave functions
inside the discrete energy spectrum are taken into
account:
.1|)(|,1|)(| 2)2(2)1( ∫∫ == dxxdxx nn ϕϕ (7)
If the energy spectra of two systems are continuous,
then one can find interdependence between their wave
functions also (here, the expressions (5) will be changed
a little):
),,(),(
),,(),(
)2()1(
)1()2(
xkAconstxk
xkAconstxk
ϕϕ
ϕϕ
+⋅=
⋅=
(8)
where ϕ(1)(k, x) and ϕ(2)(k, x) are the wave functions
for two systems with potentials V1 (x) and V2 (x). For
obtaining the exact dependence between the wave
functions in Eq. (8) one need to take into account a
condition of their normalisation (for the continuous
energy spectrum) with view of boundary conditions.
For the quantum systems with the continuous energy
spectra the SUSY QM methods allow to establish the
interdependence between the coefficients of the
penetrability and the reflection [5]. Let the potentials
V1 (x) and V2 (x) be finite at ± ∞→x , i.e. at
±=± ∞→ WxW )( (9)
we obtain:
2
21 )()( ±=± ∞→=± ∞→ WxVxV . (10)
Consider propagation of a plane wave eikx in positive
direction of x-axis in the field of the potentials V1 (x)
and V2 (x). In result of its incidence from the left we
obtain transmitted waves T1 (k') eik’x and T2 (k') eik’x, and
also reflected waves R1 (k) e-ikx and R2 (k) e-ikx. We have:
,),(
,),(
'
2,1
)2,1(
2,1
)2,1(
xik
ikxikx
eTxk
eRexk
→+ ∞→
+→− ∞→ −
ϕ
ϕ
(11)
where k and k' are defined as follows:
.', 22
+− −=−= WEkWEk (12)
Taking into account the interdependence (8)
between the wave functions for two systems with the
continuous spectra, we write:
,)'(
),)(
)((
2
''
1
2
1
TeWikNeT
ReWik
eWikNeRe
xikxik
ikx
ikxikxikx
+
−
−
−
−
+−=
++
++−=+
(13)
where N is constant, defined from the normalisation
conditions. Equating terms with the same exponent and
estimating N, we obtain:
.
'
)()(
,)()(
21
21
ikW
ikW
kTkT
ikW
ikW
kRkR
−
−
=
−
+
=
−
+
−
−
(14)
Expressions (14) establish the interdependence
between the amplitudes of penetrability and reflection
for two quantum systems. The coefficients of
penetrability and reflections of the potentials V1 (x) and
V2 (x) can be calculated as squares of modules of the
penetrability and reflection amplitudes.
3. POTENTIAL OF THE FORM
( ) 2
0)( xxconstxV −=
Let's consider superpotential of the form:
>
+
<
−=
,0,
,0,
)(
0
0
xat
xx
xat
xxxW α
α
(15)
where α > 0, x0 > 0. On the basis of Eq. (3) we find
supersymmetric potentials-partners V1 (x) and V2 (x):
+
−
=+=
−
−
=−=
<
,
2)(
)(
2
)()(
,
2)(
)(
2
)()(
0
2
0
2
2
2
0
2
1
mxxdx
xdW
m
xWxV
mxxdx
xdW
m
xWxV
xfor
αα
αα
(16)
+
+
=+=
−
+
=−=
>
,
2)(
)(
2
)()(
,
2)(
)(
2
)()(
0
2
0
2
2
2
0
2
1
mxxdx
xdW
m
xWxV
mxxdx
xdW
m
xWxV
xfor
αα
αα
(17)
23
From Eq. (16) for V1 (x) one can see that at the
condition
m2
=α (18)
potential V1 (x) becomes constant. The penetrability
coefficient relatively the propagation of the plane wave
through this potential equal to one and, in this sense, the
potential V1 (x) is reflectionless. In accordance with
Eq. (14), the penetrability coefficient of the potential
V2 (x) equals to one also:
1|||| 2
2
2
1 == TT . (19)
Note the following property: the penetrability
coefficient for the reflectionless potential is not changed
with change of x0 (at x0 > 0).
At x0 < 0 the region x ∈ ]-|x0|, +|x0|[ appears, where
the potentials have infinite high values and, in this
sense, they have absolute opacity. A case x0 = 0 is
boundary.
4. ONE-DIMENSIONAL POTENTIAL
V(x)=const / | x - x0
| n AND SPHERICALLY
SYMMETRIC POTENTIALS
V(x)=const / | r - r0
| n
Now we consider more general case with the
superpotential of the following form:
>
+
<
−
=
,0,
,0,
)(
0
0
xat
xx
xat
xx
xW
n
n
α
α
(20)
where α > 0, x0 > 0, n is a natural number. Find the
potentials-partners V1 (x) and V2 (x):
>
+
−
+
<
−
−
−
= −
−
,0,
2)(
,0,
2)(
)( 1
0
2
0
1
0
2
0
1
xat
m
xxn
xx
xat
m
xxn
xx
xV n
n
n
n
αα
αα
(21)
>
+
+
+
<
−
+
−
= −
−
,0,
2)(
,0,
2)(
)( 1
0
2
0
1
0
2
0
2
xat
m
xxn
xx
xat
m
xxn
xx
xV n
n
n
n
αα
αα
(22)
Therefore, the potential V1 (x) can be constant only
when the condition from the following one is fulfilled:
n=0 or n=1. (23)
The condition n=0 gives trivial solutions. Let's
consider another condition: n=1. In this case the
potential V2 (x) becomes reflectionless if the condition
(18) is fulfilled. If the condition (23) is not fulfilled then
one cannot reach the constancy of the potentials V1 (x)
or V2 (x) by change of the coefficients α and m. If to
change sign at W(x), then the sign at the potentials
V1 (x) and V2 (x) is changed also. Here, analysis
described above remains applicable.
Now we generalise the analysis of the one-
dimensional reflectionless potentials described above
into spherically symmetric case (at l=0). Here, one need
to use the functions W(r) and V1,2 (r) for the positive
r>0 only. At n=1 we obtain:
+
+
±=
−
+
±=
⇒
+
±=
.
2)
)(
,
2)(
)(
)(
2
0
2
2
0
1
0
mrrx
xV
mrr
rV
rr
rW
αα
αα
α
(24)
When the condition (18) is fulfilled then the
potential V1 (r) is constant, the potentials V1 (r) and
V2 (r) are reflectionless, and scattering of a particle upon
them is resonant. Note that a case n=1 is boundary
between potentials with n>1 (where an incidence of
particle upon a centre is possible) and the potentials
with n<1 (where the incidence of the particle upon the
centre is not possible).
5. CONCLUSIONS
On the basis of SUSY QM methods the condition is
found, under which the potential, having the inverse
power dependence on a space coordinate, becomes
reflectionless for wave propagation through it. The
potentials of such a type are interested in that they have
enough obvious and simple form in a comparison on a
number of potentials studying in [3,4], they are
24
expressed through elementary functions in an analytical
form in a contradiction on a majority of shape invariant
potentials studying in [5] and expressing with use of
series, and they are considered often in problems of the
scattering theory.
As further perspective, a problem of extension of a
class of the reflectionless potentials on the basis of
inverse power reflectionless potentials with use of
canonical transformations of coordinates (this method is
used for obtaining new exactly solvable potentials on
the basis of known one and described in details in [5])
can be studied.
Here, we note that solving the equation
const
dx
xdW
m
xWxV =±= )(
2
)()( 2
1
, (25)
one can find a general form of the function W(x), which
determines the reflectionless potentials. From here one
can obtain all types of the reflectionless potentials.
Here, partial solutions of Eq. (25) are:
( )
,)(
,)(tanh)(
,)(
0
0
constxW
xxBxW
xx
xW
=
−⋅=
−
±=
α
α
(26)
where B = const. The superpotential W(x) = B tanh(α(x
−x0)) is known in literature (for example, see [5]).
REFERENCES
1. V.M. Chabanov, B.N. Zakhariev. Absolutely
transparent multicannel systems. Unexpected
peculiarities // Physics Letters B. 1993, v. 319, p. 13-
15.
2. N. Saito, Y. Kayanuma. Resonant tunneling of a
composite particle through a single potential barrier
// Journ. of Phys. Condens. Matter. 1994, v. 6,
p. 3759-3766.
3. B.N. Zakhariev, V.M. Chabanov. Qualitative theory
of control of spectra, scattering, decays (Quantum
intuition lessons) (Качественная теория
управления спектрами, рассеянием, распадами
(уроки квантовой интуиции)) // Physics of
elementary particles and atomic nuclei (ЭЧАЯ).
1994, v. 25, №6, p. 1561-1597 (in Russian).
4. B.N. Zakhariev, V.M. Chabanov. About the
qualitative theory of elementary transformations of
one- and multicannel quantum systems in the inverse
problem approach (К качественной теории
элементарных преобразований одно- и
многоканальных квантовых систем в подходе
обратной задачи) // Physics of elementary particles
and atomic nuclei (ЭЧАЯ). 1999, v. 30, №2, p. 277-
320 (in Russian).
5. F. Cooper, A. Khare, U. Sukhatme. Super-symmetry
and quantum mechanics // Physics Reports. 1995,
v. 251, p. 267-385.
6. C.V. Sukumar. Supersymmetry and potentials with
bound states at arbitrary energies: II // Journ. of
Phys. A.: Math. Gen. 1987, v. 20, p. 2461-2481.
ОДНОМЕРНЫЕ ОБРАТНО СТЕПЕННЫЕ АБСОЛЮТНО ПРОЗРАЧНЫЕ ПОТЕНЦИАЛЫ
ТИПА V(x)=const / | x - x0
| n
С.П. Майданюк
С помощью методов SUSY QM получено условие, при котором обратно степенной потенциал
nxxxV 0)( −= α (α = const, x0 = const, [,] + ∞∞−∈x , n – натуральное число) становится абсолютно
прозрачным при прохождении через него плоской волны. Представлен анализ рассеяния частицы на
сферически-симметричном потенциале
nrrrV 0)( −±= α с учетом возможности абсолютной
прозрачности.
ОДНОВИМІРНІ ОБЕРНЕНО СТЕПЕНЕВІ АБСОЛЮТНО ПРОЗОРІ ПОТЕНЦІАЛИ
ТИПУ V(x)=const / | x - x0
| n
С.П. Майданюк
На основі методів SUSY QM отримано умову, при якій обернено степеневий потенціал
nxxxV 0)( −= α (α = const, x0 = const, [,] + ∞∞−∈x , n – натуральное число) стає абсолютно прозорим
при проходженні крізь його плоскої хвилі. Представлено аналіз розсіювання частинки на сферично-
симетричному потенціалі
nrrrV 0)( −±= α з врахуванням можливості абсолютної прозорості.
25
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