3 (31), 2014
- .
539.182
. . , . .
1
.
. . , . 2 3, . , , , , . , , .
. . . . , . , , , . , , .
. , , . , , , , . . - .
. -
1 .
Physics and mathematics sciences. Mathematics
159
.
. .
: , , .
V. M. Zhuravlev, V. M. Morozov
METHOD OF CONFORMAL MAPPINGS IN THE THEORY OF TWO-DIMENSIONAL QUANTUM SYSTEMS
Abstract.
Background. Development of methods for constructing exact solutions of the quantum equations is one of the major technical challenges in the field of quantum physics. Exact solutions enable not only to analyze the systems in all the subtleties, but serve as the basis for a large number of systems that are already close to zero-order approximation. From the modern point of view, of special interest are quantum systems in space dimension 2 and 3, for which there arent many methods for constructing exact solutions. Therefore, the search for approaches that provide insights not into one particular system, but several systems, using the same procedure for constructing solutions, is an urgent task. The purpose of this paper is to construct new exact solutions of two-dimensional quantum systems with special symmetry properties, which can then be used as a basis for the analysis of quantum systems in the theory of nanomaterials.
Materials and methods. In this study, to construct exact solutions of the equations of two-dimensional quantum systems the authors used the method of conformal mappings of two-dimensional coordinate space into itself. The two-dimensional coordinate space is represented by a complex plane. With the help of conformal mappings of the complex plane the researchers solved several problems of classical dynamics of a quantum particle without a spin in two-dimensional potential wells. Conformal mappings bound exactly solvable quantum problems such as the twodimensional harmonic oscillator with the quantum problem of the motion of particles in the channels. The authors present an analysis of the use of two basic types of conformal mappings, allowing to build a complete solution to the new problem on the basis of solution to the known problem, for example, such as a two-dimensional quantum oscillator. In construction of exact solutions of quantum equations the conformal mapping method is combined with the method of Darboux transformations, which expands the class of systems for which it is possible to construct exact solutions.
Results. As a result, the use of conformal mapping extends the list of methods by which it is possible to obtain exact solutions of new quantum problems that can serve as models for some real quantum systems. The systems, constructed with the help of conformal mapping, provide examples of quantum systems in which the particle wave function is constructed at the same time in the whole space, while the potential energy is such that the particle can only be in one of the regions of space that are separated by impenetrable potential barriers. The paper gives examples of such systems and examines some of their basic properties. This result is related to the problem of the Aharonov-Bohm effect.
Conclusions. Conformal mappings can be successfully used as a method of constructing solutions of quantum systems using the known results concerning the transfer of certain quantum systems on the other. The results can be used to analyze the actual quantum systems as a zero approximation.
Key words: onformal mapping, two-dimensional quantum systems, method of Darboux transformations.
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University proceedings. Volga region
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- .

, , . . . , . , , - , . , , , .
, , [1]. [2]. , , , . . , . , [3]. , , . , , - [4]. , . , , . . - . . , . .
1.
, , -
Physics and mathematics sciences. Mathematics
161
.
- . :
2
- = . (1)
2 r
r = Ix2 + 2, = 2 +-2 - .
2 2
U(r) = Uq / r , .
. (1)
*
: z = + iy, z = iy .
= 4
2
dzdz
(2)
(1) :
22 2 zz
---- = ,
7^7
(3)
0(z, z , ) = (,, ). z ^ Z(z) = 21 |1/4 Vz
2 ^(
= 4ZZ ,
(4)

4(Z, Z*, ) = ( z, z*, ), = 02.
I |1/2
-ї .
z ^ Z(z) (, ) (X ,Y), :
X =1(Z(z) + z*(z)), Y = ^(Z(z) z*(z)). (5)
2 2i
,
(4), , -
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University proceedings. Volga region
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- .
= 2V2 / :
_
2
2 ( 2
^2
2 + 72
+ 4( X 2 + 7 2) = ,
(6)
(X,Y, ) = (Z, Z , ). ,
= ( +1), n = 0,1,2,...,
:
| |1/2= 4 ,
( +1)

2
^ 20 1
=--------------
2 (n +1)2'

(X, Y) = Hk (X)Hm (Y)e~R /2Ro ,
Hn (X) - ; R2 = X2 + Y2 , Rq =yj / = V / 2V2 . n X Y , n = + , .
2.
, , , . . U(, )
-- + U (, ) = .
2
2
(7)
z, z ,
Physics and mathematics sciences. Mathematics
163
.
2h2 2 m dzdz
+ U (z, z*) = .
(8)
z ^ Z(z), , :
2h2 2 _
-------* +
m ^^
U(z,z*)
*
w(z)w(z )
=
w(z)w(z )
.
(9)
Z (z) = w(z), (Z, Z ,E) = (x, y,E). X,Y, (5), , . :
h2
2m
( -2
2 \
+
2 Y2
+ W (X, Y, ) = ,
(10)

W (X, Y, )
U (z (X ,Y), z*( X, Y)) - E + E w( z (X, Y))w(z*(X,Y)) :
(X, Y, E) = Y(Z, Z ,E), - . ,
(z (X ,Y), z*( X, Y)) = -ew(z (X ,Y)) X
xw( z*( X, Y)) + Wo( X, Y, E),
(11)

Wo (X, Y, E) = W (X, Y, E) w( z (X, Y)) w( z* (X, Y)) + E.
, , (11) Z = Z(z).
, . .
3.
, .
164
University proceedings. Volga region
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- .
3.1. 1.
2 :
2 I 2 2 2 2 \ ^
-Ay +(x +2 ) = ).
2m 2 ' '
^ 0 x :

,2 = ~((2n\ + 1)1 + (2n2 + 1)2^
nj 2 - ,
nj ,2 Hnj (x)2 ( )
2 /-) 2 2 / 2
x /2 /2
(12)
(13)
Hn () - , = ^/, = ^ / 2 :
2

2 ' 2
+ -2 (( 2 + 2) + ( 2 2)) = ,
:
,2 2
-4---- + ( | |2 +(z2 + z*2)/2) = .
2m 3z3z
+ = mojf /2, = 2 / 2 . :
z ^Z(z) = z2/2.

(14)
_^ _^_ 2 ^(
+
+
Z + Z
21 Z I
*

21 Z
:
Z (5),

2
^2
2
+
2
v 2 dY2
,1 2
+ 4| + I = -
(15)
:
W = 4 = 4 cos , R
(16)
Physics and mathematics sciences. Mathematics
165
.
- (X, Y).
= 4,
, (16) :
cos
= 2~.
R2
2 2
E = 4 = ( + 2). 2 , (13), (15) . :
2 2
Eq = ^((21 +1)W + (22 +1)2) = const, 4 = ( 2) = const.
W 2,
01
= Q0 (2n1 + 1)1 Wo--------------------
2 2n2 +1
Qq (2n1 +1) ±7^2 ((2n1 +1)2 +1) + 4p(2n2 +1)2 / m (2n2 +1)2 (2n1 +1)2
(17)
Qq =2Eq / m . , :
E = 4 = W (2n2 +1)2 + (Q0 2n1∞1 1)2
(2n2 +1)2
W (17).
X 2E W (X, Y ) = 4 X 2E R R
= 400, E = 5 . 1. , E = 250, 200, 150, 50, 0, 50, 100, 150.
, .. E Yq . . 2 Qq =1 Yq = 1 .
W:
Q = 1 + Yq 4(n2 n 1 + n2 +1).
166
University proceedings. Volga region
3 (31), 2014 - .
. 1. (15) : = 400, E = 5
)
. 2. (. . 1) Yo =1 ^ =1; - E+ ; - E-
Physics and mathematics sciences. Mathematics
167
.
. 2.
Q >0. , , = 2, . , Yo , > 2, < 2. , Yo <0 i={2 +1,2 + 2...}. 1/1 |> 4(^o - jo)(ko + jo +1), 2 .
3.2. 2.
, :
' 2
^2
+
2 2
V + (2 + 2 ) +
2 2 2 + 2
V = E V,
(18)
= X/ lo, = / lo, E = E/ Eo , = / o X, , E - , lo = , Eo = / 2, o =2 / 2 .
, . :
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University proceedings. Volga region
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- .
1 ( dRn
r
r dr I dr
+ r2 Rn + Rn = EnRn,
(19)
m - .
nm (r, ) = Rn (r)em =
2(n!)
X
-Ln
2 (r2 )r
X-1 2 e
-+im(p
(20)
( + X+) 2
X-1
Ln 2 (r2) - [5]:
Lm (x)=s
p=0
n + m
n - P
(-x )
P!
:
1
En = X + 2 + 2n, n = 0,1,2... ,

X=2+v21m2+e-2 J+4
(18) :
*
z = x + iy, z = x - iy :
(21)
+ zz + - = .
dzdz zz
*
zz z = eZ , *
. Z, Z :
2
- + e2Ze2Z + = EeZ+Z .
dZdZ
, Z = ^ + in, :
W (, ) = e4^- Ee2^, :
(22)
Physics and mathematics sciences. Mathematics
169
.
( 2
----^ +
2 ^
V
^2 2
+ 4^ + = .
= 2^
(23)
|--=1---=2... =4--=4 =5 |
. 3. (22) ; : 1 - = 1; 2 - = 2; 3 - = 3; 4 - = 4; 5 - = 5
(23) .
= X()(), X2 ,
22
- Y + X2Y = 0, -- X + (- 2 = (-- 2)X. (24)
2 ^2 1
(24) , - . , . 3. (20),
: r = ^, = ,
n = ^ n (t );
(t )
2(!) + X +
X 2
^
(25)
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University proceedings. Volga region
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- .
. W(%, E) E , . ,
(19), . . , (25) , (22). E W(%, E), :

nm
1 2 -- m + 8
m = 0,1,2...
(26)
(26) (21), X E :
X = E - 2 +.
4
(27)
(%, ) : ^ 0 % ^ . -
, , , (22). .
3
:
-
2
( -2
\2 \
+
2 2
W +
(2 + 2 ) + +
W = E W.
(28)
z = + iy, z = iy, -
-/
= 2, :
*2 -,2
- / * * * \
---4 *W + Xz +Xz + z E = EW.
2m dzdz K
%

z
2
*
zz

2
Physics and mathematics sciences. Mathematics
171
.
2

- +
Wg+T%/j*
. 2g

W = -e'F.
: = Z + in =| | :
( / _ J
-
-
2
2 2
- + -
dz2 2
Z
+
V2I
cosl -2+
Z2 +2 )4
4 2,/1
V = , (29)
= (Z, ), = , Y =| Y | ^0 .

:
2 2 \
x - y =2Z, xy = n.
, . :
, | 1 2 +2
En = | n + I +----
1 2 42
2
^ (x, y ) = CnC2
2
x++| y+1 2J V 2
X
(30)
Hn
I ! ( jJTT (
v.t Ix+* JJ Hn
\
(
l y+
V I 2, j
Cn =
1 (
V2n! V
---I. :
n = ±4
2 +2
2E - 2(2 +1)
(31)
:
x=WzWz^^n2, y=±/-Z Wz2 -n2.
(29) /21 |= 200, = 42, E = 40 . 4. (31) . , : E - (2 +1) > 0 . = 0 -
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University proceedings. Volga region
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- .
E : E > . E : n < . 5.
1
2 2
, ,
: V21 |= 200, 0 =42, E = 40
= 0
= 1
= 2
= 3



. 5. (29) . 4
4.
, ,
Physics and mathematics sciences. Mathematics
173
.
[1, 2]. , .
c
2 2 2 2
-+ ^~ * = Enn.
2 2 2
:
() = CnHn ( / 0)
2 D 2
/2
= V / , Cn =(2nn!) 1/2 () 1/4 , Hn () - :
H0() = 1, 1() = 2, 2 = 2(22 1), 3() = 4(2 3),...
-
iL ii
2m 2
(|k)+
+ Uk()
( | k ) = ( | k),
f}2
Uk () =-------, :

( 1 k )^ n +bk()

bk () = -4-ln k = / v"ln Hk( / ).
,
1( ) = ~ ln 1 = / -1, 2() = ln 2 = / -.
2 2 0
:
2 (
U1( ) =


1 1
_2 + ~2

U2( ) =
2

1
+
2 2 \2
(2 1)
(7) :
2 2 , 2 2
U1(,)= ( + ) +
2
(1 1' 2 + 2 = ( 2 + 2)
V;

- +
1
2 2
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University proceedings. Volga region
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- .
2 2 ( . mm . 2 2
U2 (, ) = (2 + 2) +
2 m
V
1
+
1
(2xz -1)z (2yz -1)
z ^Z = z2 /2 Ui(, ) -
42 1 2 E 2 1 2 E
W1 = - ----1---- = -1 (32)
m (Z-Z )2 | ZI my2 R
2 2 2
= X + X, Y - . W (32) 2
/ m = 2, E = 500 . 6.
. 6. (32)
w 2
/ m = 2, E = 500
U2(,), , - :
W2 =---0 +
2 2R
82 8X2 + 4Y2 - 4R +1 m (y2 - 4R +1)2
(33)
- - :
Physics and mathematics sciences. Mathematics
175
.
22 , , , ~ mm2 mE0
Eq = E----- = + 2 -1) = const, En =------ _
2
x0
22 ( + n2 -1)2 '
W2 (33)
2
Eq =3,1; = 0,5 . 7.
m
1
m
, . , - . , , , , . , , . , , - .

. , .
176
University proceedings. Volga region
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- .
, . . , , - . , , - [4], . , , , . , . .

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- , , , (, . , . , 42)
E-mail: zhvictorm@gmail.com
Zhuravlev Viktor Mikhaylovich Doctor of physical and mathematical sciences, professor, sub-department of theoretical physics, Ulyanovsk State University (42 Lva Tolstogo street, Ulyanovsk, Russia)
Physics and mathematics sciences. Mathematics
177
.
, (, . , . , 42)
Morozov Vitaliy Mikhaylovich
Student, Ulyanovsk State University (42 Lva Tolstogo street, Ulyanovsk, Russia)
E-mail: aieler@rambler.ru
539.182 , . .
/ . . , . . // . . - . - 2014. - 3 (31). -
. 159-178.
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University proceedings. Volga region