Kurochkin, Yu.

Otchik, V.

Ovsiyuk, E.

English

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c© 2010 Proceedings of the F&ANS-2010 Conference-School
Quantum mechanical particle in magnetic field
in the Lobachevsky space
Yu. Kurochkin
Institute of Physics, National Academy of Sciences of Belarus
V. Otchik
Institute for Command Engineers, Ministry for Emergency Services of Belarus
E. Ovsiyuk
Mozyr State Pedagogical University
Various possibilities to define analogs of the uniform magnetic field in the
Lobachevsky space are considered, using different coordinate systems in this space.
Quantum mechanical problem of motion in the defined fields is also discussed.
1. Introduction
Since Schro¨dinger [1] first solved quantum mechanical Kepler - Coulomb problem on a three-
dimensional sphere, a great number of authors have studied different aspects of this problem in
spaces of constant, positive and negative, curvature (see a concise historical review in [2]). Solu-
tions of this problem have been used to describe bound states in nuclear physics and nanophysics
[3, 4].
Much less is known about motion of particles in other configurations of electromagnetic
fields in such spaces. We mention here paper [5], where static solutions of Maxwell equations
on a three-sphere were found and applied to the study of Stark and Zeeman effects. Static
fields introduced in [5] do not allow to solve exactly the problem of motion of particles in these
fields. Recently, a simple generalization of the uniform magnetic field to the three-dimensional
Lobachevsky space has been proposed, and corresponding quantum mechanical and classical
problems have been solved [6, 7]. The generalization is based on a certain choice of solutions of
Maxwell equations which give the uniform magnetic field in the limit of zero curvature.
In this paper, we consider some alternative possibilities to define analogs of the uniform
magnetic field in the Lobachevsky space, using different coordinate systems in this space. The
quantum mechanical problem of motion in the defined fields is also discussed, and for some cases
exact solutions of Schro¨dinger equation are obtained.
48 Proceedings of the F&ANS-2010 Conference-School, (2010) 48 - 54
Quantum mechanical particle in magnetic field in . . . 49
2. Analogs of the uniform magnetic field in the Lobachevsky
space
The first generalization of the uniform magnetic field to the Lobachevsky space was done
in the paper [6]. The magnetic field in this paper was found as a solution of Maxwell equations
in the Robertson – Walker metrics with Lobachevsky space as spatial part
dS2 = c2dt2 − dl2, dl2 = ρ2[cosh2 z1(dr21 + sinh2 r1dφ21) + dz21 ]. (1)
The coordinates z1, r1, φ1 of the Lobachevsky space are defined by relations
u0 = ρ cosh z1 cosh r1, u1 = ρ cosh z1 sinh r1 cosφ1,
u2 = ρ cosh z1 sinh r1 sinφ1, u3 = ρ sinh z1, (2)
−∞ < z1 <∞, 0 < r2 <∞, ≤ φ < 2pi, u20 − u21 − u22 − u23 = ρ2,
where ua, a = 0, 1, 2, 3 are coordinates in the ambient four-dimensional pseudo-Euclidean space.
In the limit when ρ→∞, ρr1 → r, ρz1 → z, the spatial part of the line element (1) reduces
to that of Euclidean space in which r, z are cylindrical coordinates:
dS2 = c2dt2 − (dr2 + r2dφ2 + dz2). (3)
The vector potential of the uniform magnetic field in the flat space with metrics (3) can be
written in the form
At = Ar = Az = 0, Aφ = −Br
2
2
. (4)
The generalization of this potential to the Lobachevsky space reads [6]
Ar1 = Az1 = 0, Aφ1 = −Bρ2(cosh r1 − 1). (5)
The corresponding magnetic field is
Fφ1r1 = Bρ
2 sinh r1. (6)
Besides the coordinates r1, z1, φ1 (2), usually named hyperbolic, there are some other coor-
dinate systems in Lobachevsky space, which have the cylindrical coordinate system of Euclidean
space as their limit. We consider here two such systems. First one is the cylindrical coordinate
system:
u0 = ρ cosh z2 cosh r2, u1 = ρ sinh r2 cosφ2,
u2 = ρ sinh r2 sinφ2, u3 = ρ cosh r2 sinh z2, (7)
−∞ < z2 <∞, 0 < r2 <∞, ≤ φ < 2pi.
Proceedings of the F&ANS-2010 Conference-School, 2010
50 Yu. Kurochkin et al.
The line element in Lobachevsky space is here
dl2 = ρ2(dr22 + sinh
2 r2dφ
2
2 + cosh
2 r2dz
2
2),
and in the limit ρ→∞, ρr2 → r, ρz2 → z, we get the metrics of Euclidean space in cylindrical
coordinates.
Expressions for nonzero components of potential and electromagnetic field are
Aφ2 = −Bρ2 ln cosh r2, Fφ2r2 = Bρ2 tanh r2. (8)
We also consider the horospheric coordinate system in Lobachevsky space defined by rela-
tions
u0 =
ρ
2
[ez3 + (r23 + 1)e
−z3 ], u1 = ρr3e−z3 cosφ3,
u2 = ρr3e
−z3 sinφ3, u3 =
ρ
2
[ez3 + (r23 − 1)e−z3 ], (9)
−∞ < z3 <∞, 0 < r3 <∞, 0 ≤ φ < 2pi.
Metrics of Lobachevsky space in coordinates r3, z3, φ3 takes the form
dl2 = ρ2[e−2z3(dr23 + r
2
3dφ
2
3) + dz
2
3 ],
and in the limit ρ→∞, ρr3 → r, ρz3 → z, it also goes over into the metrics of Euclidean space
in cylindrical coordinates.
Nonzero components of potential and electromagnetic field are found to be
Aφ = −ρ
2Br2
2
, Fφr = ρ
2Br. (10)
Electromagnetic fields and potentials (8) and(10) were found by close analogy with approach
of paper [6], but Maxwell equations were solved in different coordinate systems. As a result, we
obtained different magnetic fields in the Lobachevsky space which reduce to the same uniform
magnetic field in Euclidean space when ρ→∞.
Besides the coordinates in the Lobachevsky space which reduce in the limit ρ→∞ to the
cylindrical coordinates of Euclidean space, we also consider a coordinate system which goes over
in the same limit into the Cartesian one. It is defined as follows:
u0 =
1
2
(ez + (1 + x23 + y
2
3)e
−z3), u1 = x3e−z3 ,
u2 = ye
−z3 , u3 =
1
2
(ez3 − (1− x23 − y23)e−z3), (11)
−∞ < z3 <∞, 0 < r3 <∞, 0 ≤ φ < 2pi,
and the line element is
dl2 = ρ2[e−2z3(dx23 + dy
2
3) + dz
2
3 ].
Proceedings of the F&ANS-2010 Conference-School, 2010
Quantum mechanical particle in magnetic field in . . . 51
Coordinates x3, y3 are related to the horospheric coordinates r3, φ3 as
x3 = r3 cosφ3, y3 = r3 sinφ3.
Transforming potential (10) to the coordinates x3, y3, z3 we find
Ax3 = −
Bρ2y3
2
, Ay3 =
Bρ2x3
2
, Az3 = 0, Fx3y3 = −Bρ2. (12)
3. The quantum mechanical problem for a particle in a magnetic
field in the Lobachevsky space
Here we consider the quantum mechanical problem for a particle in a magnetic field.
The Schro¨dinger equation in the Robertson – Walker metric
dS2 = (dx0)2 + gkldx
kdxl, g = det gkl.
in the presence of magnetic field is (we use units such that ~ =M = 1)
i∂tΨ = HΨ, H =
1
2
[(
i√−g∂k
√−g + e
c
Ak
)
(−gkl)
(
i∂l +
e
c
Al
)]
Ψ. (13)
In the hyperbolic coordinate system (2) Hamiltonian (13) with potential (5) is
H = − 1
2ρ2 cosh2 z1
[
1
sinh r1
∂
∂r1
sinh r1
∂
∂r1
− 1
sinh2 r1
(
i
∂
∂φ1
+
2eBρ2
c
sinh
r21
2
)2
− ∂
∂z1
cosh2 z1
∂
∂z1
]
. (14)
Solutions of the Schro¨dinger equation (14) were found in ([6]). Here we note that with
Hamiltonian (14) commute operators
J1 = sinφ1
∂
∂r1
+ coth r1 cosφ1
∂
∂φ1
− iBρ
2 sinh r1 cosφ1
cosh r1 + 1
,
J2 = cosφ1
∂
∂r1
− coth r1 sinφ1 ∂
∂φ1
+
iBρ2 sinh r1 sinφ1
cosh r1 + 1
, J3 =
∂
∂φ1
− iBρ2, (15)
with commutation relations of generators of group O(2,1)
[J1, J2] = −J3, [J2, J3] = J1, [J3, J1] = J2. (16)
In the cylindrical coordinate system (7) Hamiltonian (13) with potential (8) is
− 1
2ρ2
[
1
sinh r2 cosh r2
∂
∂r2
sinh r2 cosh r2
∂
∂r2
− 1
sinh2 r2
(
i
∂
∂φ2
− 2eBρ
2
c
ln cosh r2
)2
+
1
cosh2 r2
∂2
∂z22
]
. (17)
Proceedings of the F&ANS-2010 Conference-School, 2010
52 Yu. Kurochkin et al.
After substitution
Ψ = e−iEteimφ2eikz2R2(r2)
we obtain equation
− 1
sinh r2 cosh r2
d
dr2
sinh r2 cosh r2
dR2
dr2
+
1
sinh2 r2
(
m+
eB
c
ρ2 ln cosh r2
)2
R2 −
(
k2
cosh2 r2
+ 2Eρ2
)
R2 = 0.
This equation could not be solved in terms of known functions.
In the horospherical coordinate system (9) Hamiltonian (13) with potential (10) is
H =
e2z3
ρ2
[
− 1
r3
∂
∂r3
r3
∂
∂r3
+
1
r23
(
i
∂
∂φ3
− eBρ
2r23
2c
)2
− ∂
∂z3
e−2z3
∂
∂z3
]
. (18)
With Hamiltonian (18) commute operators
P1 = cosφ3
∂
∂r3
− 1
r3
sinφ3
∂
∂φ3
+
iBρ2r3 sinφ3
2
,
P2 = sinφ3
∂
∂r3
+
1
r3
cosφ3
∂
∂φ3
− iBρ
2r3 cosφ3
2
, L3 =
∂
∂φ3
.
The substitution
Ψ = e−iEteimφ3Z3(z3)R3(r3)
leads to separation of variables, and we obtain differential equations
d
dz3
e−2z3
dZ3
dz3
+ 2Eρ2e−2z3Z3 = λZ3, (19)
− 1
r3
d
dr3
r3
dR3
dr3
+
1
r23
(
m− eBρ
2r23
2c
)2
R3 = λR3. (20)
By substitution x = i
√
λez the equation (19) is reduced to the Bessel equation
d2Z3(x)
dx2
− 1
x
dZ3(x)
dx
+
(
1 +
2Eρ2
x2
)
Z3(x) = 0
with solutions
Z3(x) = C1xJ√1−2Eρ2(x) + C2xY√1−2Eρ2(x).
In the equation (20) we denote eBρ2/c = B1 and introduce a new variable y = B1r
2
3/2.
Then equation takes the form
y
d2R3(y)
dy2
+
dR3(y)
dy
− 1
4
(
y +
m2
y
− 2(B1m+ λ)
B1
)
R3(y) = 0. (21)
Proceedings of the F&ANS-2010 Conference-School, 2010
Quantum mechanical particle in magnetic field in . . . 53
We set in (21) R3(y) = y
|m/2|e−y/2F (y) and obtain a confluent hypergeometric equation [8]
y
d2F
dy2
+ (|m|+ 1− y)dF
dy
−
( |m| −m
2
− λ
2B1
)
F (y) = 0,
which has solutions Φ(α, γ; y) with
α =
|m| −m
2
− λ
2B1
, γ = |m|+ 1.
Requirement of finiteness of solutions of radial equation (20) leads to the quantization rule for
separation constant λ
λ = B1(|m| −m+ 2n+ 1), n = 0, 1, 2, ... .
In order to find solutions of Schro¨dinger equation in the coordinates (11) we take the vector
potential in the form
Ay3 = Bρ
2x3, Ax3 = Az3 = 0, (22)
which can be obtained from (12) by a gauge transformation. Then the Hamiltonian takes the
form
H =
e2z3
ρ2
[
− ∂
2
∂x2
+
(
i
∂
∂y
+
eBρ2
c
)2
− ∂
∂z3
e−2z3
∂
∂z3
]
.
Separation of variables is carried out by using the substitution Ψ = e−iEteipy3X(x3)Z(z3), and
separated equations are
d
dz3
e−2z3
dZ3
dz3
+ 2Eρ2e−2z3Z3 = λ′Z3, (23)
−d
2X
dx2
+
(
p− eBρ
2x
c
)2
X = λ′X. (24)
Equation (23) is identical to (19). In the equation (24) we introduce a new variable u =√
B1(x− p/B1). Then we obtain the harmonic oscillator equation
d2X(u)
du2
− u2X(u) + λ
′
B1
= 0. (25)
As is well known, from finiteness of solutions of equation (25) follows the quantization rule
λ′ = B1(2n+ 1), n = 0, 1, 2, ... .
Conclusion
By solving Maxwell equations in different coordinate systems in the Lobachevsky space
one may obtain alternative generalizations of the notion of the uniform magnetic field. All the
fields found in this way have the same flat space limit. In some cases, problem of the motion of
charged particles in these fields can be solved exactly. The found solutions may be of interest
for modelling physical interactions in various fields.
Proceedings of the F&ANS-2010 Conference-School, 2010
54 Yu. Kurochkin et al.
References
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(Springer, Berlin, 2006).
[3] A.A. Izmest’ev. Exactly solvable potential model for quarconiums. Yadernaya Fizika. 52, 1697-
1710 (1990).
[4] V.V. Gritsev and Yu.A. Kurochkin. Model of excitations in quantum dots based on quantum
mechanics in spaces of constant curvature. Phys. Rev. B 64, 035308 (2001)).
[5] N. Bessis, G. Bessis, and D. Roux. Space curvature effects in the interaction between atoms and
external fields: Zeeman and Stark effects in a space of constant positive curvature. Phys. Rev.
A 33, 324-336 (1985).
[6] A.A.Bogush, V.M. Red’kov, G.G. Krylov. Schro¨dinger Particle in Magnetic and Electric Fields
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[7] V.V. Kudryashov, Yu.A. Kurochkin, E.M. Ovseyuk, V.M. Redkov, The motion in the magnetic
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Proceedings of the F&ANS-2010 Conference-School, 2010


Quantum mechanical particle in magnetic field in the Lobachevsky space

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Quantum mechanical particle in magnetic field in the Lobachevsky space

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