Россия
Россия
Россия
Классическая и квантовая задачи о движении двух частиц в трехмерном пространстве Лобачевского сформулирова- ны относительно центра масс с произвольным положени- ем. Выписаны уравнения Гамильтона-Якоби и Шрёдинге- ра задачи и найдены их решения. Показано, что приве- денная масса системы зависит от относительного рассто- яния. Сформулированы и решены классическая и кванто- вая задачи жесткого ротатора в трехмерной сфере и про- странстве Лобачевского. Исследованы зависимости пери- одов колебаний ротатора от отношения масс образующих его частиц при фиксированной полной массе в случаях пространств постоянной кривизны.
задача двух тел в неевклидовом пространстве, центр масс, жесткий ротатор, пространство постоянной кривизны
Introduction
By analogy with the constructions and conclusions of
works [1, 2] and relying on the definition of the center of
mass given in works [3], we postulate its immobility in spaces
of constant curvature, in this case in the three-dimensional
Lobachevsky space, and consider the problem of two particles
with an internal interaction described by potential, depending
on the separation between particles. The essence of the
statement, which replaces the formulation of the theorem on
the center of mass in the three-dimensional Euclidean space,
is that in spaces of constant curvature: Lobachevsky, on the
3-sphere and in three-dimensional elliptical space, there is
a frame of reference in which the center of mass of the system
of particles is at rest.
1. Variables of the center of mass and relative
coordinates for a system of two particles
Since the formalism used below, despite the fact that it allows
one to unify the description of the geometries of a number
of three-dimensional and two-dimensional spaces of constant
curvature (and therefore convenient), is not widely used,
we are forced to present some calculations similar to those
used in [1-3]. The problems associated with the separation
of variables, including those in spaces of constant curvature,
can also be found in [4]. To formulate and solve the problem
in three-dimensional Lobachevsky space, instead of biquaternions
defined over double numbers, biquaternions over
complex numbers will be used.
The following definition of the center of mass coordinates
of two particles with masses m1 and m2 is used
XC = i
m1X(1) + m2X(2)
p
(m1X(1) + m2X(2))(m1 ¯X (1) + m2 ¯X (2))
.
(1)
Here the corresponding biquaternions are given over the
complex numbers, and not over the double ones, as it was
in the case with the 3-sphere [1,2].
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The three-dimensional independent coordinates of the
center of mass will be the components of the vector
qC = −i
XC
X0C
= −i
m1X(1) + m1X(2)
m1X(1)
0 + m2X(2)
0
. (2)
The coordinates of two material points in the embedding fourdimensional
space will be the components of the biquaternions:
X(1) = iX(1)
0 + X(1), X(2) = iX(2)
0 + X(2), (3)
where i2 = −1. The ends of the vectors (biquaternions) lie
on the upper field of the pseudo-Euclidean space hyperboloid,
on which the real Lobachevsky space is realized. For convenience,
the radius of space curvature is assumed to be unity.
Then
X(1) ¯X (1) = −1, X(2) ¯X (2) = −1. (4)
As independent coordinates, it is convenient to use the Beltrami
coordinates, which are components of vectors on the
sphere [5]
q(1) = −i
X(1)
X(1)
0
, q(2) = −i
X(2)
X(2)
0
(5)
with the law of addition (subtraction)
q” = ⟨q,±q’⟩ =
q ± q’ ± [q, q’]
1 ∓ (q, q’)
(6)
coinciding with the composition law of F.I. Fedorov [6]. Here,
parentheses denote the usual scalar, square brackets denote
the vector product of vectors. In variables (5), expression (2)
has the form
qC =
m1q(1)/
p
1 + (q(1))2 + m2q(2)/
p
1 + (q(2))2
m1/
p
1 + (q(1))2 + m2/
p
1 + (q(2))2
.
(7)
As noted earlier, expression (7) for the coordinates of the center
of mass coincides in form with a similar expression for
the coordinates of the center of mass in a three-dimensional
flat space, in which the expressions for constant masses
m1 and m2 are replaced by mass expressions with the dependence
of masses on coordinates m1/
p
1 + (q(1))2 and
m2/
p
1 + (q(2))2.
We also note that this definition coincides with the definition
given in [6], if we take into account that q2 = −th2r ,
where r is the distance between two points. As it follows from
the formula (7) (and shown in [7]), such a definition can be
generalized to an arbitrary number of particles. The biquaternion
analogue of the relative variable for two given particles
is the operator
Y12 = −X(2) ¯X(1), (8)
defined as
X(2) = Y12X(1). (9)
Independent three-dimensional coordinates of relative motion,
defined as components of the relative motion vector
qy =
Y12 − ¯ Y12
Y12 + ¯ Y12
=
*
−i
X(2)
X(2)
0
, i
X(1)
X(1)
0
+
=
= ⟨q(2),−q(1)⟩ =
q(2) − q(1) − [q(2), q(1)]
1 + (q(1), q(2))
. (10)
Let us also introduce four-dimensional Y1 and Y2 and threedimensional
q(1)
y , q(1)
y coordinates of points relative to the
center of mass, determined similarly to (8) and (9), namely
X(1) = Y1XC, X(2) = Y2XC, (11)
moreover
Y1 = −X(1) ¯XC, Y2 = −X(2) ¯XC. (12)
It is clear that
Y12 = Y2 ¯ Y1. (13)
Then for the first particle
q(1) = −i
X(1)
X(1)
0
=
=
*
−qy
1 + m1
m2
p
1 + q2
y
,−i
XC
X0C
+
=
= ⟨q(1)
y , qC
⟩, (14)
and for the second one we get:
q(2) = −i
X(2)
X(1)
0
=
=
*
qy
1 + m2
m1
p
1 + q2
y
,−i
XC
X0C
+
=
= ⟨q(2)
y , qC
⟩. (15)
where q(1) and q(2) are defined from Y1 and Y2 respectively.
It is easy to verify the validity of formulas (14) and (15) by direct
calculation. It should be noted that q(1) and q(2) are expressed
in terms of relative variables qy and center of mass
variables qC.
From (13) it follows that
qy = ⟨q(2),−q(1)⟩ = ⟨q(2)
y ,−q(1)
y
⟩. (16)
The variables introduced satisfy the relations
XC ¯XC = −1, Y12 ¯ Y12 = 1,
Y1 ¯ Y1 = 1, Y2 ¯ Y2 = 1. (17)
58
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2. Two material points on S1
3 . Non-relativistic
classical problem
The action for the problem of two material points in threedimensional
space, interacting with forces that depend only
on the relative variable, we write in the form [1-3]
W =
Z h1
2
m1 ˙X (1) ˙¯X(1)+
+m2 ˙X (2) ˙¯X (2)
− V (Y12)
i
dt. (18)
Here it is immediately taken into account that the operation of
differentiation and conjugation are commuting. The dot above
the letters denotes differentiation with respect to time. Expression
(18) will take a standard form if we pass to independent
variables q(1) and q(2). In this case
W =
Z h1
2
m1gab(q(1))q˙(1)
a q˙(1)
b +
+m2gab(q(2))q˙(2)
a q˙(2)
b
− ϕ(qy)
i
dt, (19)
where
gab =
1
1 + q2
"
δab − qaqb
1 + q2
#
(20)
is the metric tensor of the three-dimensional Lobachevsky
space in variables that are components of vectors on the
sphere. In expression (19), according to the accepted assumption,
we set qC = 0 and write it in spherical coordinates.
Then
W =
Z "
1
2
m1r˙2
1 + m1 sh2 r1
θ˙2
1 + sin2 θ ˙ϕ
21
+
+m2r˙2
2+m2 sh2 r2
θ˙2
2+sin2 θ ˙ϕ
22
−U(r12)
#
dt. (21)
Replacing in (21) the coordinates of individual particles with
relative variables r, θ, ϕ in accordance with formulas (14),
(15) with qC = 0, we get the following expression for the
action
W =
Z "
1
2
μ∥(r)r˙2+
+μ⊥(r) sh2 r
θ˙2 + sin2 θ ˙ϕ2
− U(r12)
#
dt, (22)
where we have introduced the longitudinal reduced mass of
two material points
μ∥ = m1
m22
+ m1m2 ch r
m21
+ m22
+ 2m1m2 ch r
!2
+
+m2
m21
+ m1m2 ch r
m21
+ m22
+ 2m1m2 ch r
!2
(23)
and the transverse reduced mass
μ⊥ =
m1m2(m1 + m2)
m21
+ m22
+ 2m1m2 ch r
. (24)
The Hamiltonian of the system is therefore equal to
H =
1
2
h
μ∥(r)r˙2+
+μ⊥(r) sh2 r
θ˙2 + sin2 θ ˙ϕ2i
+ U(r12). (25)
It is easy to check that the expression for the Hamilton function
(25) in the flat limit r −→ 0 transforms into the Hamiltonian
function of the plane problem for a reduced mass particle.
The corresponding coefficients transform into the expression
for the reduced mass
μ =
m1m2
m1 + m2
. (26)
Thus, in spaces with curvature, reduced particle masses can
be interpreted as dependent on coordinates, as it also seen
from (7) (see also [8]). The same is true for composite systems:
the reduced masses are functions of the coordinates.
Taking into account the form of the Hamilton function (25)
and the following definitions of generalized momenta
pr =
∂L
∂r˙
, pθ =
∂L
∂θ˙
, pϕ =
∂L
∂ ˙ϕ
, (27)
we consider the Hamilton-Jacobi equation
1
2μ∥(r)
∂W
∂r
!2
+
1
2μ⊥(r) sh2 r
×
×
"
∂W
∂θ
!2
+
1
sin2 θ
∂W
∂ϕ
!2#
+ U(r) +
∂W
∂t
= 0.
(28)
The last equation allows separation of variables
W = −Et +Wr(r) +Wθ(θ) +Wϕ(ϕ), (29)
and decomposes into the following equations
∂W
∂ϕ
= Mϕ, (30)
∂W
∂θ
2
+
M2
ϕ
sin2 θ
= M2, (31)
∂W
∂r
2
+
μ∥(r)
μ⊥(r)
M2
sh2 r
= 2μ∥(r)(E − U(r)). (32)
These equations are easily integrated. Wherein
Wϕ = Mϕϕ, (33)
Wθ =
Z s
M2 −
M2
ϕ
sin2 θ
dθ, (34)
Wr =
Z s
2μ∥[E − U(r)] − μ∥
μ⊥
M2
sh2 r
dr. (35)
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59
Substituting the last expressions into (29) and differentiating
with respect to constants, we obtain equations for the particle
trajectory
∂W
∂Mϕ
= ϕ1 − ϕ2−
−
Z θ2
θ1
Mϕ
sin2 θ
q
M2 − M2ϕ
sin2 θ
dθ = 0, (36)
∂W
∂M
=
Z θ2
θ1
Mdθ q
M2 − M2ϕ
sin2 θ
−
−
Z r2
r1
μ∥(r)
μ⊥(r)
Mdr
sh2 r
q
2μ∥[E − U(r)] − μ∥
μ⊥
M2
sh2 r
. (37)
The law of motion is given by the expression
∂W
∂E
= t2 − t1−
−
Z r2
r1
μ∥(r)dr q
2μ∥[E − U(r)] − μ∥
μ⊥
M2
sh2 r
. (38)
3. Schrödinger equation for two material points
in Lobachevsky space
The general formula for the classical kinetic energy of any
system is
Tcl =
1
2
X
i,j
gij(q)q˙iq˙j , (39)
where q˙i – generalized speeds, gij(q) – generalized masses.
The corresponding operator in quantum mechanics is
Tq =
−ℏ2
2
ΔBL, (40)
where ΔBL – the Laplace-Beltrami operator, which can be
obtained from the general expression
ΔBL =
√1
g
∂
∂qi
√
ggij ∂
∂qj
. (41)
Since in our case the kinetic energy expression can be seen
from (22), the Laplace-Beltrami operator has the form
ΔBL =
1
μ⊥
√
μ∥ sh2 r
∂
∂r
"
μ⊥ sh2 r
√
μ∥
∂
∂r
#
+
+
1
μ⊥ sh2 r
Δθϕ, (42)
where
Δθϕ =
1
sin θ
∂
∂θ
sin θ
∂
∂θ
+
1
sin2 θ
∂2
∂ϕ2 . (43)
Accordingly, the Schrödinger equation becomes
i
∂ψ
∂t
= Hψ, (44)
where the Hamiltonian is
H = −1
2
ΔLB + U(r). (45)
It is clear that the equation we have allows the separation of
variables
ψ = R(r)Y m
l (θ, ϕ), (46)
where Y m
l (θ, ϕ) are the spherical functions satisfying the
equation
ΔθϕY m
l (θ, ϕ) =
ΔθϕY m
l (θ, ϕ) = −l(l + 1)Y m
l (θ, ϕ) (47)
for l = 0, 1, 2, . . . and the radial part of the wave function
is the solution for the equation
d2R
dr2 +
√
μ∥
μ⊥ sh2 r
d
dr
"
μ⊥ sh2 r
√
μ∥
#
dR
dr
+
+
2μ∥(E − U) − μ∥l(l + 1)
μ⊥ sh2 r
!
R = 0. (48)
4. A particular problem of a rigid rotator in
spaces with constant curvature
As we know, in the case of a constant relative distance
between two points, a mechanical system is obtained, which
is called a rigid rotator. Despite the apparent simplicity, this
model for a flat space, both classical and quantum mechanical
[9], find interesting applications, including in the theory
of molecules and nuclear physics. From the approach developed
above, as well as in accordance with works [1-3, 9], it
follows that in spaces of constant curvature, a rigid rotator
has features associated with the dependence of the reduced
mass on the distance between points. These features are explored
below. By formulas (21), (25) and the corresponding
formulas in [2,3], the Lagrange function of a rigid rotator in
three spaces: in the Lobachevsky space, on the 3-sphere and
in the Euclidean space has the form
L = A
θ˙2 + sin2 θ ˙ϕ2
, (49)
where the quantity A has the following expressions in the
three spaces under consideration, respectively
Alob =
1
2
R2μ⊥lob sh2 r0
R
,
Asph =
1
2
R2μ⊥sph sin2 r0
R
,
Aflat =
1
2
μflatr2
0, (50)
where
μ⊥lob =
m1m2(m1 + m2)
m21
+ m22
+ 2m1m2 ch(r0/R)
, (51)
μ⊥sph =
m1m2(m1 + m2)
m21
+ m22
+ 2m1m2 cos(r0/R)
, (52)
μflat =
m1m2
m1 + m2
. (53)
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Solutions for the corresponding Hamilton-Jacobi equation
in the case of a rigid rotator have the form
θ = arccos
r
1 − M2
4AE
cos
r
E
A
t
!!
, (54)
ϕ = arctg
√
4AE
M
tg
r
E
A
t
!!
. (55)
It follows from the last formulas that the period of oscillations
T ∼
√
A. Let us analyze how the period of oscillations depends
on the ratio of the masses of particles (with a constant
total mass). Denote the mass ratio β = m1/m2 . Then
Alob =
mR2
2
sh2
r0
R
β
β2 + 1 + 2β ch(r0/R)
, (56)
Asph =
mR2
2
sin2
r0
R
β
β2 + 1 + 2β cos(r0/R)
,
(57)
A♭ =
mr2
0
2
β
(1 + β)2 . (58)
If we fix the values of the constants E = 1,M = 1 then
from the condition 1 − M2
4AE > 0 it follows that A > 1/4
and therefore, for the solutions to make sense, the mass ratio
cannot be arbitrary. Let’s take the values R = 2.5, r0 =
2,m = 1. Then the condition A > 1/4 for three spaces
leads to the following restrictions on the mass ratio of the
particles of a rigid rotator
Alob > 1/4 : 0.142 < β < 7.04; (59)
Asph > 1/4 : 0.207 < β < 4.83; (60)
Aflat > 1/4 : 0.172 < β < 5.83; (61)
We construct graphs of dependence of periods of oscillations
on the ratio of masses in the range 0.21 < β < 4.8. Figure
1 shows that the period of rotation of a rigid rotator, and
hence the magnitude of the angular momentum, depends on
the radius of space curvature R.
Figure 1. Graphs of rotation period on the ratio of the masses of the constituent
particles at a fixed total mass of the rotator. Top graph – Lobachevsky
space, average – flat space, bottom – 3-sphere.
Рисунок 1. Графики периода вращения в зависимости от отношения масс
составляющих частиц при фиксированной общей массе ротатора. Верх-
ний график – пространство Лобачевского, средний – плоское простран-
ство, нижний – трехмерная сфера.
For the equal distances between the material points of the
rotator and equal radius of the curvature for the Lobachevsky
and 3-sphere spaces, the rotation periods are maximum for
the Lobachevsky space, minimum for the 3-sphere. The corresponding
curve for flat space lies between the two mentioned
curves, with each of them tending to the flat space
curve at R −→ ∞. All three curves are similar.
In the quantum case, the Hamiltonian operator of such
a system has the form
H =
ℏ2
2I
Δθ,ϕ, (62)
where the moment of inertia of the system is
Ilob = 2mR2 sh2 r0
2R
, Iflat =
mr2
0
2
,
Isph = 2mR2 sin2 r0
2R
. (63)
The Schrödinger equationHψ = Eψ gives the energy levels
of the rigid rotator
El =
ℏ2
2I
l(l + 1), (64)
and the eigenfunctions of the Hamilton operator are equal to
the spherical functions for all three spaces ψ = Y m
l (θ, ϕ).
The levels are degenerate, since each value of the orbital
quantum number corresponds to 2l + 1 magnetic number
values. Figure 2 shows energy levels for quantum rotator in
the spaces under consideration (we set here R = 2.5, r0 =
2,m = 1).
Figure 2. Energy levels for rigid rotator in spaces of constant curvature. Top
point – 3-sphere, average – flat space, bottom – Lobachevsky space.
Рисунок 2. Уровни энергии для жесткого ротатора в пространствах посто-
янной кривизны. Верхняя точка – трехмерная сфера, средняя – плоское
пространство, нижняя – пространство Лобачевского.
Conclusion
The paper solves the classical and quantum problems of
motion of two particles in three-dimensional Lobachevsky
space, relative to the center of mass. The Hamilton-Jacobi
equation of the problem is formulated and its solutions are
found. The corresponding Schrödinger equation allows the
separation of radial and angular variables. It is shown that the
reduced masses of the system depend on the relative distance
between the particles. The classical and quantum problems
of a rigid rotator in three-dimensional Lobachevsky space are
formulated and solved. The dependences of the rotator oscillation
period on the ratio of the masses of the forming particles
at a fixed distance between them and fixed total mass
are obtained for three cases: Lobachevsky space, 3-sphere
and three-dimensional Euclidean space.
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