Россия
Уравнение Штюкельберга для частицы с двумя спиновы- ми состояниями S = 1, S = 0 исследуется в присут- ствии внешнего однородного магнитного поля. В реля- тивистском случае частица описывается 11-компонентной волновой функцией. На решениях диагонализируются опе- раторы энергии, третьей проекции полного углового мо- мента и третьей проекции линейного момента вдоль на- правления магнитного поля. После разделения перемен- ных получена система для 11 функций от одной перемен- ной. В данной системе выполнено нерелятивистское при- ближение. При этом применяется известный метод получе- ния нерелятивистских уравнений из релятивистских, ос- нованный на проективных операторах, связанных с мат- рицей Γ0 релятивистского уравнения. Нерелятивистская волновая функция оказывается четырехмерной. Получена система для четырех функций. Построены точные реше- ния в вырожденных гипергеометрических функциях. Най- дены три серии энергетических уровней, что согласуется с результатом, полученным для релятивистского уравнения Штюкельберга.
частица Штюкельберга, нерелятивистское приближение, магнитное поле, проективные операторы, точные решения, связанные состояния
Introduction
In previous paper [1], we studied the relativistic St¨uckelberg
tensor system (see the references in [1]) of 11 equations
in presence of the external uniform magnetic field. The relativistic
particle is described by 11-component wave function,
consisted of scalar, vector, and antisymmetric tensor. On solutions
there are diagonalized operators of energy, the third
projection of the total angular momentum, and the third projection
of the linear momentum along the magnetic field direction.
After separating the variables, the system of 11 radial
functions was derived, and it was solved in the terms of con-
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fluent hypergeometric functions. Three series of the energy
levels are found.
In the present paper we study the non-relativistic approximation
for this problem. We apply the well-known method
(see [2–4]) from the general theory of relativistic wave equations,
based on the minimal equation for the matrix Γ0 (in
the model under consideration, it is an 11×11-matrix). This
minimal equation allows us to introduce three projective operators
P+, P−, P0 and then expand the wave function into
three components: Ψ = Ψ+ + Ψ− + Ψ0. From the general
theory it is known that when obtaining the nonrelativistic
approximation the componentsΨ− andΨ0 should be considered
as small, and Ψ+ — as large ones. Only the component
Ψ+ enters the nonrelativistic equation. The nonrelativistic
wave function turns out to be 4-dimensional. We derive the
radial system for 4 functions. It is solved in terms of confluent
hypergeometric functions. There arise three series of energy
levels with corresponding solutions. This result agrees with
that obtained for relativistic St¨uckelberg equation.
1. Nonrelativistic approximation and projective
operators
We start with the matrixΓ0 of the basic St¨uckelberg equation
(see [1])
Γ0 =
0
BBBBBBBBB@
0 −1 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 −1 0 0 0 0 0
0 0 0 0 0 0 −1 0 0 0 0
0 0 0 0 0 0 0 −1 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
1
CCCCCCCCCA
.
(1)
It obeys the minimal equation Γ(Γ2+1) = 0, which permits
us to define three projective operators
P0 = 1 + Γ20
, P1 = P+ =
1
2
iΓ(iΓ + 1),
P2 = P− =
1
2
iΓ(iΓ − 1) (2)
with the properties
P2
0 = P0, P2+
= P+, P2−
= P−,
P0 + P+ + P− = I.
In accordance with this, the complete wave function may be
decomposed into the sum of three parts
Ψ = Ψ+ + Ψ− + Ψ0,
Ψ+ = P+Ψ, Ψ− = P−Ψ, Ψ0 = P0Ψ. (3)
It is known from the general theory that in nonrelativistic approximation
the componentΨ+ should be considered as a big
one, whereas the components Ψ−,Ψ0 — as small ones. We
readily find their explicit structure:
Ψ+ =
1
2
H − iΨ0, i(H − iΨ0),Ψ1 − iE1,Ψ2 − iE2,
Ψ3−iE3, i(Ψ1−iE1), i(Ψ2−iE2), i(Ψ3−iE3), 0, 0, 0
t
=
= (L0, iL0,L1,L2,L3, iL1, iL2, iL3, 0, 0, 0)t,
Ψ− =
1
2
H +iΨ0,−i(H +iΨ0),Ψ1+iE1,Ψ2+iE2,
Ψ3 + iE3,−i(Ψ1 + iE1),−i(Ψ2 + iE2),
−i(Ψ3 + iE3), 0, 0, 0
t
=
=
S0,−iS0, S1, S2, S3,−iS1,−iS2,−iS3, 0, 0, 0
t
,
Ψ0 = (0, 0, 0, 0, 0, 0, 0, 0,B1,B2,B3)t, (4)
where t stands for transpose. We have introduced special notations
for big and small functions.
Because when solving the relativistic problem [1], we used
the cyclic basis, now we also should transform big and small
component to this basis. Because all blocks of the matrix Γ0
preserve their form in cyclic basis,
¯Δ
0 = (1, 0, 0, 0)t, −¯G0 = (−1, 0, 0, 0),
¯K
0 =
0
B@
0 0 0 0 0 0
−1 0 0 0 0 0
0 −1 0 0 0 0
0 0 −1 0 0 0
1
CA,
¯L
0 =
0
BBBBB@
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
1
CCCCCA
,
we conclude the rule for obtaining big and small components
remains the same in this known basis:
¯ P0 = 1 + ¯Γ
20
,
¯ P+ =
1
2
i¯Γ(i¯Γ + 1), ¯ P− =
1
2
i¯Γ(i¯Γ − 1).
The formulas (4) may be written in more convenient variables
as follows:
Ψ+ =
1
2
h − ih0, i(h − ih0), h1 − iE1, h2 − iE2,
h3−iE3, i(h1−iE1), i(h2−iE2), i(h3−iE3), 0, 0, 0
t
=
= (L0, iL0,L1,L2,L3, iL1, iL2, iL3, 0, 0, 0)t,
Ψ− =
1
2
h + ih0,−i(h + ih0), h1 + iE1, h2 + iE2,
h3 + iE3,−i(h1 + iE1),−i(h2 + iE2),
−i(h3 + iE3), 0, 0, 0
t
=
=
S0,−iS0, S1, S2, S3,−iS1,−iS2,−iS3, 0, 0, 0
t
,
Ψ0 = (0, 0, 0, 0, 0, 0, 0, 0,B1,B2,B3)t. (5)
80
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Hence we can derive inverse expressions for initial variables
through big and small components:
h =
1
2
(L0 + S0), h0 = i
1
2
(L0 − S0),
hi =
1
2
(Li + Si), Ei = i
1
2
(Li − Si), (6)
i = 1, 2, 3.
Now we turn to relativistic system of equations (see in [1]),
collecting them in 4 pairs and one triple:
− iϵh0 − ikh2 +
√1
2
h
′
1
− (Br2 + 2m − 2)
2
√
2r
h1−
− √1
2
h
′
3
− (Br2 + 2m + 2)
2
√
2r
h3 = −μh,
− iϵh − ikE2 +
√1
2
E
′
1
− (Br2 + 2m − 2)
2
√
2r
E1−
− √1
2
E
′
3
− (Br2 + 2m + 2)
2
√
2r
E3 = μh0;
− √1
2
h
′ − m +√Br2/2
2r
h +
√1
2
B
′
2+
+
(Br2 + 2m)
2
√
2r
B2 − ikB3 + iϵE1 = μh1,
+
√1
2
h
′
0 +
(Br2 + 2m)
2
√
2r
h0 − iϵh1 = μE1;
ikh + iϵE2 − √1
2
B
′
1
− (Br2 + 2m + 2)
2
√
2r
B1−
− √1
2
B
′
3 +
(Br2 + 2m − 2)
2
√
2r
B3 = μh2,
−ikh0 − iϵh2 = μE2;
√1
2
h
′ − m +√Br2/2
2r
h +
√1
2
B
′
2
−
− (Br2 + 2m)
2
√
2r
B2 + ikB1 + iϵE3 = μh3,
−√1
2
h
′
0 +
Br2 + 2m
2
√
2r
h0 − iϵh3 = μE3;
−√1
2
h
′
2 +
Br2 + 2m
2
√
2r
h2 + ikh3 = μB1,
−ikh1 − √1
2
h
′
2
− Br2 + 2m
2
√
2r
h2 = μB3,
+
√1
2
h
′
1
− Br2 + 2m − 2
2
√
2r
h1+
+
√1
2
h
′
3 +
Br2 + 2m + 2
2
√
2r
h3 = μB2.
Eliminating the small components B1, B2, B3 with the help
of three last equations we get
−iϵh0−Br2 + 2m − 2
2
√
2r
h1−ikh2−Br2 + 2m + 2
2
√
2r
h3+
+
√1
2
dh1
dr
− √1
2
dh3
dr
= −μh,
− iϵh − ikE2 +
√1
2
dE1
dr
− Br2 + 2m − 2
2
√
2r
E1−
− √1
2
dE3
dr
− Br2 + 2m + 2
2
√
2r
E3 = μh0;
iϵμE1 − (Br2 + 2m)μ
2
√
2r
h+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 h1+
+
(Br2 + 2m)ik
2
√
2r
h2+
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 h3−
− √μ
2
dh
dr
+
1
2r
dh1
dr
+
√ik
2
dh2
dr
+
+
Br2 + 2m + 1
2r
dh3
dr
+
1
2
d2h1
dr2 +
1
2
d2h3
dr2 = μ2h1,
(Br2 + 2m)μ
2
√
2r
h0 − iϵμh1 +
√μ
2
dh0
dr
= μ2E1;
iϵμE2 + ikμh − (Br2 + 2m − 2)ik
2
√
2r
h1−
−(Br2 + 2m)2
4r2 h2 − (Br2 + 2m + 2)ik
2
√
2r
h3+
+
√ik
2
dh1
dr
+
1
r
dh2
dr
− √ik
2
dh3
dr
+
d2h2
dr2 = μ2h2,
−ikμh0 − iϵμh2 = μ2E2;
iϵμE3 − (Br2 + 2m)μ
2
√
2r
h+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 h1+
+
(Br2 + 2m)ik
2
√
2r
h2+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 h3+
+
√μ
2
dh
dr
− Br2 + 2m − 1
2r
dh1
dr
−
−√ik
2
dh2
dr
+
1
2r
dh3
dr
+
1
2
d2h1
dr2 +
1
2
d2h3
dr2 = μ2h3,
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(Br2 + 2m)μ
2
√
2r
h0 − iϵμh3 − √μ
2
dh0
dr
= μ2E3.
Let us take into account the formulas (2), this results in
pair I
ϵ(L0 − S0) − Br2 + 2m − 2
2
√
2r
(L1 + S1)−
−ik(L2 + S2) − Br2 + 2m + 2
2
√
2r
(L3 + S3)+
+
√1
2
d(L1 + S1)
dr
− √1
2
d(L3 + S3)
dr
= −μ(L0 + S0),
−ϵ(L0 + S0) − ik(L2 − S2) +
√1
2
d(L1 − S1)
dr
−
−Br2 + 2m − 2
2
√
2r
(L1 − S1) − √1
2
d(L3 − S3)
dr
−
−Br2 + 2m + 2
2
√
2r
(L3 − S3) = μ(L0 − S0);
pair II
−ϵμ(L1 − S1) − (Br2 + 2m)μ
2
√
2r
(L0 + S0)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 (L1+S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 (L3 + S3)−
−√μ
2
d(L0 + S0)
dr
+
1
2r
d(L1 + S1)
dr
+
+
√ik
2
d(L2 + S2)
dr
+
Br2 + 2m + 1
2r
d(L3 + S3)
dr
+
+
1
2
d2(L1 + S1)
dr2 +
1
2
d2(L3 + S3)
dr2 = μ2(L1 + S1),
(Br2 + 2m)μ
2
√
2r
(L0 − S0) − ϵμ(L1 + S1)+
+
√μ
2
d(L0 − S0)
dr
= μ2(L1 − S1);
pair III
− ϵμ(L2 − S2) + ikμ(L0 + S0)−
− (Br2 + 2m − 2)ik
2
√
2r
(L1 + S1)−
− (Br2 + 2m)2
4r2 (L2 + S2)−
− (Br2 + 2m + 2)ik
2
√
2r
(L3 + S3)+
+
√ik
2
d(L1 + S1)
dr
+
1
r
d(L2 + S2)
dr
−
−√ik
2
d(L3 + S3)
dr
+
d2(L2 + S2)
dr2 = μ2(L2 + S2),
−ikμ(L0 − S0) − ϵμ(L2 + S2) = μ2(L2 − S2);
pair IV
−ϵμ(L3 − S3) − (Br2 + 2m)μ
2
√
2r
(L0 + S0)+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 (L1 + S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 (L3+S3)+
+
√μ
2
d(L0 + S0)
dr
− Br2 + 2m − 1
2r
d(L1 + S1)
dr
−
−√ik
2
d(L2 + S2)
dr
+
1
2r
d(L3 + S3)
dr
+
+
1
2
d2(L1 + S1)
dr2 +
1
2
d2(L3 + S3)
dr2 = μ2(L3 + S3),
(Br2 + 2m)μ
2
√
2r
(L0 − S0) − ϵμ(L3 + S3)−
− √μ
2
d(L0 − S0)
dr
= μ2(L3 − S3).
Within each pair, let us sum and subtract equations, this results
in:
pair I
Br2 +√2m − 2
2r
L1 + 2ikL2 +
Br2 +√2m + 2
2r
L3−
−
√
2
dL1
dr
+
√
2
dL3
dr
+ 2ϵS0 = 2μS0,
2ϵL0 +
−Br2 − 2m + 2 √
2r
S1−
− 2ikS2 − Br2 +√2m + 2
2r
S3+
+
√
2
dS1
dr
−
√
2
dS3
dr
= −2μL0;
82
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pair II
(Br2 + 2m)μ
2
√
2r
(L0 − S0) − ϵμ(L1 + S1)+
+
√μ
2
dL0
dr
− dS0
dr
− ϵμ(L1 − S1)−
−(Br2 + 2m)μ
2
√
2r
(L0 + S0)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 (L1+S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 (L3 + S3)−
−√μ
2
dL0
dr
+
dS0
dr
+
1
2r
dL1
dr
+
dS1
dr
+
+
√ik
2
dL2
dr
+
dS2
dr
+
Br2 + 2m + 1
2r
dL3
dr
+
dS3
dr
+
+
1
2
d2L1
dr2 +
d2S1
dr2
+
1
2
d2L3
dr2 +
d2S3
dr2
= 2μ2L1,
−ϵμ(L1 − S1) − (Br2 + 2m)μ
2
√
2r
(L0 + S0)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 (L1+S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2) +
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 (L3 + S3)−
−√μ
2
dL0
dr
+
dS0
dr
+
1
2r
dL1
dr
+
dS1
dr
+
+
√ik
2
dL2
dr
+
dS2
dr
+
Br2 + 2m + 1
2r
dL3
dr
+
dS3
dr
+
+
1
2
d2L1
dr2 +
d2S1
dr2
+
1
2
d2L3
dr2 +
d2S3
dr2
−
−(Br2 + 2m)μ
2
√
2r
(L0 − S0) + ϵμ(L1 + S1)−
−√μ
2
dL0
dr
− dS0
dr
= 2μ2S1;
pair III
−ϵμ(L2 − S2) + ikμ(L0 − S0)−
−(Br2 + 2m − 2)ik
2
√
2r
(L1 + S1)−
−(Br2 + 2m)2
4r2 (L2 + S2)−
−(Br2 + 2m + 2)ik
2
√
2r
(L3 + S3)+
+
√ik
2
dL1
dr
+
dS1
dr
+
1
r
dL2
dr
+
dS2
dr
−
−√ik
2
dL3
dr
+
dS3
dr
+
d2L2
dr2 +
d2S2
dr2
−
−ikμ(L0 − S0) − ϵμ(L2 + S2) = 2μ2L2;
−ϵμ(L2 − S2) + ikμ(L0 + S0)−
−(Br2 + 2m − 2)ik
2
√
2r
(L1 + S1)−
−(Br2 + 2m)2
4r2 (L2 + S2)−
−(Br2 + 2m + 2)ik
2
√
2r
(L3 + S3)+
+
√ik
2
dL1
dr
+
dS1
dr
+
1
r
dL2
dr
+
dS2
dr
−
−√ik
2
dL3
dr
+
dS3
dr
+
d2L2
dr2 +
d2S2
dr2 +
+ikμ(L0 − S0) + ϵμ(L2 + S2) = 2μ2S2;
pair IV
−ϵμ(L3 − S3) − (Br2 + 2m)μ
2
√
2r
(L0 + S0)+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 (L1 + S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 (L3+S3)+
+
√μ
2
dL0
dr
+
dS0
dr
−Br2 + 2m − 1
2r
dL1
dr
+
dS1
dr
−
−√ik
2
dL2
dr
+
dS2
dr
+
1
2r
dL3
dr
+
dS3
dr
+
+
1
2
d2L1
dr2 +
d2S1
dr2
+
1
2
d2L3
dr2 +
d2S3
dr2
+
+
(Br2 + 2m)μ
2
√
2r
(L0 − S0) − ϵμ(L3 + S3)−
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−√μ
2
dL0
dr
− dS0
dr
= 2μ2L3,
−ϵμ(L3 − S3) − (Br2 + 2m)μ
2
√
2r
(L0 + S0)+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 (L1 + S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 (L3+S3)+
+
√μ
2
dL0
dr
+
dS0
dr
−Br2 + 2m − 1
2r
dL1
dr
+
dS1
dr
−
−√ik
2
dL2
dr
+
dS2
dr
+
1
2r
dL3
dr
+
dS3
dr
+
+
1
2
d2L1
dr2 +
d2S1
dr2
+
1
2
d2L3
dr2 +
d2S3
dr2
−
−(Br2 + 2m)μ
2
√
2r
(L0 − S0) + ϵμ(L3 + S3)+
+
√μ
2
dL0
dr
− dS0
dr
= 2μ2L3.
The parameter μ relates to physical (positive) mass by the
formula
μ = −M.
Let us separate the rest energy M by formal change
ϵ = M+E, whereE is nonrelativistic energy of the particle.
The above equations become simpler:
pair I
Br2 +√2m − 2
2r
L1 + 2ikL2 +
Br2 +√2m + 2
2r
L3−
−
√
2
dL1
dr
+
√
2
dL3
dr
+ 2(M + E)S0 = −2MS0,
2(M + E)L0 +
−Br2 − 2m + 2 √
2r
S1 − 2ikS2−
−Br2 +√2m + 2
2r
S3 +
√
2
dS1
dr
−
√
2
dS3
dr
= 2ML0;
pair II
−(Br2 + 2m)M
2
√
2r
(L0 − S0) + (M + E)M(L1 + S1)−
−M√
2
dL0
dr
− dS0
dr
+ (M + E)M(L1 − S1)+
+
(Br2 + 2m)M
2
√
2r
(L0 + S0)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 (L1+S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 (L3 + S3)+
+
M√
2
dL0
dr
+
dS0
dr
+
1
2r
dL1
dr
+
dS1
dr
+
+
√ik
2
dL2
dr
+
dS2
dr
+
+
Br2 + 2m + 1
2r
dL3
dr
+
dS3
dr
+
+
1
2
d2L1
dr2 +
d2S1
dr2
+
1
2
d2L3
dr2 +
d2S3
dr2
= 2M2L1,
(M + E)M(L1 − S1) +
(Br2 + 2m)M
2
√
2r
(L0 + S0)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 (L1+S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 (L3 + S3)+
+
M√
2
dL0
dr
+
dS0
dr
+
1
2r
dL1
dr
+
dS1
dr
+
+
√ik
2
dL2
dr
+
dS2
dr
+
Br2 + 2m + 1
2r
dL3
dr
+
dS3
dr
+
+
1
2
d2L1
dr2 +
d2S1
dr2
+
1
2
d2L3
dr2 +
d2S3
dr2
+
+
(Br2 + 2m)M
2
√
2r
(L0 − S0) − (M + E)M(L1 + S1)+
+
M√
2
dL0
dr
− dS0
dr
= 2M2S1;
pair III
(M + E)M(L2 − S2) − ikM(L0 + S0)−
−(Br2 + 2m − 2)ik
2
√
2r
(L1+S1)−(Br2 + 2m)2
4r2 (L2+S2)−
−(Br2 + 2m + 2)ik
2
√
2r
(L3+S3)+
√ik
2
dL1
dr
+
dS1
dr
+
+
1
r
dL2
dr
+
dS2
dr
− √ik
2
dL3
dr
+
dS3
dr
+
+
d2L2
dr2 +
d2S2
dr2 + ikM(L0 − S0)+
+(M + E)M(L2 + S2) = 2M2L2,
(M + E)M(L2 − S2) − ikM(L0 + S0)−
−(Br2 + 2m − 2)ik
2
√
2r
(L1+S1)−(Br2 + 2m)2
4r2 (L2+S2)−
84
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−(Br2 + 2m + 2)ik
2
√
2r
(L3+S3)+
√ik
2
dL1
dr
+
dS1
dr
+
+
1
r
dL2
dr
+
dS2
dr
− √ik
2
dL3
dr
+
dS3
dr
+
+
d2L2
dr2 +
d2S2
dr2
− ikM(L0 − S0)−
−(M + E)M(L2 + S2) = 2M2S2;
pair IV
(M + E)M(L3 − S3) +
(Br2 + 2m)M
2
√
2r
(L0 + S0)+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 (L1 + S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 (L3 + S3)−
−M√
2
dL0
dr
+
dS0
dr
−Br2 + 2m − 1
2r
dL1
dr
+
dS1
dr
−
−√ik
2
dL2
dr
+
dS2
dr
+
1
2r
dL3
dr
+
dS3
dr
+
+
1
2r
d2L1
dr2 +
d2S1
dr2
+
1
2r
d2L3
dr2 +
d2S3
dr2
−
−(Br2 + 2m)M
2
√
2r
(L0 − S0) + (M + E)M(L3 + S3)+
+
M√
2
dL0
dr
− dS0
dr
= 2M2L3,
(M + E)M(L3 − S3) +
(Br2 + 2m)M
2
√
2r
(L0 + S0)+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 (L1 + S1)+
+
(Br2 + 2m)ik
2
√
2r
(L2 + S2)+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 (L3+S3)−
−M√
2
dL0
dr
+
dS0
dr
−Br2 + 2m − 1
2r
dL1
dr
+
dS1
dr
−
−√ik
2
dL2
dr
+
dS2
dr
+
1
2r
dL3
dr
+
dS3
dr
+
+
1
2
d2L1
dr2 +
d2S1
dr2
+
1
2
d2L3
dr2 +
d2S3
dr2
+
+
(Br2 + 2m)M
2
√
2r
(L0 − S0) − (M + E)M(L3 + S3)−
−M√
2
dL0
dr
− dS0
dr
= 2M2S3.
Let us neglect small components, so we obtain
Br2 +√2m − 2
2r
L1 + 2ikL2 +
Br2 +√2m + 2
2r
L3−
−
√
2
dL1
dr
− dL3
dr
+ 2S0(M + E) = −2MS0,
2EL0 +
−Br2 − 2m + 2 √
2r
S1 − 2ikS2−
−Br2 +√2m + 2
2r
S3 +
√
2
dS1
dr
− dS3
dr
= 0,
(Br2√+ 2m)M
2r
S0 +
√
2M
dS0
dr
+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 L1+
+
(Br2 + 2m)ik
2
√
2r
L2+
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 L3+
+
1
2r
dL1
dr
+
√ik
2
dL2
dr
+
Br2 + 2m + 1
2r
dL3
dr
+
+
1
2
d2L1
dr2 +
1
2
d2L3
dr2 + 2MEL1 = 0,
(Br2√+ 2m)M
2r
L0 +
√
2M
dL0
dr
− 2(M + E)MS1+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m − 1)2
8r2 L1+
+
(Br2 + 2m)ik
2
√
2r
L2+
+
B2r4 + 4B(m + 1)r2 + 4m2 − 4
8r2 L3+
+
1
2r
dL1
dr
+
√ik
2
dL2
dr
+
Br2 + 2m + 1
2r
dL3
dr
+
+
1
2
d2L1
dr2 +
1
2
d2L3
dr2 = 2M2S1,
2MEL2 − 2ikMS0 − (Br2 + 2m − 2)ik
2
√
2r
L1−
−(Br2 + 2m)2
4r2 L2 − (Br2 + 2m + 2)ik
2
√
2r
L3+
+
√ik
2
dL1
dr
+
1
r
dL2
dr
− √ik
2
dL3
dr
+
d2L2
dr2 = 0,
−2(M+E)MS2−2ikML0− (Br2 + 2m − 2)ik
2
√
2r
L1−
−(Br2 + 2m)2
4r2 L2 − (Br2 + 2m + 2)ik
2
√
2r
L3+
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+
√ik
2
dL1
dr
+
1
r
dL2
dr
− √ik
2
dL3
dr
+
d2L2
dr2 = 2M2S2,
(Br2√+ 2m)M
2r
S0 −
√
2M
dS0
dr
+ 2ML3E+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 L1+
+
(Br2 + 2m)ik
2
√
2r
L2+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 L3−
−Br2 + 2m − 1
2r
dL1
dr
− √ik
2
dL2
dr
+
+
1
2r
dL3
dr
+
1
2
d2L1
dr2 +
1
2
d2L3
dr2 = 0,
−
√
2M
dL0
dr
+
(Br2√+ 2m)M
2r
L0 − 2(M + E)MS3+
+
B2r4 + 4B(m − 1)r2 + 4m2 − 4
8r2 L1+
+
(Br2 + 2m)ik
2
√
2r
L2+
+
−B2r4 + (−8k2 − 4Bm)r2 − 4(m + 1)2
8r2 L3−
−Br2 + 2m − 1
2r
dL1
dr
− √ik
2
dL2
dr
+
1
2r
dL3
dr
+
+
1
2
d2L1
dr2 +
1
2
d2S1
dr2 +
1
2
d2L3
dr2 = 2M2S3.
We assume that nonrelativistic energy may be neglected in
comparison with rest energyM +E ≈ M. Also from equations
(1), (4), (6), (8) we express the small variables S0, S1,
S2, S3 and substitute them in equations (2), (3), (5), (7). In this
way, we obtain equations which contain only the big components
d2
dr2 +
1
r
d
dr
+ 2ME − k2 − (m − 1)2
r2
−
−B2r2
4
− Bm
L1 = 0, L1 = N1f1;
d2
dr2 +
1
r
d
dr
+ 2ME − k2 − m2
r2
−
−B2r2
4
− Bm
L2 = 0, L2 = N2f3;
d2
dr2 +
1
r
d
dr
+ 2ME − k2 − (m + 1)2
r2
−
−B2r2
4
− Bm
L3 = 0, L3 = N3f2;
d2
dr2 +
1
r
d
dr
+ 2ME − k2−
−m2
r2
− B2r2
4
− Bm
L0+
+
1
2
√
2
d
dr
− m + Br2/2 − 1
r
L1+
+
1
2
√
2
d
dr
+
m + Br2/2 + 1
r
L3 = 0.
The last equation may be re-written differently
d2
dr2 +
1
r
d
dr
+ 2ME − k2−
−m2
r2
− B2r2
4
− Bm
L0+
+
1
2
√
2
(N1bm−1f1 + N3am+1f2) = 0. (7)
It is evident that L0 = constf3, then eq. (7) takes on the
form
N1bm−1f1 + N3am+1f2 = 0. (8)
There exist differential constraints (see in [1])
bm−1f1 = C1f3, am+1f2 = C2f3,
they permit us to transform the previous relation to the following
form (see [1])
N1bm−1f1 + N3am+1f2 =
= (N1C1 + N3C2)f3 = 0 ⇒ N1C1 + N3C2 = 0.
Thus we have obtained 4 separate equations (only 3 equations
are different, their solutions will be found in the next
section)
d2
dr2 +
1
r
d
dr
+ 2ME − k2 − (m − 1)2
r2
−
−B2r2
4
− Bm
L1 = 0, L1 = N1f1;
d2
dr2 +
1
r
d
dr
+ 2ME − k2 − m2
r2
−
−B2r2
4
− Bm
L2 = 0, L2 = N2f3;
d2
dr2 +
1
r
d
dr
+ 2ME − k2 − (m + 1)2
r2
−
−B2r2
4
− Bm
L3 = 0, L3 = N3f2;
d2
dr2 +
1
r
d
dr
+ 2ME − k2 − m2
r2
−
−B2r2
4
− Bm
L0 = 0, L0 = N0f3;
and the algebraic constraint
N1C1 + N3C2 = 0.
86
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Recall that (see [1])
C1 =
√
X − B, C2 =
√
X + B,
X = 2BN = 2ME − k2. (9)
General solution on the nonrelativistic equation consists of
three components (the general multiplier eimϕeikz is omitted):
Ψ = e
−iE1tN1
0
B@
1
0
0
0
1
CA
f1 + e
−iE2tN3
0
B@
0
0
1
0
1
CA
f2+
+e
−iE3tN2
0
B@
0
1
0
0
1
CA
f3 + e
−iE3tN0
0
B@
0
0
0
1
1
CA
f3(r).
We have 3 different series of energy levels, E1, E2, E3
and 3 different wave functions. This result agrees with that
obtained in relativistic case.
2. Solving the differential equations
The above three equations let us transform to the variable
x = Br2/2, B > 0:
d2L1
dx2 +
1
x
dL1
dx
+
−1
4
+
1
2
2ME − k2 − Bm
Bx
−
−1
4
(m − 1)2
x2
L1 = 0, (10)
d2L2
dx2 +
1
x
dL2
dx
+
−1
4
+
1
2
2ME − k2 − Bm
Bx
−
−1
4
m2
x2
L2 = 0, (11)
d2L3
dx2 +
1
x
dL3
dx
+
−1
4
+
1
2
2ME − k2 − Bm
Bx
−
−1
4
(m + 1)2
x2
L3 = 0. (12)
Consider eq. (10):
L1 = Xa1eb1xF1,
x
d2F1
dx2 + (2a1 + 1 + 2b1x)
dF1
dx
+
+
1
4
(4b21
− 1)x +
1
4
4a21
− (m − 1)2
x
+
+
1
4
8a1b1B + 4b1B + 4ME − 2k2 − 2Bm
B
F1 = 0.
Imposing restrictions a1 = ±1
2
|m − 1|, b1 = −1
2
we obtain
the equation of confluent hypergeometric type
x
d2F1
dx2 + (2a1 + 1 − x)
dF1
dx
+
+
1
2
−(2a1 + 1)B + 2ME − k2 − Bm
B
F1 = 0
with parameters
α1 =
(2a1 + 1)B − 2ME + k2 + Bm
2B
, γ1 = 2a1 +1.
In order to get the bound states
a1 = +
1
2
|m − 1|, γ1 = |m − 1| + 1,
α1 =
k2 + B(|m − 1| + m + 1) − 2ME
2B
;
that polynomial condition α1 = −n1 gives
E1 − k2
2M
=
B
M
n1 +
|m − 1| + m + 1
2
. (13)
Two other equations lead to similar results. Thus, we get
L1 = x
|m−1|/2F
−x/2F(−n1, |m − 1| + 1, x),
E1 − k2
2M
=
B
M
n1 +
|m − 1| + m + 1
2
,
L2 = x
|m|/2F
−x/2F(−n2, |m| + 1, x),
E2 − k2
2M
=
B
M
n2 +
|m| + m + 1
2
, (14)
L3 = x
|m+1|/2F
−x/2F(−n3, |m + 1| + 1, x),
E3 − k2
2M
=
B
M
n3 +
|m + 1| + m + 1
2
.
Discussion
The nonrelativistic wave function for St¨uckelberg particle
turned out to be 4-dimensional. We have derived the
corresponding radial system for 4 functions. It has been
solved in terms of confluent hypergeometric functions. There
arise three series of energy levels with corresponding solutions.
This result agrees with that obtained for the relativistic
St¨uckelberg equation.
It may be noted that the similar nonrelativistic study for
St¨uckelberg particle in presence of the external Coulomb field
was done in [5]. The energy spectrum was found. Besides, a
general Pauli-like equation was derived for this particle in
presence of arbitrary electromagnetic field.
1. Ovsiyuk, E.M. St¨uckelberg particle in external magnetic field, and the method of projective operators / E.M. Ovsiyuk, A.P. Safronov, A.V. Ivashkevich, O.A. Semenyuk // Известия Коми НЦ УрО РАН. - 2022. - № 5 (57). - С. 69-78.
2. Богуш, А.А. Нерелятивистский предел в общековариантной теории векторной частицы / А.А. Богуш, В.В. Кисель, Н.Г. Токаревская, В.М. Редьков // Известия НАН Беларуси. Серия физ.-мат. наук. - 2002. - № 2. - С. 61-66.
3. Bogush, A.A. Duffin-Kemmer-Petiau formalism reexamined: nonrelativistic approximation for spin 0 and spin 1 particles in the Riemannian space-time / A.A. Bogush, V.V. Kisel, N.G. Tokarevskaya, V.M. Red’kov // Annales de la Fondation Louis de Broglie. - 2007. - Vol. 32. - № 2-3. - P. 355-381.
4. Редьков, В.М. Поля частиц в римановом пространстве и группа Лоренца / В.М. Редьков. - Минск: Белорусская наука, 2009. - 486 с.
5. Ovsiuyk, E.M. St¨uckelberg particle in the Coulomb field, nonrelativistic approximation, wave functions and spectra / E.M. Ovsiyuk, O.A. Semenyuk, A.V. Ivashkevich, M. Neagu // Nonlinear Phenomena in Complex Systems. - 2022. - Vol. 25. - № 3. - P. 352-367.
