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[Ja , {Pb , Gb , Jb }] = i?abc {Pc , Gc , Jc }.
[m, Ja ] = [m, Pµ ] = [m, Ga ] = 0,
Algebra {Pµ , Ja , Ga , m} has three invariant (Casimir) operators
P2
C1 = P0 ? (mJa ? ?abc pb Gc )2 . (3.5)
, C2 = m, C3 =
2m s

The eigenvalues of the operators C1 , C2 and C3 are associated with internal energy,
mass and square of the spin of a particle, described by eq.(3.1).
So the problem of finding GIWE in the form (3.1) reduces to the determination of
explicit expressions of operators Hs , satisfying the relations (3.2), (2.6), (3.3). In [7,
9, 10] the Hamiltonians Hs have been obtained in such a form:
1
(3.6)
Hs = ?1 m + 2?3 Sa pa + Cab pa pb ,
2m
where
Cab = ?ab ? 2(?1 ? i?2 )(Sa Sb + Sb Sa ), (3.7)
?a are 2(2s + 1)-row Pauli matrices, commuting with Sa (3.3). One can make sure
directly that the operators (3.6), (2.6), (3.3) satisfy conditions (3.2), and the operators
(3.5) eigenvalues are equal to
c1 = ±m, c3 = m2 s(s + 1).
c2 = m,
Therefore, one concludes that eqs.(3.1), (3.6) are Galilean invariant and describe a
nonrelativistic partic1es of mass m and spin s.
370 W.I. Fushchych, A.G. Nikitin

The motion equation for a charged particle in an external electromagnetic field can
be obtained from (3.1), (3.6) via standard substitution (2.11). As a result one obtains
? 1
(3.8)
i ?= ?1 m + 2?3 Sa ?a + Cab ?a ?b + eA0 ?.
?t 2m
Equation (3.8) (as well as the first-order equation (2.1) after substitution (2.11))
will be invariant under Galilei transformations [10] if the vector potential will be
similtaneously transformed according to [4]
A > A = RA, A0 > A0 = A0 + V · A. (3.9)
It is demonstrated in [7, 9] that eq.(3.8) describes dipole, quadrupole and spin-
orbit coupling of a charged particle with an external field. In the case s = 1 such a
2
description is in good accordance with that given by Dirac equation.

4. Energy spectrum of any spin particle in Coulomb field
In this section GIWE are applied to solve the problem of the description of any
spin particle movement in Coulombic field.
After minimal substitution (2.11) one comes from (2.1) to the equation
L(?) = ?µ ? µ + ?5 m. (4.1)
L(?)? = 0,
Here we find the exact solutions of eqs.(4.1), (2.7), (2.10) for the case in which the
external field is reduced to Coulomb potential
ze
A0 = ? . (4.2)
A = 0,
x
Simultaneonsly we obtain an approximate solution of Schr?dinger type equation (3.8),
o
(4.2).
To simplify eqs.(4.1), (2.7), (2.10), it is convenient to use the transformation
? > ? = U ?1 ?, L(?) > L (?) = U L(?)U, (4.3)
where U = exp[i((? · p)/m)]. As a result, using Campbell–Hausdorf formula and
taking into account relations (2.4), one obtains the following equivalent equation:
p2 ?ezx
e
L (?)? ? ? ?0 ? + ?·E E=
0
(4.4)
+ ?5 m ? = 0, .
x3
2m m
Let us use the notation
? = column (?1 , ?2 , ?3 , ?1 , ?2 ), (4.5)
where ?a are (2s + 1)-component functions, ?? are (2s + 1)-component ones. Substi-
tuting (4.5) into (4.4), one obtains using (2.7), (2.8), (2.10)
?
p2 ze2 g S · x
ze2 a
? ? (4.6)
p0 + ?1 , g= .
2sm x3
x 2m 2

According to (4.4), (2.7), (2.10), ?1 = ?2 = 0 and the functions ?2 , ?3 are
expressed via ?1 :
p2
1 1 m
?2 = ? ?1 , ?3 = ? ?0 ? ?2 (4.7)
?1 .
a 2m 2m a
On one- and two-particle Galilei-invariant wave equations for any spin 371

So eqs.(4.1), (4.2), (2.7), (2.10) reduce to eq.(4.6) for the (2s + 1)-component wave
function ?1 . It is demonstrated in [7, 9] that Schr?dinger-type equations (3.8), (4.2)
o
also may be reduced to eq.(4.6) (with g = ±s) by the consequent approximate
transformations of Foldy–Wouthuysen [13] type.
The solutions of eq.(4.6), which correspond to states with energy ?, can be written
in a form ?1 = exp[?i?t]?(x). Taking into account the symmetry of eq.(4.6) under
group O3 , it is convenient to represent ?(x) as a linear combination of spherical
spinors
?(x) = ?? (x)?s j??m ,
j
(4.8)
? = ?s, ?s + 1, . . . , ?s + 2nsj , nsj = min(s, j),
where ?s j??m = ?s j??m (x/x) are the eigenfunction of the operators J 2 , J 3 and
j j
? ? L + S) with eigenvalues j(j + 1), m, and (j ? ?)(j ? ? + 1).
L (J = x ? p + S
2

Substituting (4.8) into (4.6), one obtains the following equations for radial functions
?? (x):
D?? (x) = x?2 b?? ?? , (4.9)
where
d2
? 2d j(j + 1)
?
D = 2m ? + + 2+ ,
x2
x dx x dx (4.10)
g?
b?? = ? ? ?(2j + 1) ??? +
2 2
a?? , ? = ze ,
s
a?? are the matrix elements of operator S · x/x in basis {?s j??m }, determined by
j
the relation
S·x s
?i j??m = a?? ?s j?? m . (4.11)
j
x
The values of a?? for s = 1 are well known (see, e.g., [14]). These values for spin
2
are calculated in the appendix and are
1
a?? = ? (?? ? +1 a?+s + ?? ? ?1 a?+s+1 ),
2
(4.12)
1/2
µ(dj ? µ)(ds ? µ)(djs ? µ)
aµ = ,
(dsj ? 2µ ? 1)(dsj ? 2µ + 1)
where
ds = 2s + 1, dj = 2j + 1, dsj = ds + dj ,
(4.13)
µ = s + ? = 0, 1, 2, . . . , 2nsj , nsj = min(s, j).
The matrix b?? commutes with the operator D (4.10) and is diagonalizable, so
the system (4.9) can be reduced to the system of noncoupled equations
D? = x?2 bsj ?, (4.14)
where D is operator (4.9), bsj are the matrix b?? eigenvalues. Any equation (4.14)
in ones turn reduces to the well-known equation [15]
d2 y dy k2
z
+ ?? ? (4.15)
z + y = 0,
dz 2 dz 4 4z
372 W.I. Fushchych, A.G. Nikitin

where
v
v ?m 2
z = 2 ?2m?x, k 2 = d2 + 4bsj . (4.16)
y= z?, ?= ?, j
2?
The eq.(4.15) solutions for coupled states |? < 0| are expressed via Laguerre polino-
mials, and parameter ? takes the values [15]
k+1
(4.17)
?= +n, n = 0, 1, 2, . . . .
2
From (4.16), (4.17) one obtains
m?2
?=? (4.18)
2.
12 1
+ bsj + n +
j+ 2 2


Formula (4.18) gives the energy levels of a nonrelativistic spinning particle in
Coulomb field. Parameter bsj in (4.18) takes the values determined as the roots of the
matrix (4.10) characteristic equation
?g
det b?? ? bsj ??? ? det ?2 ? ?dj ? bsj ??? + (4.19)
a?? = 0,
s
where a?? are given in (4.12). The eq.(4.18) solutions and the analysis of spectrum
(4.18) are given in the next section.

5. Discussion of formula (4.18)
Formula (4.19) determines an algebraic equation of order 2nsj + 1. This equation
can be resolved in radicals only for s ? 3 or j ? 3 . To analyse the spectrum (4.18)
2 2
for arbitrary s and j it is convenient represent the eq.(4.19) solutions in such a form:
bsj = ?2 ? ?dj + (g?)2 bsj + o(g?)4 , (5.1)
where we suppose that ? 1. By using (4.12), (4.19), (5.1), it is not difficult to
obtain the explicit expressions for
a2 a2
1
bsj ?+s
? ?+s+1 (5.2)
=2 ,
j+1?? j??
?
8s
where aµ are the coefficients (4.12).
Using (5.2) and expanding the function (4.18) in powers of ?2 , one obtains

mg 2 ?4 bsj
m?2
?=? 2 + 2 6
?
1 +o ? ,
n n l+ 2 (5.3)
n = n + j ? ? + 1 = 1, 2, . . . , l = j ? ? = 0, 1, . . . , n ? 1.
Formula (5.3) determines the fine structure of the energy spectrum of any spin
particle in Coulomb field. The parameters bsj in (5.3) are easily calculated by formulae
?
(5.2), (4.12).
The first member on the r.h.s. of (5.3) gives the well-known Schr?dinger energy
o
levels of a nonrelativistic particle in Coulombic field. The second term gives the
correction of order ?4 which is connected with the existence of particle spin. As will
On one- and two-particle Galilei-invariant wave equations for any spin 373

be shown below, this correction corresponds to spin-orbit and Darwin coupling of a
particle with a field.
According to (5.2), (5.3) any energy level, corresponding to possible value of main
quantum number n, is splitted to n?1 sublevels, corresponding to possible values of l.
Besides any level with fixed n and l is additionally splitted into sublevels, the number
of which is 2nsj + 1, nsj = min(s, j). In contrary to the relativistic case the energy
levels of a nonrelativistic particle of spin 1 in Coulomb field are nondegenerated.
2
Consider the spectrum (5.2), (5.3) for s ? 1 and j ? 1. Using (4.12), one obtains
from (5.2)
1
1
b? = 2?d?1 ,
j
b0j = 0, ?=± ;
2
? = 0; j
?
2
2(1 ? ?2 ) ?1, j = 0,
dj + ?
b1j ?
=? , ?=
?1, 0, 1, j = 0;
2dj (dj ? ?) d2 ? 1
?
j
1 (5.4)
s1
b?2 = (?1)s+?+1 (2sds )?1 ,
? = ?s; ? = ?s, ?s + ;
bs0 = 0,
?
2
(s + ? ? 1)(ds + s + ? ? 1) (s + ? ? 1)2 ? 1
bs1 = + ,
2sds (ds ? s ? ? + 1)
?
2s2 (s + 1)
? = ?s, ?s + 1, ?s + 2.
For s = 0 formulae (5.3), (5.4) give the well-known energy spectrum of a spinless
nonrelativistic particle in Coulomb field. For s = 1 one obtains from (5.3), (5.4)
2

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