ñòð. 116 |

Here

n2 d

? ? = K 2 ? S 2 ? 2S3 + nS2 + + 2iS2 (1 ? w2 )1/2 ?

2

4 dw

(3.5)

1 n n

?2S1 + S3 ?

L3 + + S3 w ,

(1 ? w2 )1/2 2 2

where

d2 d

K 2 = (1 ? w2 ) ? 2w ?

dw2 dw

[L3 + (n/2 + S3 ) ? (n/2 + S3 )w]2 2

n (3.6)

? ? + S3 ,

1?w 2 2

w = cos ?.

The solutions of eq.(3.4) can be represented as an expansion in Jacobi polino-

mials [14] and eigenfunctions of the operator L3

F? (r)Pn/2+j,n/2+? (w) exp[?i(j ? ?)?],

k

(3.7)

?=

?

k

where Pn/2+j,n/2+? is the complete set of the normalized eigenfunctions of the

commuting operators J3 = L3 + S3 , S3 and K 2 (3.6) which correspond to the ei-

genvalues j, ? and ?k(k + 1), respectively, moreover

k? and k ?

n n n

are integers,

+? , +j +?

2 2 2

(3.8)

? = ?nsk , ?nsk + 1, . . . , nsk , nsk = min(s, k).

Using recurrent relations [14]

? ?µw

d

(1 ? w2 )1/2 k

+ P? µ (w) =

dw (1 ? w2 )1/2 (3.9a)

= [(k + µ )(k ? µ + 1)]1/2 P? µ ?1 (w),

k

512 W.I. Fushchych, A.G. Nikitin, W.M. Susloparow

? ?µw

d

(1 ? w2 )1/2 ? k

P? µ (w) =

dw (1 ? w2 )1/2 (3.9b)

= ?[(k ? µ )(k + µ + 1)]1/2 P? µ +1 (w),

k

and formulae for the matrices S1 and S2 in Gel’fand–Zeytlin basis [21], we come to

the following equations for radial function F? (w):

DF? (r) = r?2 A?? F? (r), (3.10)

where

k(k + 1) ? n2 /4

d2 2d

D = (? ? m ) + 2 + ?

2 2

(3.11)

,

r2

dr r dr

1 ? 2s

A?? = s(s + 1) ? 2? 2 + n? ??? ? ??? ,

2s

? = ? ± 1, (3.12)

??? = 0,

1/2

n n

= ? (s + ?)(s ? ? + 1) k + + ? k? ??+1

????1 = ???1? .

2 2

Since the matrix A?? is diagonalizable, system (3.10) can be reduced to the

system of noncoupled equations

DF? (r) = r?2 B? F? (r),

? sk ? ? (3.13)

F? = u?? F? ,

sk

where B? are the matrix A?? eigenvalues, which coincide with the roots of the

characteristic equation

det A?? ? B? ???

sk

(3.14)

= 0,

u?? is the operator diagonalizing the matrix A?? .

Each of eqs.(3.13) by the replacement of the variable ? = (?2 ? m2 )1/2 r reduces to

the wen-known one [14]

? ? k(k + 1) ? n2 /4 + B? ?

d2 F sk

2 dF

+ 1? (3.15)

+ F = 0,

d?2 ?2

? d?

the solution of which (limited at the point ? = 0) is expressed via Âåssål’s function

1

F = v Jv(k+n/2+1/2)(k?n/2+1/2)+B sk (?),

? (3.16)

? ?

where k satisfies of the conditions (3.8).

One can make sure by the direct verification that at least for s < 3 , (k + n/2 +

2

1/2)(k ? n/2 + 1/2) + B? > 0. This means that ? > m, and so particle with spin

sk

s < 3 in magnetic-pole field has continuous energy spectrum and has not coupled

2

states. In the ñàses s = 0 and s = 1 the absence of coupled states was demonstrated

2

by Dirac [12] and Harish-Chandra [14].

Relativistic particle of arbitrary spin 513

According to the above the explicit solution of the wave eq.(2.4) for the case in

which the external field source is a magnetic monopole has the form

N

?s (t, r) = exp[?i?t] exp[i(S2 cos ? ? S1 sin ?)?]?

(?2 ? m2 )1/2 r (3.17)

(?)Jv(k+n/2+1/2)(k?n/2+1/2)+B sk (

?)?]u?1 Pn/2+j,n/2+?

? exp[?i(j ? ?2 ? m2 · r),

k

??

?

where N is the normalization constant. Solutions of the starting system (2.1) may be

expressed through the function (3.17) with the help of the relations (2.3c).

Let us give the explicit expressions for B? and usk , if s ? 1:

sk

??

1/2

1 n1 n1

1

2k

=± k? +

B k+ + ,

±1 4 2 2 2 2

2

?p/3 cos[(? + ??)/3],

1k

B? = 2

q 3 1

p = ?(2k + 1)2 + n2 ? ,

cos ? = ,

?(p/3)2 4 3

2

8 16

q = ? k(k + 1) ? , ? = 0, ±1;

3 27

?c1 1

1 c1

k

?, ? = ± ;

2

u?? = ,

c2 c2 2

? ?

p1 p2

? ?1 ?

1k 1 1k ? n/2 1

? ?

n/2 + B1 B1

? ?

? ?

p1 p2

=? ?, ?, ? = 0, ±1;

? ?2 ?2

u1k ? ?

1k 2 B0 ? n/2

1k

?? n/2 + B0

? ?

? ?

p1 p2

? ?3 ?3

1k 3 B?1 ? n/2

1k

n/2 + B?1

1/2 1/2

n n1 n n1

p1 = ? 2 k ? p2 = ? 2 k + k? +

k+ + , .

2 2 2 2 2 2

Íere c1 , c2 , ?1 , ?2 , ?3 are arbitrary nonzero constants.

4. Arbitrary spin particle in the Coulomb field

In the case of Coulomb potential A0 = Ze/r, A = 0, eq.(2.4) in spherical co-

ordinates takes the following form:

2

1 ? 2? 1 ? i?

? m2 ? = ? (S · r)?, (4.1)

r ? + 2 ?? + ?+

r2 ?r ?r sr3

r r

where ? = Ze2 , ? is an angular part of Laplace operator.

Equation (4.l), as eq.(3.2), has exact solutions in separated variables. In [9] eq.(4.1)

is solved by using of the spherical spinor basis. Here we shall obtain the expressions

of eq.(4.1) solutions through Jacobi polinomials which are more convenient basis in

more general case of combination of the Coulomb and magnetic-pole potentials.

514 W.I. Fushchych, A.G. Nikitin, W.M. Susloparow

In such a way, as was done in previous section, we shall pass to the representation,

in which the matrix S · r/r3 is diagonal. Using for this purpose the transformation

operator (3.3), we obtain

2

1 ? 2? 1 ? i?

? + 2? ? ? + ? m2 ? = ? (4.2)

r ?+ S3 ?,

n=0

2 ?r sr2

r ?r r r

where ? ? is the operator (3.5) with n = 0.

n=0

Representing the solutions of eq.(4.2) in the form (3.7) one comes to eq.(3.10) for

the radial wave function, where

d2

2

? 2d k(k + 1)

? m2 + ? (4.3)

D = ?+ + ,

dr2 r2

r r dr

i?

A? = s(s + 1) ? 2? 2 ? ? ??? ? ?? , (4.4)

?? ??

s

?? is the matrix (3.12) corresponding to n = 0.

??

The matrix A? is diagonalizable, so the system of equations (3.10), (4.3), (4.4) is

??

sk

equivalent to noncoupled eqs.(3.13), (4.3), where B? are the roots of the matrix (4.4)

sk

characteristic equation (for explicit expressions for the coefficients B? see [9]). Each

of these equations in its turn is reduced to the well-known equation of the for [22]

d2 y dy l2

z

+ ?? ? (4.5)

ñòð. 116 |