ñòð. 87 |

?=? ?=± . (5.5)

+3 ,

n l+ 1 l+ 1 +?

n2 2

2 2

It is interesting to compare (5.5) with the fine-structure formula for a Dirac electron

interacting with Coulomb field. One can make sure itself that for g 2 = ?1 the energy

levels (5.5) may be represented as

p4

? = ?D ? (5.6)

,

8m2

where ?D gives the energy levels of a Dirac electron in Coulombic field, and average

is taken in Schr?dinger wave functions.

o

According to (5.6), formula (5.5) takes into account all “relativistic” corrections

predicted by Dirac equation, except the relativistic kinetic-energy correction p4 /8m2 .

It means that formula (5.5) takes into account the contributions of spin-orbit and

Darwin couplings, and so these couplings may be described in frame of Galilei-

invariant theory.

Let us give for the completeness the exact solutions of eqs.(4.19) for s ? 1 and

j?1

11

1

±

b0j = 0, b2j = d2 + 4(g?)2 ,

j

42

v

c 1 ?

1

= + 2 ?c cos ? = 0, ±1, j = 0,

b1 2 ?+? , (5.7)

3 3 2

2

12 1 g?

1

(ds ? 3) ±

bs0 = 0, bs 2 = d2 + ,

s

4 2 s

374 W.I. Fushchych, A.G. Nikitin

v

d 1 ?

bs1 = s(s + 1) ? 2 + + 2 ?d cos µ = 0, ±1, s = 0,

?+µ ,

3 3 2

where

b 2 1 1 4

cos ? = v (g?)2 + d2 ? , a = ?(g?)2 ? ? b,

, b=

3 j 27

?c3 3 27

2 2

f 2 g? 1 1 g? 4

cos ? = v + d2 ? , d=? ? ? f.

, f=

3 s 27

?d3 3 s s 27

Contrary to the approximate formulae (5.3), (5.4) relations (4.18), (5.7) give the

exact values of the energy levels predicted by eqs.(4.1), (4.2). Using (4.18), (4.19) it

is not difficult to obtain the exact spectrum also for s = 3 and any j, and for j = 3

2 2

and any s. We do not give the corresponding cumber-some formulae here.

6. Two-particle equations

The Breit equation [12] is an important and often used one in the quantum-

mechanical two-body problem. Besides a lot of incontrovertible merits, this equation

has the shortcoming of principle — it is not invariant either under Poincar? or under

e

Galilei group. So the Breit equation does not satisfy any relativity principle accepted

in physics.

In this section we find two-particle wave equation, which describes the system of

electrically charged spin- 1 particles with the same accuracy as the Breit equation,

2

but is Galilei invariant.

Starting from one-particle Schr?dinger-like GIWE (3.1), one may write the equa-

o

tion for a system of two noninteracting particles in such a form:

?

? t, x(1) , x(2) = Hs(1) + Hs(2) ? t, x(1) , x(2) , (6.1)

i

?t

where ? t, x(1) , x(2) is the 2(2s(1) + 1) ? 2(2s(2) + 1)-component wave function,

Hs(1) and Hs(2) are the Hamiltonians of the first and second particle — i.e. differential

operators of form (3.6). Here and below we use the indices (1) and (2) to distinquish

the quantitics related to the first and second particle.

Equation (6.1) is manifestly invariant under Galilei group. The Galilei group

generators on the set of eq.(6.1) solutions are represented as a direct sum of single-

particle generators (2.6), (3.3).

It is convenient to arrive at (6.1) from individual variables x(1) and x(2) to c.m.

ones. Previously we transform eq.(6.1) to such a representation, in which the internal

energy operator

P2

?

C1 = Hs(1) + Hs(2)

2(m(1) + m(2) )

does not depend on total momentum P = p(1) + p(2) . Using for this purpose the

transformation operator (compare (4.3))

i? · P

? = ?(1) + ?(2) , (6.2)

U = exp , M = m(1) + m(2) ,

M

On one- and two-particle Galilei-invariant wave equations for any spin 375

(?) (?)

where ?(?) = ?1 ? i?2 S (?) , ? = 1, 2, one obtains from (6.1) the following

equivalent equation:

? = U HU ?1 = P + E,

2

? ? (6.3)

i ? = H?, ? = U ?, H

?t 2M

where E is the internal energy operator

(1) (2) (1) (2)

E = ?1 m(1) + ?1 m(2) + 2?3 S (1) · p ? 2?3 S (2) · p+

p2

1 1 1

(1) (1) (2) (2)

? ?1 ? i?2 ? ?1 ? i?2

+ , (6.4)

µ m(1) m(2) 2

m(1) p(1) ? m(2) p(2)

m(1) m(2)

p=

µ= , .

M M

So to describe the system of free nonrelativistic particles of spins s(1) and s(2)

one may use the motion equation (6.1) in single particle variables, or eq.(6.3) in c.m.

ones. The Galilei group generators on the sets of this equation solutions have the

form

P = p(1) + p(2) ,

P0 = Hs(1) + Hs(2) ,

J = x(1) ? p(1) + x(2) ? p(2) + S (1) + S (2) , (6.5)

G = tP ? m(1) x(1) ? m(2) x(2) + ?

for eq.(6.1), and

? +E+ P ,

2

?

P = ?i

1

P0 =H ,

2m ?X

(6.6)

J = X + P + S, S = x ? p + S (1) + S (2) ,

G = tP ? M X, X = m(1) x(1) + m(2) x(2) M 1 , x = x(1) ? x(2) ,

for eq.(6.3).

Starting from (6.1) or (6.3) one may look for motion equation for interacting

particles in such a form:

?

(6.7)

i ? = [Hs(1) + Hs(2) + V ]?,

?t

or

? ? (6.8)

i ? = [H + V ]?,

?t

?

where V and V are interaction Hamiltonians. The requirement of Galilei invariance

?

reduces to commutation of operators V and V with generators (6.5) and (6.6). It

means that the operators can depend upon internal variables x and p only and be

scalars under spatial rotations. Besides that V must satisfy the condition

[V, ?] = 0. (6.9)

376 W.I. Fushchych, A.G. Nikitin

It follows from (6.3), (6.9) that eq.(6.7) may be reduced to the from (6.8) using

the transformation operator (6.2). So formula (6.8) gives a wider class of GIWE, as

soon as eq.(6.8) in general is not reducible to the form (6.7).

So the condition of Galilei invariance give a wide choice of interaction Hamilto-

nians. Consider some examples which are interesting from the physical point of view.

?

1) Central potential V = If (x), where I is the unit matrix. The corresponding

equation (6.3) in the c.m. frame takes the form

?

?(t, x) = H(x, p)?(t, x), (6.10)

i

?t

where H(x, p) = E + If (x), E is operator (6.4).

To analyse eq.(6.10) we suppose the momentum p to be small enough: p2 m2 .

Applying to (6.10) the standard approximate diagonalization procedure of Barker–

Glover–Chraplivy (BGC) [16], one comes to Hamiltonian

p2

(1) (2)

H= ?1 m(1) + ?1 m(2) + + If (x)+

2µ

(6.11)

(1) S ·x?p (2) S · x ? p ?f

(1) (2)

1

+ ?1 + ?1 .

x 2m(1) 2m(2) ?x

?

In spite of the spin independence of V , the approximate Hamiltonian (6.11) con-

tains terms, which correspond to-spin-orbit coupling.

2) Breit potential

? ?

S (1) · x S (2) · x

2 2

e (1) (2) e ? (1) ?.

VB = ? S · S (2) + (6.12)

+ 2?3 ?3

x2

x x

? ?

The BGC reduction for Hamiltonian HB = E + VB , where E and V are given in (6.4),

(6.12), leads to the following result:

P2

H> + H int , (6.13)

2M

where

(1) (2)

p2 e2 e2 ?1 ?1 1 1

(1) (2)

? ? p · p + p · x 2x · p +

int

H = ?1 m(1) + ?1 m(2) +

2µ x 2µM x x

(1) (2)

e2 ?1 ?1 1 1

S (1) + S (2) + S (1) + 2S · x ? p?

(2)

+3 2

x 2m(1) m(2) 2m(1) 2m(2)

(1) (2)

3S (1) · xS (2) · x

e2 ?1 ?1

? S ·S ?

(1) (2)

+

2m(1) m(2) x3 x2

(1) (2)

2 ?1 ?1 1 1

S (1) · S (2) +

2

+4?e + ?(x).

8m2 8m2

3 m(1) m(2) (1) (2)

On one- and two-particle Galilei-invariant wave equations for any spin 377

(1) (2)

On the set of functions, satisfying ?1 ? = ?1 ? = ?, Hamiltonian (6.14) may be

represented as

p4 1 1

H int = HB + 3 + m3 ,

8 m(1) (2)

where HB is the approximate Breit Hamiltonian in the c.m. frame [16]. So GIWE

(6.8) with potential (6.12) leads in approximation 1/m2 to the results, which are

analogous to the ones predicted by Breit equation, but do not take into account the

ñòð. 87 |