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?

?a pa ?

?

(1 ? ?0 )? = ? (4.16)

?0 ?.

2m

The system of equations (4.15), (4.16) is completely equivalent to (4.3), (4.12).

Thus we have obtained Galilean invariant equations (4.15), (4.16) for a particle

with arbitrary spin s, moreover, the wave function has 2(2s + 1) components. As in

Section 2, the subsidiary condition (4.16) is not the only one which can be added

to (4.15) in order to remove the redundant components of the wave function ?. For

instance, it is possible to postulate that the wave function ? satisfy instead of (4.16)

the following equation

? ? 1?

?

2 ?2

(k Sp )2 ? (k Sp )2 ? ?

p2 p2

1?

?? = ?, ? = ? H ? , H?

a a

? . (4.17)

? ? + +

? ?

2 2m 4m 2m 4m

+

On the non-relativistic motion equations in the Hamiltonian form 437

Equations (4.15), (4.17) are Galilean invariant. On the set of the solutions of these

equations the following representation of the algebra G is realized

? ? ?

Pa = pa = ?i Jab = xa pb ? xb pa + Sab ,

P 0 = p0 = i , ,

?t ?xa

Xa = xa + [U † , xa ]U,

Ga = tpa ? mXa , (4.18)

?1/2

?? ? ? ?? ? ?

?4 = 2?0 ? 1.

U = 1 + ?4 ? 2 + ?4 ? + ??4 ,

The generators (4.18), as (3.12), are Hermitian with respect to the usual scalar

product (3.2).

5. The non-relativistic particles in an external electromagnetic field

It is known from the relativistic equation theory that the equation of motions

which are mathematically equivalent in the case of a free particle, leads to different

physical consequences after the introduction of an interaction. It means that various

mathematically equivalent representations for the equations are physically non-equi-

valent. The classical example of such a situation is the equation for an electron in

the Dirac and in the Foldy–Wouthuysen (FW) [3] representations. If one introduces

the minimal interaction into the free equation in the Dirac representation, the result

is obtained which is in a good accordance with experimental data. If, however, one

introduces the interaction into the free equation in the FW representation, any sensi-

ble result will not be obtained. Another example is the Kemmer–Duffin equation

which does not lead to the spin-orbit and to the Darwin couplings by introducing

the minimal interaction into the original free equation, but describes these couplings

if one introduces the interaction into the mathematically equivalent equation in the

Hamiltonian form [9].

It turns out that the analogous situations takes place also for the non-relativistic

equations. We shall see, that equations (2.12), (4.15) in contrast to (2.1) and (4.3),

(4.13), lead to the spin-orbit and to the Darwin couplings.

First we consider equation (2.12). After the replacement pµ > ?µ = pµ ? eAµ one

obtains

(?a ?a )2

?

i ? = H(?)? = (1 ? ?0 ) ? ?a ?a ? (1 ? ?0 )2m + eA0 ?. (5.1)

?t 2m

In order to obtain from (5.1) the equation for the 2(2s + 1) component wave

function it is necessary to remove the “odd” terms ?a ?a in (5.1), i.e. to diagonalize

the operator H(?). In the presence of the interaction such a problem may be solved

only approximately as in the relativistic case [3]. We shall solve this problem up to

terms of order 1/m2 with the help of a set of successive unitary transformations.

After the first transformation

?a ?a

U1 = exp ??4 ?4 = 2?0 ? 1 (5.2)

,

2m

one obtains

(?a ?a )2

†

H(?) > H (?) = ? m(1 ? ?4 ) + eA0 ?

(1)

U1 H(?)U1

= (1 + ?4 )

4m

(5.3)

ie ie 1 1

? ?4 (?a Ea ) ? (?a ?a )3 + O

[?a ?a , ?b Eb ]? + ,

8m2 12m2 m3

2m

438 W.I. Fushchych, A.G. Nikitin, V.A. Salogub

?A0 ?Aa

Ea = ? ? ,

?xa ?t

where O(1/m3 ) possesses the terms of a power 1/m3 .

After the second transformation

ie

U2 = exp ? (5.4)

(?a Ea )

4m2

the operator H (2) (?) becomes

(?a ?a )2

†

? (1 + ?4 ) ? m(1 ? ?4 ) + eA0 +

(2) (1)

H (?) = U2 H (?)U2

4m

(5.5)

1 ie 1

(?a Ea )3 ?

+ [? ? , ?b Eb ]? + O .

2 aa

2 m3

12m 8m

At least, with the help of the operator

1

? (? ? )3 , (5.6)

U3 = exp 34 aa

24m

one obtains the final form of H (3) (?):

(?a ?a )2

†

?

H (3) (?) = U3 H (2) (?)U3 = (1 + ?4 )

4m

(5.7)

ie 1

?(1 ? ?4 )m + eA0 ? [?a ?a , ?b Eb ]? + O .

8m2 m3

It follows from (2.2) that the Hamiltonian H (3) (?) is a completely even (commuti-

ng with ?4 ) operator. On the set of ?+ , satisfying the condition

1

(1 + ?)?+ = ?+ , (5.8)

2

the Hamiltonian H (2) (?) has the form

2

?a SH

?e + eA0 ?

(3) +

H (?)? =

2m 2ms

ie e (5.9)

?S · (E ? ? ? ? ? E) ? div E ?+ ,

2s 2

8m 8m

i

Ha = ? ?abc [?b , ?c ]? .

2

Thus, starting from the non-relativistic equation (5.1), we have obtained the

SH

approximate Hamiltonian (5.9) which describes not only the dipole ?e , but

2ms

ie e

also the spin-orbit ? 2 S(? ? E ? E ? ?) and the Darwin ? 2 div E inte-

8m s 8m

raction of a charged particle with an external electromagnetic field. For the spin

On the non-relativistic motion equations in the Hamiltonian form 439

s = 1/2 particle the Hamiltonian (5.9) coincides with the one, obtained by Foldy and

Wouthuysen [3] from the relativistic Dirac equation.

Now we appeal to equation (4.15) and introduce to it the minimal interaction

pµ > ?µ . It leads to the Hamiltonian

?

2 2

?a ?0 (?a ?a ) ? ?0 k 2 S H + ?a ?a + (1 ? ?0 )2m + eA0 .

? ? ?

? ?? (5.10)

H(?) =

2m 2m 4m

This Hamiltonian, as well as (5.1), cannot be diagonalized exactly. By the analogy wi-

th (5.2)–(5.7), one can diagonalize (5.10) approximately with the help of the operator

?

U = exp(iB3 ) · exp(iB2 ) · exp(iB1 ), (5.11)

where

?

?4 ?a ?a , ? ?

B1 = ?? ?4 = 2?0 ? 1, (5.12)

2m

?

?a Ea

B2 = ?e (5.13)

,

4m2

1 i? ? ?

? ?4 (?a ?a )3 ? [?a ?a , ?a ]? ?

2

B3 =

8m3 k

(5.14)

ek 2 ? ek 2 ? ?

? [?a ?a , Sb Hb ]? ? [(?a ?a ), ?4 (Sb Hb )]? .

4 4

As a result one obtains

2 2 2

?a 1 ? (k S?) + k (1 + ?4 ) eS H ?

?

? (1 ? ?4 )

H (3) (?) =

2m 2 2m 2 4m

(5.15)

ie 2 1

?

?(1 ? ?4 )m + eA0 ? k [S?, S E]? + O .

8m2 m3

?

On the set of ?+ = ?4 ?+ this Hamiltonian takes the form

2

ek 2

?a 2 SH

?k e + eA0 ? i 2 [S?, S E]?

(3) +

?+ . (5.16)

H (?)? =

2m 4m 8m

Using the identity

i ?Ea

[S?, S E]? ? ? (3[Sa , Sb ]+ ? 2?ab s(s + 1)) ?

6 ?xb

(5.17)

ie i

? s(s + 1) div E ? S(E ? ? ? ? ? E),

3 2

one can rewrite equation (5.16) in the form

2

k2

?a 2 SH ?Ea

? ek + eA0 ? Qab ?

(3) +

H (?)? =

2m 4m 24 ?xb

(5.18)

2

e ek

? k 2 s(s + 1) div E ? S(E ? ? ? ? ? E) ?+ ,

2 2

24m 16m

440 W.I. Fushchych, A.G. Nikitin, V.A. Salogub

where

e

{3[Sa , Sb ]+ ? 2?ab s(s + 1)} (5.19)

Qab =

2m2

is the tensor of the quadrupole interaction.

Thus the non-relativistic equation without redundant components (4.15) allows us

to obtain the description of a motion of the spin s particle in an external electromag-

netic field. Such a description is in a good qualitative accordance with experimental

data. For s = 1/2 (5.18) coincides with the FW Hamiltonian if one puts an arbitrary

constant k = 1/s.

6. Conclusion

So we have demonstrated that the non-relativistic Hamiltonian equations (2.12),

(4.15) give the consistent description of a charged particle of any spin in external

fields. Thus we have shown that the spin-orbit, the Darwin and the electric quadrupole

interactions are not the truly relativistic effects but may be considered within the

framework of the non-relativistic mechanics.

It is interesting to compare the obtained results with the ones predicted by the

relativistic theory. One can make sure that there is not only the qualitative but

also the quantitative accordance between them. We have demonstrated this fact for

the case s = 1/2. If one puts into (5.18) k = ±2, the resulting equation completely

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