ñòð. 68 |

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W.I. Fushchych, Scientific Works 2000, Vol. 1, 301–305.

On the equations of motion for particles with

arbitrary spin in nonrelativistic mechanics

W.I. FUSHCHYCH, A.G. NIKITIN, V.A. SALOGUB

It is well known that the electron motion in the external electromagnetic field is

described by the relativistic Dirac equation. In this case, in the Foldy–Wouthuysen

representation, the Hamiltonian includes the terms corresponding to the interaction

of the particle magnetic moment with a magnetic field (? (1/m)(?H)) and the terms

which are interpreted as a spin-orbit coupling (? (?/m2 ){(p ? eA) ? E)). Apart from

these constituents the Hamiltonian includes the Darwin term (? (1/m2 ) div E) [1].

Such a description is in good accordance with the experimental data.

It was shown by Bargman [2] that it is possible to introduce the particle spin

in the nonrelativistic quantum mechanics by performing the central extension of the

Galilei group. In connection with this Bargman result the problem of finding the

motion equations, which are invariant with respect to the extended Galilei group G,

arises naturally.

Such a problem has been considered in [3–5]. The equations obtained in [5] have

redundant components and, besides, these equations do not describe the spin-orbit and

the Darwin couplings if one makes the replacement pµ > ?µ = pµ ? eAµ in them.

The aim of this note is to find such motion equations for a particle with spin which

are invariant relative to the group G, have no redundant components and describe the

spin-orbit and the Darwin couplings of the particle with the field. It is reached by the

supposition that the free nonrelativistic particle Hamiltonian has two energy signs

just as the Dirac Hamiltonian. This is equivalent to the requirement that the theory

(equations) be invariant under such a transformation:

t > ?t, (1)

T ?(t, x) = ? T ?(?t, x),

where ? is some unitary matrix. In terms of group-theoretical language this means

some extension of the group G to the group G ? G, which includes the transforma-

tion (1).

In order to find the motion equations in the Schr?dinger form

o

?

? = Hs ?(t, x), (2)

i

?t

which are invariant under the group G and the transformation (1), we have used the

method of the work [6], where the same problem has been solved for the full Poincar?

e

group. Equations (2) with an unknown operator function Hs will be invariant under

the group G if the following relations are satisfied:

[Hs , Pa ] = [Hs , Jab ] = 0, [Hs , Ga ] = iPa , [Ga , Gb ] = 0,

[Jab , Ga ] = i?ac Gb ? i?bc Ga ,

[Pa , Gb ] = i??ab m,

(3)

[Jab , Jcd ] = i(?ac Jbd + ?bd Jac ? ?ad Jbc ? ?bc Jad ),

[?, Pa ] = [?, Jab ] = [?, Ga ] = 0,

Lettere al Nuovo Cimento, 1975, 14, ¹ 13, P. 483–488.

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

P0

[T, Jab ] = {T, ?} = {T, Ga } = [T, Pa ] = 0, ?= ,

|P0 |

where Pa , P0 = Hs , Jab and Ga are the generators of the Galilei group, T is the

operator of time reflection.

To determine all possible (up to unitary transformations independent of the particle

momentum) operators Hs , which satisfy relations (3), we use the following realization

of the generators Pa and Jab :

?

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

,

?xa

(4)

sab 0

Sab = , a, b = 1, 2, 3,

0 sab

where sab are the generators of the irreducible representation D(s) of the O3 group.

We require the Hamiltonian Hs to be the differential and Hermitian operator with

respect to the usual scalar product

d3 x ?† (t, x)?2 (t, x), (5)

(?1 , ?2 ) = 1

where ? is the 2(2s + 1)-component function, which satisfies eq. (1).

Expanding the operator Hs in a complete system of the ortoprojectors

s

Sp ? r S12 p3 + S31 p2 + S23 p1

(6)

?r = , Sp = ,

r ?r |p|

r = ?s

r =r

and using the results of the work [6], we have obtained that the Hamiltonians Hs

which satisfy conditions (3) are represented by the formulae

v Sp

p2 p2 p2

H 1 = ?1 ? sin2 ? 1 (7)

m+ + ?2 sin 2? 1 + ?3 2 sin ? 1 ,

2m m 2m s

2 2 2 2

v Sp

p2 (Sp)2 (Sp)2

H1 = ?1 m + ?2 sin2 ?1 + ?2 (8)

sin 2?1 + ?3 2 sin ?1 ,

2ms2

2m sm s

p2 (Sp)2

H 3 = ?1 ?2 sin2 ? 2 +

m+ 3

2m sm

2

p2 1 1

? sin 2? 3 ? sin ? 2 9 ? sin2 ? 2 + (9)

+?2 3 3

2m 8 4

2

v Sp

(Sp)2 9 1

sin 2? 3 ? sin ? 2 9 ? sin ?

2

+ + ?3 2 sin ? 2 ,

3 3 3

2ms2 8 4 s

2 2

p2 3

Hs = ?1 (10)

m+ , s> ,

2m 2

where ?s are arbitrary parameters, ?a are the 2(2s + 1)-dimensional Pauli matrices.

The Hamiltonians (7)–(10) are the square roots of the operators

p4

H2 = m2 + p2 + (11)

.

4m2

On the equations of motion for particles with arbitrary spin 303

The operator H 1 is the nonrelativistic analogue of the Dirac Hamiltonian for an

2

electron. As will be shown in what follows, the parameters cos 2?s must be interpreted

as the anomalous magnetic moment of the particle.

If ? = ?/4, eq. (2) with the Hamiltonian (7) may be written in the compact form

p4

µ

(12)

[m + ?µ p ]? = i?4 ?,

2m

where ?µ are the 4 ? 4 dimensional Dirac matrices

Sa

?0 = ?1 , ?a = i?2 , ?4 = ?3 .

s

The operators H1 and H 3 are the nonrelativistic Hamiltonians for the particles

2

with spins s = 1 and s = 3 respectively.

2

The Hamiltonians (7)–(9) are not diagonal. They may be diagonalized by the

unitary transformation

Hs > Hs = UHs U † , (13)

where

E + ?1 Hs p2

U= p2 + m2 +

?r , E= .

4m2

p2 2r 2

1? sin2 ?s

r 2E E + m + s2

2m

For the case s > 3 the only trivial Hamiltonians (10) which result in no spin

2

effects are possible in our statement of the problem.

The description of the behaviour of the nonrelativistic particle with spin in the

external electromagnetic field is made by the replacement pµ > ?µ in eq. (1). In

order to preserve the explicit Hermiticity of the Hamiltonians it is necessary to

symmetrize previously the formulae (7)–(9) using the identity

1

(14)

pa p b = (pa pb + pb pa ).

2

After such a symmetrization and the replacement pµ > ?µ in (7)–(9) we obtain

?2 (S?)2 e(SH)

Hs = ?1 m + ? sin2 ?s + sin2 ?s + e?+

2 2

2m ms 2ms

(15)

v S?

?2 (S?)2 e(SH)

?

+?2 as + bs bs + ?3 2 sin ?s ,

2ms2 4ms2

2m s

where Ha = i?abc [?a , ?c ] is the magnitude of the magnetic field,

a 1 = sin 2? 1 , b 1 = 0, a1 = 0, ba = sin 2?1 ,

2 2 2

1 3 1

a 3 = ? sin 2? 3 ? sin ? 3 1? sin2 ? 2 ,

3

8 4 9

2 2 2

9 3 1

sin 2? 3 ? sin ? 3 1? sin2 ? 2 .

b3 = 3

8 4 9

2 2 2

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

For s ? 3 it is impossible to diagonalize the Hamiltonians Hs for the interacting

2

particles exactly. The diagonalization of the Hamiltonians (15) to terms of the power

1/m2 is made by the unitary operator

U = exp [i(B3 )] exp [i(B2 )] exp [i(B1 )] ,

s s s

where

v

2

B1 = ?

s

sin ?s ?2 (S?),

2ms

v

(S?)2

1 ebs (SH) e 2 sin ?s

?

s 2

B2 = ?3 as ? + bs + SE ,

4m2 s2 2s2 s

v v (16)

2 sin3 ?s

1 2 sin ?s

{S?, ? } + {S?, SH}?

s 2

B3 = ?2

8m3 s3

s

v

4 2 sin3 ?s iebs

? (S?)3 ? ieas [? 2 , ?] ? 2 [(S?)2 , ?] .

s3 s

After the transformation (16) the Hamiltonian (15) takes the form

v

?2 e(SH) e 2 sin ?s bs

Hs = ?1 m + ?as + 2 ?

2

+ sin ?s + e? +

2 2s

2m 2ms 4m 4s

ñòð. 68 |