fulltext

Pauli equation for fermion-dyon dynamics

P. C. PANT, V. P. PANDEYand B. S. RAJPUT

Department of Physics, Kumaun University - Nainital 263001, India (ricevuto il 25 Luglio 1997; approvato il 16 Settembre 1997)

Summary. — By investigating the behaviour of a fermion moving in the field of a

dyon, the generalized Dirac equation has been solved for energy eigenvalues and the Hamiltonian of this system has been shown to involve spin contribution. The generalized spin momentum of the fermion has been shown to behave as extra energy source for fermion interacting with generalized electromagnetic field. The interaction of spin and orbital momenta of fermion moving in the generalized electromagnetic field has been analyzed. By introducing suitable spinors the Pauli equation for fermion moving in the field of dyon has been analysed and it is shown that the ad hoc introduction of spin in this system perceptibly modifies the energy eigenvalue and eigenfunction of the bound state of the system.

PACS 11.10.Cd – Axiomatic approach.

PACS 03.65.Ge – Solutions of wave equations and bound states.

1. – Introduction

By keeping in view the current potential importance of monopoles [1] and dyons [2], a self-consistent quantum field theory of generalized electromagnetic field associated with dyons has been constructed [3-5] using two four-potentials to avoid the use of arbitrary string variables and assuming the generalized charge, generalized four-current and generalized four-potential associated with dyons as complex quantities with their real and imaginary parts as electric and magnetic constituents. Extending this work in the present paper, we have undertaken the study of the behaviour of a fermion moving in the generalized electromagnetic field of a dyon and derived the energy eigenvalues by solving the corresponding Dirac equation. It has been shown that the interaction of spin and the generalized potential leads to an extra energy which is expressible in terms of generalized spin moment of the particle con-cerned [6]. Analysing the gauge and relativistic invariance properties of generalized Dirac equation, the problem of interaction of spin and orbital angular momenta of this system has been investigated and the expression for Hamiltonian has been derived. We have also undertaken the study of Pauli equation for a fermion moving in the field of a dyon by introducing suitable spinors and it has been shown that bound-state energy eigenvalues perceptibly change from the usual bound-state energy eigenvalues and eigenfunctions of dyon-fermion bound states.

2. – Behaviour of a fermion in the field of a dyon

In order to investigate the behaviour of a fermion in the generalised electromagnetic field of a dyon we start with the following free-particle Dirac equation in Feynman notation

(PC_{1 x) c(x) 4 0 ,} (2.1)

where x 4mcOˇ. The corresponding Hamiltonian in usual notation is given as

H 4ca× Q p× 1bmc2_.

(2.2)

m is the effective rest mass of dyon. The plane-wave solution for bispinor c (x) may

be written as follows: c(x) 4

u

j × h ×

v

4 (P m , sm) exp

k

₂ i hP m xm

l

, (2.3)

where j× and h× are two-component functions. In order to derive the wave equation of a fermion moving in the field of an electromagnetic dyon, let us consider the four-potential of external field as ]Vm

( 4 ]Am( 2 ]Bm(, where ]Am( and ]Bm( are electric and magnetic four-potentials, respectively. The wave equation in this case may be obtained by using the following transformations for energy and momentum operators in free Dirac’s equation:

P×_{K P}×_{2 K V}K, _{e Ke2KV *}0

(2.4)

in units of ˇ 4C41 with K a constant defined by

K 4eigi, (2.5)

or

Pm_{K (P}m_{2 K V}m *) . (2.5a)

Thus we get the following equations for the two-component spinors j× and h×: [s_{× Q ]P}×_{2 K V}K_{* (] h× 4 [e2KV}0* 2m] j× ,

(2.6)

[s_{× Q ]P}×_{2 K V}K_{* (] j}×_{4 [e 2 K V}0* 1m] h× .

(2.7)

Restricting ourselves to the case of non-relativistic motion in a weak field and considering only positive energy solutions by using

e 4E1m; NE 2 K V0* Nbm ,

we get the following energy eigenvalue equation in terms of spinor j× : [ 1 O2m]P×2 K VK* (2 1 K V *0 2 NKNO2 m(s× Q g×) ] j×4 Ej× , (2.8) where g 4¯×x](K/NKN) VK* ( 4¯×x(e2iu_VK_{* )} (2.9)

Equation (2.8) is analogous to Pauli equation for an electron moving in an electromagnetic field. It has the following extra contribution in energy gained by fermion, while moving in generalized electromagnetic field due to its spin

E 842NKN/2m(s× Qg×) .

(2.10)

This equation may also be written as

E 842mDg× 42u 8D(s× Q g×) ,

(2.11) where

m 8D4 NKNO2 m

is defined as the magnetic moment associated with a system of a dyon and fermion, and

m

×_{D4 m 8Ds}×

as generalized spin moment of this system. Consequently the extra energy term in the Hamiltonian may be interpreted as the energy of interaction of generalized spin moment of the fermion with the vector field resulting from the space rotation of generalized four-potential. The third component of the generalized spin moment operator for fermion may be written as

(m×

D)34 (NKNO2 m) s×3,

(2.12)

the eigenvalues of which are 6NKNO2m46mD.

Equation (2.10) clearly demonstrates that the value for magnetic moment for the system of a fermion moving in the field of a dyon is very large when compared to the value of Bohr magneton as magnetic charge plays a major role in this system. General Dirac equation for a fermion moving in the generalized electromagnetic field of a dyon is given as (in Feynman notations):

[PC×_{2 K V}C

K

1x] c(x) 4 0 . (2.13)

It can be demonstrated that this equation is relativistically invariant and gauge covariant under the gauge transformation, VmK Vm1 ¯mf , where f is an arbitrary function accompanied by the following unitary transformation of the wave function:

c Kc8 exp [Kf ] .

(2.14)

3. – Spin-orbit coupling for a fermion and a dyon

Let us consider the motion of a fermion in the field of a dyon retaining the terms up to the order of v2

. Substituting V 40 and V0(r) 4Vd(r) 4KV *0 in eqs. (2.6) and (2.7) we

get

[E 2Vd(r) ] j×4 (s× Q p×) h×, (3.1)

[E 12m2Vd(r) ] j×4 (s× Q p×) j× , (3.2)

where we have used e Ke1m. These equations in first approximation yield the following energy eigenvalue solutions in terms of spinor:

[E 2KV0* ] j×4 s × Q P× 2 m

k

1 2 1 2 m]E 2 K V0* (

l

(s× Q P×) j× , (3.3)

which on further simplification gives the following expression for energy operator (Hamiltonian) in first approximation:

H 4KV *0 1

k

1 2 1 2 m]E 2 K V0* (

l

P ×2 2 m 1 (3.4) 1 1 4 m2[s× Q ]K¯× V0* 3P × (] 2 1 2 m2[k¯× V *0 3 P×] .

In order to derive expression for Hamiltonian, in second approximation we use in eq. (3.2), instead of a function j× , another function f× given by

f

× 4 u×j× ,

the normalization of which up to second order leads to the following value of factor u :

u

× B12 P

2

8 m2 .

Using this value of u× _{(and hence of f), we get the relativistic expression for}

corresponding Hamiltonian (in second approximation) as follows:

H × 4 u×H×u×21 4

k

K V *0 1 P×2 2 m

l

2 1 2 m[E 2KV *0 ] 2 ]* 1 (3.5) 1 1 4 m2[s× Q ]KV0* 3P × (] 1 1 8 m2[¯ 2_{K V} 0* ] or H × 4

k

P 2 2 m 1 K V0*

l

2 1 2 m[E 2KV0* ] 2 2 1 8 m2[K]div c*D(] 2 (3.6) 2 1 4 m2[s× Q Re ]Kc*D(] 44 H × 01 H×11 H×21 H×34 H×01 H×I, where c*D4 2¯×V *0 . H ×

0 corresponds to the non-relativistic term of the Hamiltonian, while HI is the

relativistic correction term to the Hamiltonian, various parts of which arise due to different relativistic interactions.

4. – Pauli equation for a fermion moving in the generalized electromagnetic field of a dyon

In order to undertake the study of the behaviour of a fermion of charge eimoving in the field of a dyon with charge qj4 ej2 igj in nonrelativistic framework with inclusion of a spin effects, we start with the following Schrödinger equation of a spinning dyon in the field of another dyon

k

2 1 2 mp 2 1 V(r) 1 F(r) L× Qs×

l

u

c1 c2

v

4 E

u

c1 c2

v

, (4.1)

where the spin-orbit interaction F(r) L× Qs× will be treated as a small perturbation. Though the non-relativistic equation (4.1) is not sufficiently complex to yield precise values for fine structure of energy levels of this system as they are also affected by the relativistic corrections to the kinetic energy operators (21O2m) p×2 _{as much as by} specific spin-orbit interaction, it can be regarded as a useful guide to an understanding of the role of spin in the bound-state of a fermion and a dyon.

The unperturbed Hamiltonian

H×04 2 1 2 mp 2 1 V(r)

(

where V(r) 42KOr1P2 O2 mr2

)

(4.2)

represents the familiar central force problem for the system of a fermion moving in the generalized electromagnetic field of a dyon (6) and the spin-orbit interaction energy H8 is given by H 84 K 2 m2

o

1 r3

p

L× QS×1 P2 2 m4

o

1 r4

p

L× QS× . (4.3)

In this equation P 4eiej and other symbols have there usual meaning. We know

that

J2

4 L21 S21 2 L Q S , so the Pauli operator for H8 is given as

(H 8)P4 K 4 m2

o

1 r3

p

[ (J 2₎ P2 (L2)P2 (S2)P] 2 P 4 m4

o

1 r4

p

[ (J 2₎ P2 (L2)P2 (S2)P] (4.4)

and the Pauli equation becomes

(HP) cP4 [ (H0)P1 (H 8 )P] cP4 W(P) , (4.5) where (H0)P4

u

H0 0 0 H0

v

_P 4 (4.6) 4

u

2( 1 O2 m) p ×2 2 KOr 1 P2O2 mr2 0 0 2( 1 O2 m) p22 KOr 1 P2O2 mr2

v

and

cP4

u

c₁ c2

v

P (4.7)

represents the Pauli wave function. The Pauli wave equation for unperturbed Hamiltonian is given as

u

H0 0 0 H0

v

u

c₁( 0 ) c2( 0 )

v

4 W( 0 )

u

c1( 0 ) c2( 0 )

v

(4.8) or H0c6 ( 0 ) 4 W( 0 )c( 0 ) 6 . (4.9)

This equation can be solved by the method of separation of variables by writing the wave function as

c 4 U(r)

r Yeigj, l 8, m 8(u , f) ,

(4.10)

where Yeigj, l8, m8(u , f) are dyon harmonics (7) and the radial function U(r) Or4R(r)

satisfies the equation

r2

{

1 rR(r) d2 dr2(rR) 12m(E2V)

}

4 2 LYeigj, l 8, m 8(u , f) Yeigj, l 8, m 8(u , f) (4.11) with L 4 1 sin u ¯ ¯u

g

sin u ¯ ¯u

h

1 1 sin2_u ¯2 ¯f2 (4.12)

and V is defined by eq. (4.2).

Substituting the value of V from (4.2) into (4.11), we get 1 r2 d dr

k

r 2 dR dr

l

1 2 m

y

E 1 K r 2 P2 2 mr2 2 l 8(l 811) 2 mr2

z

R(r) 40 . (4.13)

By substituting the dimensionless variable

r 4ar ,

this equation (4.13) becomes 1 r2 d dr

g

r 2 dR dr

h

1

g

2 mE a2 1 2 mK ar 2 l 8(l 811)2P2 2 ma2 r2

h

R(r) 40 , (4.13a) where a2 4 8 mNEN 4 2 8 mE .

Equation (4.13a) may be written as 1 r2 d dr

g

r 2 dR dr

h

1

y

l r 2 1 4 2 s(s 11) r2

z

R(r) 40 , (4.14) s(s 11) 4 [l 8(l 811)2P2]

o

m 22 E (4.15) and l 4K

o

m 22 E ;

eq. (4.14) yields the energy eigenvalue for the system of a fermion spinning around a dyon as follows:

En4 22 K2m[ ( 2 n 11)1 ](2l11)21 4 P2(1 O2]22, (4.16)

where n 40, 1, 2, R and c06will be the wave functions describing the behaviour of a

fermion moving in the field of a dyon and these are simply Rnl(r), Yeigj, l8, m8(u , f) where Yeigj, l8, m8(u , f) are dyon harmonics (7). Thus the Pauli wave function for spin up and

spin down states would be given as

(c( 0 )₁₎ P4 c(n , l 8 , m 8 , ms4 11 O2 ) N Hb 4 Rnl 8Yeg , l , mN Hb 4

u

R_{nl 8}Yeigj, m ,8 l 8 0

v

(4.17) and (c( 0 )₂₎ P4 c(n , l 8, m 8, ms4 11 O2 ) N Ib 4 Rnl 8Yeigj, l 8, m 8N Ib 4

u

0 R_{nl 8}Yeigj, l 8, m 8

v

. (4.18)

In the absence of the spin-orbit interaction both the wavefunctions correspond to the same energy. In order to determine splitting due to spin-orbit interaction we should choose a representation in which H8 is diagonal

(f1)P4 f(n , l 8 , j 4 l 1 1 O2 , mj) 4

o

l 81mj1 1 O2 2 l 11 Y(n , l 8, ml4 mj2 1 O2 , ms4 11 O2 )1 (4.19) 1

o

l 2mj1 1 O2 2 l 11 c(n , l 8, ml4 mj1 1 O2 , ms4 21 O2 )4

u

o

l 1mj1 1 O2 2 l 11 RnlYeigj, l 8 mj2 1 O2

o

l 2mj1 1 O2 2 l 11 RnlYeigj, l 8 m11O2

v

.

Then the first-order perturbation due to spin-orbit interaction would be given by Ws( 1 )4



dt f†(H)Pf 4 K 4 m2



_dt 1 r3f 1_{[ (J}2₎ P2 (L2)P2 (S2)P] f 1 (4.20) 1 P 2 4 m4



dt 1 r4f †_{[ (J}2₎ P2 (L2)P2 (s2)P] f 4 or Ws( 1 )4 K 4 m2

km

j( j 11)2l(l11)2 3 4

nl

d 1 r3

y

l 6mj1 1 O2 2 l 11 NRnlN 2_Q (4.21) Q NYeigj , l , mj 21O2N 1 l Z mj1 1 O2 2 l 11 NRnlN 2 NYeigj, l , mj2 1 O2N

z

1 1 P 2 8 m4[]j( j11)2l(l11)23O4(]



_dt 1 r4

y

l 6mj1 1 O2 2 l 11 Q QNRnlN2NYeigjmj1 1 O2N 2 1 l Z mj1 1 O2 2 l 11 NRnlN 2 NYeigj, j 8, m11O2N 2

z

_,

where the plus and minus signs correspond to j 4l11O2 and j4l21O2, respectively. After integration we get

Ws( 1 )4

.

`

/

`

´

K 4 m2l

o

1 r3

p

1 P2 8 m4l

o

1 r4

p

for j 4l11O2 , K 4 m2(l 11)

o

1 r3

p

1 P2 8 m4(l 11)

o

1 r4

p

for j 4l21O2 , (4.22) where

.

`

/

`

´

o

1 r3

p

4



0 Q 1 r3NRnlN 2_r2 dr 4 1 n3_{l(l 11O2)(l11)} Q 1 a0 2 ,

o

1 r4

p

4



0 Q 1 r4NRnlN 2_r2 dr 4 3 25n 3 (l 11O2) a2 0 n5_{(l 21O2)(l13O2) a}04 . (4.23)

The splitting of energy levels corresponding to quantum number n is

W 4W( 0 ) 1 W( 1 )4 (4.24) 4

.

`

/

`

´

En2 EnK 2 m2_n3 (l 11)(2l11) a2 0 2 EnP 2 l [ 3 25n3(l 11O2) ] a02 m4 ( 2 l 21)(2l11)(2l13) a4 0 for j 4l11O2 , En1 EnK 2 m2_{nl( 2 l 11) a}02 2 EnP 2_a2 0 m4n5_{( 2 l 21)(2l11)(2l13) a}04 j 4l21O2 ,

where Enis given by eq. (4.16) and the Bohr radius for this system is given as a04 P2 1 1 mk . (4.25)

Equation (4.24) gives the splitting of energy levels corresponding to quantum number n for j 4l11O2 and j4l21O2, respectively. It shows that spin-orbit interaction modifies the energy spectrum of bound-state of a fermion and dyon perceptibly from the usual case of bound-state of a fermion and dyon [7]. In this way the ad hoc introduction of spin becomes important since the relativistic Dirac equation for a fermion moving in the field of a dyon cannot be solved exactly [7] due to a term vanishing more rapidly than r21 _{in the potential of the system.}

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