1. Art and Diseño

Exact solution of the Dirac equation for a Coulomb and a scalar
Potential in the presence of an Aharonov-Bohm and a magnetic
monopole elds
Vctor M. Villalba
Centre de Physique Theorique C.N.R.S.
PostScript〉 processed by the SLAC/DESY Libraries on 8 Mar 1995.
Luminy- Case 907-F-13288 Marseille Cedex 9, France
Centro de Fsica, Instituto Venezolano de Investigaciones Cientcas
IVIC, Apdo. 21827, Caracas 1020-A Venezuela
Abstract
In the present article we analyze the problem of a relativistic Dirac electron
in the presence of a combination of a Coulomb eld, a 1=r scalar potential as
well as a Dirac magnetic monopole and an Aharonov-Bohm potential. Using
the algebraic method of separation of variables, the Dirac equation expressed
in the local rotating diagonal gauge is completely separated in spherical coordinates, and exact solutions are obtained. We compute the energy spectrum
and analyze how it depends on the intensity of the Aharonov-Bohm and the
HEP-TH-9503051
magnetic monopole strengths.
Typeset using REVTEX
e-mail address: [email protected]
1
I. INTRODUCTION
The Dirac equation is a system of four coupled partial dierential which describes the
relativistic electron and other spin 1/2 particles. Despite the remarkable eort made during the last decades in order to nd exact solutions for the relativistic Dirac electron the
amount of solvable congurations is relatively scarce, being the Coulomb problem perhaps
the most representative example and also one of the most discussed and analyzed problems
in relativistic quantum mechanics. Among the dierent approaches available in the literature for discussing the Dirac-Coulomb problem in the presence of other interactions like the
Aharonov-Bohm eld or any other electromagnetic potential we have the quaternionic approach proposed by Hautot , the Stackel separation method developed by Bagrov et al , the
algebraic method of separation of variables ; , the shift operator method , and the algebraic
method proposed by Komarov and Romanova .
Recently, Lee Van Hoang et al have solved the Dirac-Coulomb problem when an
Aharanov-Bohm and a Dirac magnetic monopole elds are present. The authors use, for
tackling the problem, a two dimensional complex space which results after applying the
Kustaanheimo-Stiefel transformation on the three space variables, reducing in this way the
Kepler problem to an oscillator problem. This idea lies on the utilization of a SU(2) dynamical algebra for computing the resulting energy spectrum , which, like the spinor solution,
is expressed in terms of intrinsic coordinates appearing after using the complex space. The
utilization of dierent techniques for studying the Dirac-Coulomb eld in the presence of an
Aharonov-Bohm eld or a magnetic monopole could give rise to the idea that this problem
is not soluble without introducing new variables or additional conserved quantities. Here it
is shown that using the algebraic method of separation of variables it is possible to solve
the Dirac equation in the presence of a Coulomb eld and a scalar 1/r potential with an
Aharonov-Bohm and a magnetic monopole elds. The advantage of this approach is that
does not require the introduction of non bijective quadratic transformations, also it becomes
clear the role played by the Dirac magnetic monopole as well as the Aharonov-Bohm eld
1
2
34
5
6
7
8
9
2
in the angular dependence of the spinor (~r) solution of the Dirac equation.
The article is structured as follows: In Sec. II, applying a pairwise scheme of separation,
we separate variables in the Dirac equation expressed in the local rotating frame, we separate
the radial dependence from the angular one. In Sec. III. the angular dependence is solved
in terms of Jacobi Polynomials. In Sec IV, the separated radial equation is solved and the
energy spectrum is computed. In Sec. V, we discuss the inuence of the Aharonov-Bohm
eld and the magnetic monopole on the energy spectrum.
II. SEPARATION OF VARIABLES
In this section we proceed to separate variables in the Dirac equation when a Coulomb
eld, a scalar 1/r potential as well as a Dirac magnetic monopole and a Aharonov-Bohm eld
are present. For this purpose, we write in spherical coordinates the covariant generalization
of the Dirac equation
n
o
~ (@ iA) + M + V~ (r) = 0
(1)
where ~ are the curved gamma matrices satisfying the relation, f~ ; ~ g = 2g , and
are the spin connections . with V~ (r) as the scalar 1/r eld
+
10
0
V~ (r) = r
(2)
where 0 is a constant, and the vector potential A reads
A = Amon + ACoul + AAB
(
)
(
)
(
)
(3)
where the components of the Coulomb potential ACoul take the form
(
)
A Coul = V (r) = r ; AiCoul = 0; i = 1; 2; 3
(
0
)
(
)
(4)
the Aharonov Bohm potential AAB reads
(
)
F e^
A AB = r sin
#'
(
)
3
(5)
and the Dirac monopole eld Amon is
(
)
cos #) e^
A mon = g (1 r sin
# '
(
(6)
)
where, following the Dirac prescription for quantizing the magnetic charge, g takes integer
or half integer values
11
g = 2j ; j = 0; 1; 2::::
(7)
If we choose to work in the xed Cartesian gauge, the spinor connections are zero and
the ~ matrices take the form
h
i
~ = = ; ~ = ( cos ' + sin ') sin # + cos # = ;
h
i
~ = 1r ( cos ' + sin ') cos # sin # = r ;
(8)
1 ( sin ' + cos ') = ~ = r sin
#
r sin #
where are the standard Minkowski gamma matrices, and the Dirac equation in the xed
Cartesian tetrad frame (8) takes the form
(
)
(@t + iV (r)) + @r + r @# + r sin # (@' iF i(1 cos #)g) + M +V~ (r) Cart = 0
0
0
0
2
1
1
1
2
2
3
0
1
2
3
1
2
1
3
3
2
3
(9)
where we have introduced the spinor Cart, solution of the Dirac equation (9) in the xed
tetrad gauge. In order to separate variables in the Dirac equation, we are going to work in
the diagonal tetrad gauge where the gamma matrices ~d take the form
1 ~d = ; ~d = ; ~d = 1r ; ~d = r sin
#
0
0
1
1
2
2
3
3
(10)
Since the curvilinear matrices ~ and ~d satisfy the same anticommutation relations, they
are related by a similarity transformation, unique up to a factor. In the present case we
choose this factor in order to eliminate the spin connections in the resulting Dirac equation.
The transformation S can be written as
12
4
S = r(sin1#) = exp( '2 ) exp( #2 )a = S a
1
2
3
1
where a is a constant non singular matrix given by, a = ( applied on the gamma's acts as follows
1
2
(11)
0
1 2
1
+ + I ) which
2
1
3
2
a a = ; a a = ; a a = ;
1
1
3
2
1
1
3
1
3
(12)
2
the transformation S acts on the curvilinear ~ matrices, reducing them to the rotating
diagonal gauge as follows
S ~S = g = ~d (no summation)
(13)
1
then, the Dirac equation in spherical coordinates, with the radial potential V (r), in the local
rotating frame reads
)
(
(@t + iV (r)) + @r + + r @# + r sin # (@' iF i(1 cos #)g) + M + V~ (r) rot = 0
(14)
0
3
2
1
where we have introduced the spinor rot, related to Cart by the expression
Cart = S rot = S arot
(15)
0
and are the standard Dirac at matrices.
Applying the algebraic method of separation of variables ; , it is possible to write eq. (14)
as a sum of two rst order linear dierential operators K^ , K^ satisfying the relation
h^ ^ i
n^ ^ o
K ; K = 0;
K +K =0
(16)
34
1
1
2
1
2
2
K^ = k = K^ 2
(17)
1
then, if we separate the time and radial dependence from the angular one, we obtain
"
#
^
K = i @# + sin # (@' iF i(1 cos #)g) = k
(18)
2
2
3
0
5
1
h
i
K^ = ir @t + @r + M + i V (r) + V~ (r) = k
(19)
rot = (20)
0
1
1
0
0
1
with
0
1
where the operator K^ as well as K^ have been chosen to be Hermitean, therefore the
constant of separation k appearing in (18) and (19) is real. Notice that if we drop out the
Aharonov-Bohm and the magnetic charge contributions in eq. (18) we obtain the Brill and
Wheeler angular momentum K^ , and separation constant k takes integer values.
Now we proceed to decouple the equation (18) governing the angular dependence of
the Dirac spinor In order to simplify the resulting equations, we choose to work with the
auxiliary spinor related to as follows
1
2
10
= a
(21)
Since the operators K^ and K^ commute with the projection of the angular momentum
i@, with eigenvalues m, we have that equation (18) takes the form
"
#
(22)
@# i sin # (m + F + (1 cos #)gt) = ik
In order to reduce the equation (22) to a system of ordinary dierential equations, we
choose to work in the following representation for the gamma matrices,
0
1
0
1
i
B 0 CC
B i 0 CC
i = B
(23)
@ i A ; = B@
A;
0
0 i
Then, substituting (23) into (22) we obtain,
#
"
@# i sin # (m + F + (1 cos #)g) = ik
(24)
"
#
@# + i sin # (m + F + (1 cos #)g) = ik
(25)
2
1
2
1
0
3
13
0
1
2
1
2
with
1
1
2
0
= eim B
B@ 1
2
6
1
CC
A
2
(26)
III. SOLUTION OF THE ANGULAR EQUATIONS
In this section we are going to solve the systems of equations (24) and (25). It is not
dicult to see that the spinor can be written as G(r; t) , where G(r; t) is an arbitrary
function, this property allows us to consider only the system (24). Using the standard Pauli
matrices , we have that (24) reduces to
3
2
1
(m F (1 cos #)g) + sin # d
d# = 0
(27)
k sin # + (m F (1 cos #)) + sin # d
d# = 0
(28)
k sin #
1
2
2
with
1
0
B
= Qeim B
@
1
1
2
2
1
1
CC
A
(29)
where Q is a function depending on the variables t, r, to be determined after a complete
separation of variables. In order to solve the coupled system of equations (27)-(28) we make
the following ansatz,
= (sin #2 )a(cos #2 )bf (#); = (sin #2 )c(cos #2 )dq(#)
2
1
(30)
where a, b, c, and d are constants to be xed in order to obtain solutions of the governing
equations for f (#) and q(#) in terms of orthogonal special functions. Substituting (30) into
(27) and (28) we obtain,
(x) = 0
kq(x) ( m2 F2 + a2 + 12 )f (x) + (1 x) dfdx
(31)
kf (x) + ( d2 g m2 F2 )q(x) + (1 + x) dqdx(x) = 0
(32)
where we have made the change of variable x = cos #; and we have simplied the resulting
equations (31) and (32) by imposing c = m F; and b = m F 2g: If we set a = m F +1;
and d = m F + 1 2g;and make the change of variables u = (1 x)=2, we obtain that the
coupled system of equations (31)-(32) takes the form
7
(u) = 0
kq(u) (m F + 12 )f (u) u dfdu
kf (u) + (m F 2g + 12 )q(u) + (u 1) dqdu(u) = 0
(33)
(34)
from (33) and (34) we obtain
q(u) + [(m F + 1 ) 2(m F g + 1 )u] dq(u) +
u(1 u) d du
2
2 du
+ [k (m F + 12 )(m F 2g + 21 )]q(u) = 0
f (u) + [(m F + 3 ) 2(m F g + 1)u] df (u) +
u(1 u) d du
2
du
+ [k (m F + 12 )(m F 2g + 21 )]f (u) = 0
2
2
2
(35)
2
2
2
(36)
the solution of the equation (35) can be expressed in terms of the hypergeometric function
F (a; b; c; u) as follows
14
q(u) = c F (a; b; c; u)
(37)
1
where c is a constant, and a; b; and c are
1
q
k +g
a = m F g + 21
2
q
b = m F g + 12 + k + g
2
2
(38)
2
(39)
c = m F + 12
(40)
then, with the help of (34) and (37) we nd that f (u) reads
f (u) = c m kF + F (a; b; c + 1; u)
1
1
2
Since we are looking for normalizable solutions according to the product
Z 2 k yk d# = kk
2
0
0
8
0
(41)
(42)
we have that the series associated with the hypergeometric function F (a; b; c; u) should be
truncated reducing it to polynomials. This one is possible if a = n or b = n in (37) where
n is a no negative integer value. Then using the relation between the Jacobi Polynomials
and the functions F (a; b; c; x)
Pn; (x) = n(n! +(++1)1) F ( n; n + + + 1; + 1; 1 2 x )
we have that (41) and (37) reduce respectively to
(
)
f (x) = c0Pnm
(
q(x) = c0 g +
F +1=2;m F 2g 1=2) (x)
pk + g
Pnm
k
2
2
(
(43)
(44)
F 1=2;m F 2g+1=2) (x)
(45)
where c0 is a constant, and n reads
q
n = m + F + g 12 + k + g
Then, the components and of the spinor (29) can be written as
= c0(sin #2 )m F (cos #2 )m F g Pnm F = ;m F g = (cos #)
2
1
+1
2
pk + g
# )m F (cos # )m
(sin
k
2
2
2
(46)
2
1
2 = c0 g +
2
2
(
+1 2
2
1 2)
F 2g+1 P (m F 1=2;m F 2g+1=2) (cos #)
n
Notice that the orthogonality relation for the Jacobi Polynomials
Z
(1 x)(1 + x) Pn; (x)Pm; (x)dx =
2 (n + + 1) (n + + 1) 2n + + + 1 (n + 1) (n + + + 1) mn
1
(
)
(
(47)
(48)
)
1
+ +1
(49)
imposes some restrictions on the values of m; F and g in (47) and (48) In fact, from (49)
we have that > 1; > 1 and consequently we have that
15
m F + 21 > 0; m F 2g + 1=2 > 0
(50)
aords the required condition of orthogonality. However, some values considered in the
inequalities given by (50) fail in fullling a condition which should be also took into account
9
in selecting the possible values of m; F and g: : All the expectation values associated with
the separating operators should exist. In fact, if we consider the operator K^ dened by the
R
R
relation (18) we have that the integrals d# @# as well as d# @# should be nite.
This one imposes some restrictions on m; F and g. Looking at the convergency when # ! 0;
and # ! we obtain the relations
2
1
1
2
2
m F > 0; m F 2g > 0
(51)
which are weaker than those given in (50)
In order to be able to consider other relations among the parameters m; F and g dierent
from (46) we are going to consider a second solution of the equation dierential equation
(35)
q(u) = u c (1 u)c
1
a b F (1
a; 1 b; 2 c; u)
(52)
then, using the recurrence relations for the hypergeometric functions, we nd that a solution
for the system (33)-(34) reads
f (u) = c u
2
= m+F (1
u)
1 2
a; 1 b; 21 m + F ; u)
m+F +2g+1=2 F (1
(53)
k
= m F (1 u) m F g = F (1 a; 1 b; 3 m + F ; u)
u
m+F
2
where c is a constant. From (53) and (54) we obtain that and take the form
q(u) = c
1 2
2 1
2
+
2
1
= c(sin #2 )
1
2
m+F (cos # ) m+F +2g+1 P ( 1=2 m+F;2g m+F +1=2) (cos #)
n
2
= c pk +kg g (sin #2 ) m F (cos #2 ) m F g Pn = m F; g m
with c as a constant of normalization, and in the present case n is
q
n = m F g 12 + k + g
2
1
2
(54)
+ +2 +1 2
+
+ +2
(1 2
+
2
2
2
(55)
F 1=2) (cos #)
(56)
+
(57)
2
Following the same reasoning used for deriving (51), we have that the expressions for and
given by (55) and (56) are valid when
1
2
10
m + F > 0; 2g m + F > 0
(58)
A third possible solution of the equation (35) can be written as
q(u) = (1 u)c
a b F (c
a; c b; c; u)
(59)
then, substituting (59) into (34) we nd that f (u) and q(u) take the form:
q
q
f (u) = c (1 u) m F g = F (1 + g + k + g ; 1 + g k + g ; m F + 23 ; u) (60)
3
+ +2 +1 2
2
q(u) = c m Fk + 1=2 (1 u)
2
m+F +2g 1=2 F (g +
3
2
2
q
q
k + g ; g k + g ; m F + 12 ; u)
(61)
2
2
2
2
where c is a constant. Using the relation between the hypergeometric function F (a; b; c; x)
and the Jacobi Polynomials Pn; (x) given by eq. (43) we arrive at
3
(
= c(sin #2 )m
1
)
F +1 (cos # ) m+F +2g+1 P (m F +1=2; m+F +2g+1=2) (cos #)
n
2
= c n +k 1 (sin #2 )m F (cos #2 )
2
m+F +2g P (m F 1=2; m+F +2g 1=2) (cos #)
n+1
where c is a constant of normalization and n is given by
q
n= k +g g
2
(62)
(63)
(64)
2
In this case, we have to impose the following restrictions on the values of m; F and g:
m F > 0; m + F + 2g > 0
(65)
Finally, considering as solution of the equation (35) the expression
q(u) = u cF (a c + 1; b c + 1; 2 c; u)
1
then, in the present case the functions f (u) and q(u) read
q
q
f (u) = u = m F F ( g k + g ; g + k + g ; 12 m + F ; u)
1 2
+
2
2
11
2
2
(66)
(67)
q(u) =
k
=
u
m+F
1 2
1
2
m+F F (
= c (sin #2 )
1
4
= c nk (sin #2 )
with n given by
2
4
1
g
q
q
k + g + 1; g + k + g + 1; 23 m + F ; u) (68)
2
2
2
2
m+F (cos # )m F 2g P ( 1=2 m+F;1=2+m F 2g) (cos #)
n
2
m+F (cos # )m F 2g+1 P (1=2 m+F; 1=2+m F 2g) (cos #)
n 1
2
q
n= k +g +g
2
2
(69)
(70)
(71)
and c is a constant of normalization. The solutions (69) and (70) are well behaved according
to the Dirac inner product as well as to the expectation value of the operator of angular
momentum (18) if
4
m + F > 0; m F 2g > 0
then, we have that the results (64) and (71) can be gathered as follows
q
n= k +g jg j
2
2
(72)
(73)
when the condition on m, F and g
F + g+ j g j> m > g j g j +F
(74)
is satised.
Regarding the eigenvalues m of the projection of the angular momentum operator i@'
we have that since the transformation (11), relating the Dirac spinors rot and Cart in the
local (rotating) and the Cartesian tetrad frames transforms after a rotation as follows
Sz (' + 2) = Sz (')
(75)
and the spinor Cart is single valued, then we obtain
rot(' + 2) = rot(')
and therefore m takes half integer values
m = N + 21 ; N = 0; 1; 2:::
12
(76)
(77)
IV. SOLUTION OF THE RADIAL EQUATION
Now, we are going to solve the system of equation (19), governing the radial dependence
of the spinor rot solution of the Dirac equation. This equation can be written in the form,
!
k
@t + @r + (M + V~ (r)) i V (r) + i r = 0
(78)
3
0
0
3
3
Using the representation for the gamma matrices given by (23), and the fact that eq. (78)
commutes with the energy operator @t with eigenvalues E , we obtain the following system
of equations
!
k
(79)
dr + r (E V (r) M V~ (r)) = 0
3
1
!
k
dr + r 2
(E V (r) + M + V~ (r)) = 0
3
2
1
(80)
From (24)-(25),(79)-(80) and the fact that the Dirac equation (14) commutes with i@ and
i@t, we have that the spinor can be written as follows
0
1
BB A(r) CC
BB
CC
A
(
r
)
B
CC
= c ei m' Et B
(81)
BB
C
BB c B (r) CCC
@
A
c B (r)
1
0
(
2
)
1
2
where c is a constant, and A(r) and B (r) satisfy the system of equations
!
k
dr + r A(r) (E V (r) V~ (r) M )B (r) = 0
!
k
dr + r B (r) (E V (r) + V~ (r) + M )A(r) = 0
(82)
(83)
notice that in this way we have xed the values of the functions Q(r; t) = e iEtA(r), and
G = B (r)=A(r) appearing during the process of separation of variables. Substituting into
(82) and (83) the form of the scalar and the Coulomb potentials we get
13
!
d + k A(r) E M + 1 ( + 0) B (r) = 0
dr r
r
(84)
!
k B (r) + E + M 1 ( 0) A(r) = 0
r
r
(85)
A = E + M; B = M E; ^ = + 0; = 0 (86)
d
dr
Introducing the notation
and the new variable ; related to r as follows
p
= Dr = M
p
E r = ABr
2
2
(87)
we have that the system of equations (84)-(85) reduces to
!
!
d + k A(r)
B + ^ B (r) = 0
d D !
d k B (r) + A
d D
!
^ A(r) = 0
We shall look for solutions of the system (88)-(89) in the form of power series
1
X
A() = e s a
+
=0
1
X s+ b
=0
B () = e
(88)
(89)
;
16 17
(90)
(91)
Substituting (90)-(91) into (88)-(89) we nd
(s + k)a
0
^ b = 0
(92)
(s k)b
0
^ =0
a
(93)
0
0
and
Bb
[(s + ) + k] a ^b + D
14
1
a = 0
1
(94)
^ + A a
[(s + ) k] b a
D
From (92)-(93) it follows that
p
s= k
+ 0
2
2
b = 0
1
(95)
1
(96)
2
where we have dropped out (96) the negative root because we are looking for wavefunctions
regular at the origin of coordinates. From (94)-(95) we have that
8
9
8s
9
s
<
=
<
A ^ a = A ^ [(s + ) k]= b
[(
s
+
)
+
k
]
(97)
:
; B ; : B
The series (90) and (91) will have a good behavior at innity if they terminate for a nite
value N. Putting aN = bN = 0 in (94)-(95) with aN 6= 0 and bN 6= 0; we arrive at
bN = A
(98)
aN D
Substituting (98) into (97), and taking into account (86) we get
+1
+1
p
(s + N ) M
E = E + 0M
2
(99)
2
where s is given by (96)
Finally, we obtain the energy spectrum:
8
v
!
u
>
0
<
0
u
t
E = M > (s + N ) + (s + N ) + :
9
=
(s + N ) >
(100)
;
[(s + N ) + ] >
Here two particular cases could be considered: a) 0 = 0 which corresponds to the Coulomb
potential. In this case the energy spectrum reduces to:
"
# =
p
E = M 1 + (s + N )
; s= k (101)
where the negative root has been dropped out because it is not compatible with the relation
(99) A second possibility is given by b) = 0; which is the scalar V 0(r) = 0=r potential.
In this case the energy spectrum takes the form
"
#=
0
p
E = M 1 (s + N )
; s = k + 0
(102)
notice that in the present case states with negative energy are possible, here we do not have
critical behavior like in the Coulomb case.
02
2
2
2
2
2
2
2
1 2
2
2
2
2
1 2
2
2
2
15
2
2 2
V. DISCUSSION OF THE RESULTS
In this section we are going to discuss the inuence of the Aharonov-Bohm potential
and the Dirac magnetic monopole charge on the energy spectrum. Here we have mention
that the Aharonov-Bohm as well as the magnetic monopole contributions are present in the
expression (100) via the factor s given by (96), since the explicit form of k depends on the
relation among m; F and g: In fact, we have that when the inequalities (65) or (72) are valid,
the expression for s takes the form
q
s = (n+ j g j) g + 0
(103)
2
2
2
2
and no contribution of the Aharonov Bohm potential is observed in (100). The values of m
for which (103) takes place are given by the expression (74)
It is worth mentioning that when the magnetic monopole contribution is absent, the
inequalities given by the expression (74) never take place, and consequently, the expression
for s given by eq. (103) is not applicable. In this case the energy spectrum can be computed
by substituting the expression
q
s = (n+ j m F j +1=2) + 0
(104)
2
2
2
into (100). Analogously, the energy spectrum when F =
6 0 and g6= 0 can be obtained after
substituting into (100) the following value of s
q
s = (n + m F g + 1=2) g + 0
(105)
2
2
2
2
when m F > 0 and m f 2g > 0: Otherwise, when m F < 0 and m F 2g < 0,
the value of s to substitute into (100) reads
q
s = (n m + F + g + 1=2) g + 0
(106)
2
2
2
2
Perhaps the most interesting and puzzling result of the present paper is the non dependence
of the energy spectrum on the Aharanov-Bohm potential for a range of values given by the
inequality (74). Despite this phenomenon was already pointed by Hoang et al there are
7
16
some discrepancies between their results and those ones present in this paper. Basically the
problems lie on the criteria for establishing the boundary conditions and the normalizability
of the wave functions. It is worth mentioning that since the Aharonov-Bohm contribution
F can take non integer values, then the parameters and in the Jacobi Polynomials
Pn; (x) can be negative provided that > 1; and > 1: Obviously if the Aharonov
Bohm potential is absent, from the results presented in Sec. 3 we have that 0 and
0 (or in the case when F is an integer). Hoang et al consider that and are always
positive restricting in the way the range of validity of the solutions. A second point to
remark is that we not only impose the normalizability of the wave functions but also the
existence of the expectation value of the angular momentum operator, which is equivalent
R
to say that y K^ d#d' < 1; in this way we have to impose that the spinor components
and presented in Sec. 3 should satisfy R ; @# ; < 1 . Regarding the assertion made
in about the existence of quantum states forbidden for the Dirac particle in the presence
of the Coulomb plus the Dirac monopole potentials, we have that the boundary conditions
imposed on the wave function in the present article avoid such a anomalous behavior and
consequently the spinor is well dened for any value of the parameters m; F and g:
(
1
)
2
12
12
7
ACKNOWLEDGMENTS
The author wishes to express his indebtedness to the Centre de Physique Theorique for the
suitable conditions of work. Also the author wishes to acknowledge to the CONICIT of
Venezuela and to the Fundacion Polar for nancial support.
17
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