On the Delta Funtion Normalization of the
Wave Funtions of the Aharonov-Bohm
Sattering of a Dira Partile
Vanilse S.Araujo 1
, F.A.B.Coutinho 2
, and J. Fernando Perez 1
1
Instituto deFsiadaUSP,CaixaPostal66318,05315-970, S~aoPaulo,SP,Brazil
2
FauldadedeMediinadaUSP,01246-903,S~aoPaulo,SP,Brazil
Reeivedon20February,2002
In a previous paper, we found the most general boundary onditions for the Aharonov -Bohm
sattering ofaDirapartile. Wefoundtheresulting wavefuntionsbutwedidnotworryabout
deltanormalizingthem.Asiswellknow,inpratie,itisnoteasytoevaluatethedivergingintegrals
ourringintheproess. Thepurposeofthis paperistoevaluatethoseintegralsandpresent the
resulting deltanormalizedeigenfuntions.
I Introdution
Inapreviousartile[1℄weonsideredtheHamiltonian
operatorH of aDirapartile ofmassm>0,moving
in twodimensionsin thepresene ofaninnitely thin
magnetiuxtubeattheorigin,formallydened as
H = h
!
p + e
!
A i
!
+m (1)
where !
p =(p
x ;p
y ),
!
=(
1 ;
2 ),
i =
i 0
0
i
; i=1;2;
and
=
3 0
0
3
; (2)
where
1 ;
2 and
3
arethePaulimatries. Thevetor
potential,in theCoulomb,Gaugeis
e
!
A =
r 2
( y;x): (3)
Weonsideredalsotheheliityoperatorgivenby
= h
!
p + e
!
A i
!
; (4)
where !
=(
1 ;
2 ) ,and
i =
0
i
i 0
; i=1;2: (5)
Inthis previousartile[1℄,wefound themost
gen-operator. The Hamiltonian operator and the
Heli-ity operator admit a four parameter family of
self-adjoint extensions in one-to-one orrespondene with
the boundary onditions(BC's) to be satisedby the
eigenfuntionsattheorigin. TheationsoftheHeliity
operator and the Hamiltonianoperator H ommute
beforespeiationoftheBC's. Although thisours,
to ensure ommutativity and onsequently to obtain
ommon eigenfuntions,it isnotsuÆient to take the
same BC's for bothoperators as laimed in referene
[2℄. This fat ours beausebothoperator H and ,
whenatingin aommondomain, donotletit
invari-ant.
The Heliity onservation an be obtained by the
imposing a formal ondition that leads to ertain
re-lations between the parameters of the extensions. In
otherwordstheformalonditionweimposedenesnew
domainswiththeparametersobeyingertainrelations.
These newdomainswefoundarethemostgeneral
do-mains where both operators H and are self-adjoint
and eetively ommute with onsequently ommon
eigenfuntions. In referene [1℄, we wrotedown these
most general ommon eigenfuntions, but wedid not
delta normalize them. In this paper we present this
alulation,thatompletestheresultsoftheartile[1℄.
This paper is organized as follows. In Setion II, we
presenttheomputationofthenormalizationonstant
ofthemostgeneralommoneigenfuntionsofHand
that satisesthemoregeneralBC's[1℄. InSetionIII
ondi-boundary ondition that makes H self-adjoint. This
boundary ondition depends on one parameterand is
theboundaryonditionobtainedinreferene[2℄. This
isnotaoinidene,butisrelatedtothefatthata
self-adjointoperatorpossessesompletesetoforthonormal
eigenfuntions. Unfortunately this proedure an not
be used to obtain the most general boundary
ondi-tions,but onlytheabovementionedspeialase.
II Normalization of the most
general eigenfuntions of H
and
The general form of the ommon eigenfuntions
E;
(kr) of H and operators before speiation of
thedomainsare[1℄
+jEj;k (kr)=
1
N 0
B
B
J
(kr)+
+ (k)J
(kr)
ik
jEj+m (J
1
(kr)
+ (k)J
1+ (kr))
k
jEj+m ( J
(kr)+d
+ (k)J
(kr))
i(J
1
(kr) d
+ ( k)J
1+ (kr))
1
C
C
A
; (6)
jEj;k (kr)=
1
N 0
B
B
J
(kr)+ (k)J
(kr)
+ik
jEj m (J
1
(kr) (k)J
1+ (kr))
k
jEj m (J
(kr)+d (k)J
(kr))
i(J
1
(kr) d (k)J
1+ (kr))
1
C
C
A
; (7)
where jEjarethepositiveandnegativeeigenvaluesofHand=k (k= p
E 2
m 2
)aretheeigenvaluesof.
By imposing the most general onditions of self-adjointness and ommutativity for H and operators, the
oeÆients
(k)and d
(k)mustobey[1℄
(k)=d (k)=
+ (k)=d
+
(k)=+ p
2
2
( 1) n
1 +n
2
sin
( 1 ( 1) n
1
ossin')
k
p
2m
2jj 1
; (8)
for=+k,and
(k)=d (k)=
+ (k)=d
+ (k)=
p
2
2
( 1) n
1 +n
2
sin
( 1 ( 1) n1
ossin')
k
p
2m
2jj 1
; (9)
for= k.
d
Forfutureusewedene
a= p
2
2
( 1) n
1 +n
2
sin
( 1 ( 1) n1
ossin')
1
p
2m
2jj 1
: (10)
Theparameters and' aretheparametersof the
extensionsthat mustsatisfysomerelations(see
equa-tions4.9and4.10ofreferene[1℄).
Fororthonormalitywemusthave
1
Z
0 rdr
E; (kr)
E 0
; 0
(k 0
r)= 1
p
kk 0
Æ( k k 0
): (11)
Using theformuladevelopedby Ausdreth, Jasper
andSkarzhinsky[3℄,
1
R
0 rdrJ
(kr)J
(k
0
r)= 1
p
k k 0
Æ(k k 0
)os+
2
sin
2 0
2 k
k 0
; (12)
andthewell-knownformula[4℄
1
Z
0 rdrJ
(kr)J
(k
0
r)= 1
p
kk 0
Æ(k k 0
); (13)
we must show rst of all that the non-Æ ontribution
termsthatomefromequation(12)mustvanishinthe
omputationofequation(11).
Todo this let us onsider the twoases: a) when
=k and 0
=k 0
( or= k and 0
= k 0
) and b)
when =k and 0
= k 0
(or = k and 0
=k 0
):
Consideringtheforms of
E;
( kr) given by equations
(6)and(7),therossingtermsofequation(11)arethe
1 N 2 1 R 0 rdrf[ (k)J (kr)J (k 0 r)+ (k 0 )J (k 0 r)J (kr)℄ k k 0 (E+m)(E 0 +m) [ (k)J +1 (kr)J 1 (k 0 r)+ (k 0 )J +1 (k 0 r)J 1 (kr)℄+ + k k 0 (E+m)(E 0 +m) [d (k)J (kr)J (k 0
r)+d
(k 0 )J (k 0 r)J (kr)℄ [d (k)J +1 (kr)J 1 (k 0
r)+d
(k 0 )J +1 (k 0 r)J 1 (kr)℄g: (14)
Usingtheformulagivenbyequation(12)thenonÆontributiontermsoftheaboveequationare
1
N 2
2
sin(E+E 0 ) n k k 0 k 2 k 0 2 [ (k) 1 E 0 +m +d (k) 1 E+m k k 0 ℄ k 0 k k 2 k 0 2 [ (k 0 ) 1 E+m +d (k 0 ) 1 E 0 +m k 0 k ℄ o : (15) Taking (k)=d
(k)and
(k
0
)=d
(k
0
)givenbyequations(8), (9)and(10)fortheasea,weseethat the
resultingnonÆ-ontributionoftheaboveequationvanishesasitshould,
1
N 2
2
sin(E+E 0 ) k k 0 k 2 k 0 2 f 1 (E 0 +m)k + 1 (E+m)k 0 a 1 (E+m)k 0 + 1 (E 0 +m)k
ag= 0: (16)
Consideringtheforms of
E;
(kr)given byequations(6)and (7),therossingtermsof equation(11)are,for
theaseb, thefollowing:
1 N 2 1 R 0 rdrf[ (k)J (kr)J (k 0 r)+ (k 0 )J (k 0 r)J (kr)℄ k k 0 (E+m)(E 0 +m) [ (k)J +1 (kr)J 1 (k 0 r)+ (k 0 )J +1 (k 0 r)J 1 (kr)℄+ k k 0 (E+m)(E 0 +m) [d (k)J (kr)J (k 0
r)+d
(k 0 )J (k 0 r)J (kr)℄ +[d (k)J +1 (kr)J 1 (k 0
r)+d
(k 0 )J +1 (k 0 r)J 1
(kr)℄g: (17)
Usingtheformulagivenbyequation(12)thenonÆontributiontermsoftheaboveequationare
1
N 2
2
sin(E+E 0 ) n k k 0 k 2 k 0 2 h (k) 1 E 0 +m d (k) 1 E+m k k 0 i k 0 k k 2 k 0 2 h (k 0 ) 1 E+m d (k 0 ) 1 E 0 +m k 0 k io : (18) Taking (k)=d
(k)and
(k
0
)=d
(k
0
)givenbyequations(8), (9) and(10)fortheaseb, weseethat the
resultingnonÆ-ontributionoftheaboveequationvanishesasitshould:
1
N 2
2
sin(E+E 0 ) k k 0 k 2 k 0 2 n 1 (E 0 +m)k 1 (E+m)k 0 a + 1 (E+m)k 0 1 (E 0 +m)k a o
=0: (19)
d
Soweseethatthemostgeneralommon
eigenfun-tions given by equations (6) to (10) of referene [1℄
arenormalizable,sinethenon-Æfuntionontribution
vanishes in the omputation of equation (11). Let us
The Æ funtion ontribution of the rossing terms of
equation (11),aftersomemathematialmanipulations
using equations(8), (9), (10)and (12)anbewritten
as
1
N 2
os
1
p
k k 0
Æ(k k 0
)f1+ k k
0
(E+m)(E 0
+m) gf
(k)+
(k
0
)g
+ 1
N 2
os
1
p
k k 0
Æ( k k 0
)f1+ k k
0
(E+m)(E 0
+m) gfd
(k)+d
(k
0
)g
= 1
N 2
4E
E+m [2a(k)
2jj 1
℄ os
k
Æ(k k 0
): (20)
Colletingnowthediret termsofequation(11)wehave
1
N 2
1
R
0 rdrf[J
j j (kr)J
j j (k
0
r)+
(k)
(k
0
)J
jj (k
0
r)J
j j (kr)℄
+ k k
0
(E+m)(E 0
+m) [J
jj 1 (kr)J
j j 1 (k
0
r)+
(k)
(k
0
)J
1 jj (k
0
r)J
1 jj (kr)℄+
+ k k
0
(E+m)(E 0
+m) [J
jj (kr)J
j j (k
0
r)+d
(k)d
(k
0
)J
jj (k
0
r)J
j j (kr)℄
+[J
jj 1 (kr)J
j j 1 (k
0
r)+d
(k)d
(k
0
)J
1 jj (k
0
r)J
1 j j
(kr)℄g: (21)
Usingtheformula(13)theaboveequation,aftersomemathematialmanipulationsusingequations(8),(9)and
(10)gives
1
N 2
4E
E+m [1+a
2
(k) 4jj 2
℄
Æ(k k 0
)
k
: (22)
Consideringthe Æ ontribution of the rossingterms given by equation (20) and of thediret termsgiven by
equation(21),thenormalizationonditionofequation(11)turnsouttobe
N = r
E+m
4E
1
[1+2aos()(k) 2jj 1
+a 2
( k) 4jj 2
℄ 1
2
: (23)
d
III The orthonormality
ondi-tion and the one parameter
family of self-adjoint
exten-sion for H
We an obtain a one parameter family of self-adjoint
extensions ofHoperator (theBC'sofreferene[2℄)by
imposingorthonormalityforthe eigenfuntionsofthis
operator.Itisnotneessarytodotheompliated
al-ulations of refs.[1℄and [2℄. Thisis nota oinidene,
butitoursbeauseaself-adjointoperatoralwayshas
a omplete set of orthonormal eigenfuntions. Let us
onsiderthegeneralformofaneigenfuntionofHgiven
byequations(6)and(7). Thenon-Æontribution
ross-ing terms of the upper omponent spinor in the
om-byequation(12),gives
1
N 2
2
sin( )
(k)
k
k 0
k 2
k 0
2 f1+
E m
E 0
+m g
1
N 2
2
sin( )
(k
0
) k
0
k
k 2
k 0
2 f1+
E 0
m
E+m
g: (24)
Imposing theorthonormalityonditionthis
ontri-butionmustvanish. Thenwehave
(k)=
k 2jj
E+m tan
1
m 2jj 1
; (25)
where theonstant 1
m 2jj 1
was introdued for
dimen-sionalreasonsand tan isafree parameterof the
ex-tension.
afterusingtheformulagivenbyequation(12),gives:
1
N 2
2
sin()d
(k)
k
k 0
k 2
k 0
2 f
k
k 0
+ k k
0
(E 0
+m)(E+m) g
1
N 2
2
sin()d
(k
0
) k
0
k
k 2
k 0
2 f
k 0
k +
k k 0
(E 0
+m)(E+m) g:
(26)
Imposingthat thisontributionvanishweget
d
(k)=
k 2jj
E m
tg 1
m 2jj 1
; (27)
where the onstant 1
m 2jj 1
was introdued for
dimen-sionalreasonsandtan isafreeparameter.
The resultsof equations (25) and (27) orrespond
totheBC'sofreferene[2℄andalsoofreferene[5℄. In
theaseofreferene[2℄theboundaryonditionsforthe
twotopomponentsbeomedeoupledfromthe
bound-ary onditionsfor thetwobottonomponents. In the
ase of referene[5℄, only the two top omponentsare
onsidered.
We an also obtain the normalization onstant in
thisase,byomputingalltheÆ funtionontribution
termsbytherossinganddirettermsandthen
impos-ing the normalizabilityof equation (11). Forthe two
omponentsofreferene[5℄,weget
N = r
E+m
4E
1
[1+2tanos() k
2j j
E+m 1
m 2jj 1
+tan 2
k
4j j
(E+m) 2
1
m 4j j 2
℄ 1
2
: (28)
d
Oneanhekthatthisresultisthesamepresented
bySousaGerbert forthetwoomponentspinorin
ref-erene[5℄
In our moregeneral ase, imposing the
ommuta-tivityonditionforH andthat is
(k)=d
(k)for
allk,wehave
N = r
E+m
4E
: (29)
Referenes
[1℄ V.S.Araujo, F.A.B.Coutinho,and J.Fernando Perez,
J.Phys.A:Math. Gen.34,1(2001).
[2℄ F.A.B. Coutinhoand J.FernandoPerez,Phys.Rev.D
49,2092(1994).
[3℄ J.Ausdretsh,U.Jasper,andV.D.Skarzhinsky,J.Phys.
A:Math.Gen28,2359(1995).
[4℄ M.AbramowitzandI.A.Stegun,Handbook
Mathemat-ialFuntions,DoverPubliations,NewYork(1968).
