The Linear Potential Propagator via
Wave Funtion Expansion
Ant^onio B. Nassar, (1;2)
Jose M. F. Bassalo, (2)
Paulo T. S. Alenar, (2)
Jose L. M. Lopes, (2)
Jose I.F. de Oliveira, (3)
and Mauro S.D. Cattani (4)
(1)
ExtensionProgram-DepartmentofSienes, UniversityofCalifornia,LosAngeles, California90024, USA
(2)
Departamento deFsiadaUFPA,66075-900, Belem, Para,Brazil
(3)
DepartamentodeFsiadaUNAMA,66050-000, Belem,Para,Brazil
(4)
Institutode FsiadaUSP,C.P.66318,05315-970, S~aoPaulo,SP, Brazil
Reeivedon3June,2002
Weevaluatethequantumpropagatorforthemotionofapartileinalinearpotentialviaareently
developpedformalism[A.B.Nassaret al.,Phys. Rev. E56, 1230, (1997)℄. Inthis formalism, the
propagatoromesabout asa typeof expansionofthe wavefuntionoverthe spaeof theinitial
veloities.
I Introdution
Thequantumpropagatorforthemotionofapartilein
alinearpotentialhasbeenreentlydisussedbyseveral
authors[1-4℄using dierenttehniques. Thisarsenalof
tehniques has an immediate pedagogial appeal and
anbevaluable in somefuture alulation whih may
requiretheuseofanonstandardapproah.
Here we present an alternative approah for the
evaluationofthelinearpotentialpropagatorviaanew
formalismdevelopedbyNassaretal.[5℄Inthis
formal-ism theentralresultis
K(x;t;x
o ;t
o
) = (m=2 ~) Z
+1
1 dv
o
F(x;t;x
o ;t
o ;v
o )e
i
~ S
q uantum (x;t;x
o ;t
o ;v
o )
; (1)
where
F(x;t;x
o ;t
o ;v
o ) =
a(t)
a
o
1=2
exp h
ima(t)_
2~a(t)
1
4a 2
(t)
[x X(t)℄ 2
+ im
_
X(t)
~
[x X(t)℄ i
; (2)
S
quantum (x;t;x
o ;t
o ;v
o ) =
Z
t
o dt
0
1
2 m
_
X 2
(t 0
) V[X(t 0
)℄
~ 2
4ma 2
(t 0
)
; (3)
X(t)isthesolutiontothelassialequation
X = 1
m V
0
[X(t);t℄; (4)
anda(t)isthesolutionto thequantumequation
a +
1
V 00
[X(t);t℄
a = ~
2
2 3
Theterms onthe right-hand side ofEquations(4)
and (5) are due the presene of the lassial and the
quantum potential,respetively.
Equation(1) provides thepropagator asa typeof
expansion of the wave funtion over the spae of the
initial veloities. It is worth notiing the presene of
the quantum potentialin the quantum ation
(Equa-tion(3))andintheexpansionoeÆient(Equation(2))
throughthefuntiona(t):
II Linear Potential
Inwhat follows,wealulate in somedetailthe
quan-tum propagator for the linearpotential V = f x
through the formalism given by Equations (1,2,3),
therefore demonstrating its validity and usefulness as
anewtoolforfutureinvestigations.
First,wendthesolutionstoEquations(4)and(5),
subjettotheinitial onditions
X(0) = x
o ;
_
X
o = v
o
; a(0)=a
o
; a(0)_ = 0; (6)
to be
X(t) = f
2m t
2
+ v
o t + x
o
; (7)
a(t) = a
o p
1 + (t=) 2
; (8)
where = 2ma
2
o
~ .
Equations(7)and(8)allowustoarryoutinfulltheargumentsoftheexponentofEquations(1,2,3)asfollows.
TherstandtheseondargumentsofEquation(2)anbeexpanded,respetively,as
ima(t)_
2~a(t)
1
4a 2
(t)
[x X(t)℄ 2
=
1
4a 2
o
[1 + (t=) 2
℄ [
it~
2a 2
o m
1℄
h
(x x
o )
2
+ f
2
t 4
4m 2
+ v 2
o t
2
+ ft
2
m v
o
t (x x
o )
f t 2
m
2v
o t
i
; (9)
im _
X(t)
~
[x X(t)℄ = im
~
[(x x
o )v
o v
2
o t
+(x x
o )[
f t
m f
2
t 3
2m 2
3f t 2
2m
℄: (10)
Likewise, thequantumation(Equation(3))anbeexpressedas
Z
t
o dt
0
1
2 m
_
X 2
(t 0
) + f X(t 0
)
~ 2
4ma 2
(t 0
)
=
= 1
2 mv
2
o
+ f x
o +
f 2
t 2
m
+ 2f tv
o
~ 2
4ma 2
o
[1 + (t=) 2
℄
: (11)
ThepropagatorinEquation(1)isofthegeneralform
K = D(t) Z
+1
1 dv
o exp
A(t) v 2
o
+ B(t)v
o
+C(t)
=
= D(t) r
A(t) exp
B 2
(t)
4A(t)
+ C(t)
;
where theoeÆientsA(t),B(t), C(t)andD(t)are,respetively:
A(t) = h
t 2
4a 2
+
imt
~[1 + (t=) 2
℄ i
; (12)
B(t) = h
t
2 +
im
2 ih
(x x
o )
f t 2
i
C(t) = im
2~ v
2
o t
i
2 tan
1
(t=) + i
~ f x
o t +
i
~ f
2
t 3
3m +
i
~ v
o f t
2
; (14)
D(t) = m
2 ~ [
a(t)
a
o ℄
1=2
: (15)
Thetermsin(x x
o )
2
,(x+ x
o
),andindependentofx andx
o
are,respetively,
i
~ m
2t
(x x
o )
2
; (16)
i
2~
f t(x+x
o
); (17)
i
~ f
2
t 3
24m i
2 tan
1
(t=): (18)
Thisallowsustowrite
K(x;t;x
o ;0) =
m
2 i ~t
1=2
exp h
i
~ [
m
2t
(x x
o )
2
+ f t
2
(x + x
o )
f 2
t 3
24m ℄
i
; (19)
whihistheresultfoundin Refs. [1-4℄.
d
Wehavepresented anewapproahfor the
evalua-tionofthelinearpotentialpropagator. Inthis
formal-ism,thepropagatoromesaboutasatypeofexpansion
ofthewavefuntionoverthespaeoftheinitial
velo-ities. Thisnewapproahhasapedagogialappealand
posesanalternativerouteforfutureresearh
investiga-tions.
Referenes
[2℄ G.P.Arrighini,N.L.DuranteandC.Guidotti, Am.
J.Phys.64,1036(1996).
[3℄ R.W.Robinett, Am.J.Phys.64,803(1996).
[4℄ L. S. Brown and Y. Zhang, Am. J. Phys. 62, 806
(1994).
[5℄ A. B. Nassar, J. M. F. Bassalo, P. T. S.Alenar, L.
S.G.Canela andM. Cattani, Phys.Rev.E56, 1230
