Con inuum coupling in one-dimensional sca e ing using a ans o med ha monic oscilla o basis
I. Ma el,1F. Pe
´ ez-Be nal,1M. Rod ı
´guez-Galla do,2J. M. A ias,2and J. Go
´mez-Camacho2
1Depa amen o de Fı
´sica Aplicada, Uni e sidad de Huel a, 21071 Huel a, Spain
2Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , Facul ad de Fı
´sica, Uni e sidad de Se illa, Apa ado 1065,
41080 Se illa, Spain
共Recei ed 11 Decembe 2001; published 24 Ap il 2002兲
The coupling o he con inuum is s udied in a one-dimensional p oblem ha desc ibes he in e ac ion o a
weakly bound composi e objec wi h a wall in a semiclassical app oach. A ans o med ha monic oscilla o
basis is in oduced o p o ide an app op ia e disc e e and ini e basis o ea ing he con inuum pa o he
spec um. The con e gence o he sca e ing magni udes is in es iga ed as he numbe o s a es in he basis is
inc eased. The ole o bound- o-con inuum and con inuum- o-con inuum coupling is in es iga ed.
DOI: 10.1103/PhysRe A.65.052708 PACS numbe 共s兲: 03.65.Nk, 03.65.Sq, 24.10.Eq, 34.10.⫹x
I. INTRODUCTION
A composi e quan um-mechanical objec is desc ibed by
an in e nal Hamil onian ha includes he kine ic ene gy o
he cons i uen s, as well as he in e ac ions be ween hese.
The eigens a es o he in e nal Hamil onian will be gi en, in
gene al, by a ini e 共o , a leas , disc e e兲numbe o bound
s a es, and a con inuum o b eakup s a es, which can be cha -
ac e ized by he ela i e momen um o he agmen s. When
such an objec , ini ially in i s g ound s a e, which is bound,
unde goes a sca e ing p ocess om a s uc u eless a ge he
dynamics o he sys em is go e ned by he o al Hamil onian
ha includes he in e nal Hamil onian o he objec plus he
in e ac ion wi h he a ge . As a esul , he objec may be
exci ed o o he bound s a es, o o he con inuum o b eakup
s a es. E en i he objec ends up in he g ound s a e, he
sca e ing magni udes will be a ec ed by he coupling o
bound and b eakup s a es.
The e ec o coupling o bound s a es can be desc ibed by
means o a coupled-channels calcula ion. In a ime-
independen o malism, i in ol es he solu ion o a ini e
numbe o second-o de coupled di e en ial equa ions on he
ela i e coo dina e, which appea as a esul o p ojec ing he
Sch o
¨dinge equa ion on he bound wa e unc ions. In a
semiclassical ime-dependen o malism, one has o sol e a
ini e numbe o i s -o de coupled di e en ial equa ions on
he ime a iable. In bo h cases, he p ocedu e is a he
s aigh o wa d, al hough i may be compu a ionally di icul ,
i many bound s a es a e conside ed.
The e ec o coupling o b eakup s a es is mo e di icul
o desc ibe. The con inuum wa e unc ions ha e an in ini e
ange and a e no no malizable. Thus, he coupling po en ials
om bound s a es o he con inuum s a es ha e a e y long
ange, and he coupling po en ials om con inuum- o-
con inuum s a es ha e an in ini e ange. Tha makes i nec-
essa y o use some disc e iza ion p ocedu e o subs i u e he
con inuum o b eakup s a es by a ini e numbe o no maliz-
able s a es, which, in he adequa e limi , should ep esen he
e ec o coupling o he ue con inuum. Se e al me hods
ha e been p oposed o his pu pose. The R-ma ix me hod
关1兴sol es he many-body p oblem in a box and hen make
he ma ching wi h he adequa e bounda y condi ions. The
S u mian basis 关2–4兴uses bound s a es o scaled po en ials,
which a e o hogonal when weigh ed wi h he po en ials. The
Siege pseudos a e o mula ion 关5兴p o ides a ini e basis
ep esen a ion o he ou going wa e solu ions o he adial
Sch o
¨dinge equa ion o cu o po en ials. The Gamow
s a es 关6兴a e non-no malizable solu ions o he Sch o
¨dinge
equa ion co esponding o ou going bounda y condi ions
cha ac e ized by complex ene gies. The me hod o con-
inuum disc e iza ion coupled channels 关7兴disc e izes he
con inuum by means o aking ixed in e als, o bins, o k
alues in he con inuum s a es. Finally, a comple e basis o
single pa icle wa e unc ions, such as he ha monic oscilla-
o , can be used o expand bo h bound and sca e ing s a es
关8兴.
We ha e ecen ly p oposed he use o a ans o med ha -
monic oscilla o 共THO兲basis o desc ibe he e ec o he
con inuum 关9,10兴. The basic idea is o de ine a local scale
ans o ma ion 关11–13兴, which is such ha con e s he
g ound-s a e wa e unc ion o he weakly bound composi e
objec
␺
B(x) in o a ha monic oscilla o wa e unc ion
␾
0
HO(s)关9,10兴. The unc ion s(x), which de ines he local
scale ans o ma ion, is gi en, o a one-dimensional p ob-
lem, by
冕
⫺⬁
x
兩
␺
B共x⬘兲
兩
2dx⬘⫽
冕
⫺⬁
s
兩
␾
0
HO共s⬘兲
兩
2ds⬘⫽1⫹e 共s兲
2.
共1兲
Then, one gene a es a se o o hogonal wa e unc ions
␾
n
THO(x)⫽Hn„s(x)…
␺
B(x), such ha he s a e wi h n⫽0 co-
incides wi h he g ound s a e, and he s a es wi h n⬎0 de-
sc ibe he con inuum, o o he bound s a es i hey exis .
Then, one akes a ini e basis, which is uniquely de e mined
by he numbe N⫹1 o THO s a es conside ed, and diago-
nalizes he Hamil onian in his basis. The esul ing eigen-
s a es and eigen alues a e aken as ep esen a i es o he
con inuum. We showed ha , as he numbe o s a es in he
THO basis inc eases, he eigens a es appea mo e densely
packed close o he b eakup h eshold, al hough he e a e
eigens a es ha appea a highe ene gies. Besides, we dem-
ons a ed ha global s uc u e magni udes ela ed o he cou-
pling o he con inuum, such as sum ules, we e e y accu-
a ely desc ibed using ela i ely small THO bases. Ou
pu pose in his pape is o s udy he adequacy o he THO
PHYSICAL REVIEW A, VOLUME 65, 052708
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basis o desc ibe he e ec o he con inuum in he sca e ing
p ocesses. We make use o he semiclassical app oxima ion
in which he ela i e mo ion o p ojec ile and a ge is de-
sc ibed by classical ajec o ies. This app oach is alid when
he wa eleng h associa ed o he ela i e mo ion is small
compa ed o he ange o he in e ac ion 关14兴. This is he case
o collisions o hea y nuclei, a oms, and molecules on a
wide ange o ene gies.
In his pape we make use o he THO basis o desc ibe
he e ec o coupling o he con inuum in a model one-
dimensional p oblem. In Sec. II we p esen he model Hamil-
onian, we in oduce he semiclassical app oxima ion o de-
sc ibe he sca e ing, and o mula e he adiaba ic and sudden
app oxima ions, which allow o an exac solu ion. In Sec.
III we p esen he sca e ing calcula ion in he THO basis,
and in es iga e he con e gence o elas ic and b eakup p ob-
abili ies as he numbe o s a es in he THO basis is in-
c eased. In Sec. IV we in es iga e he e ec o including o
neglec ing he e ec o con inuum- o-con inuum coupling.
Sec. V is o he summa y and conclusions.
II. ONE-DIMENSIONAL SCATTERING MODEL
In his wo k we discuss he applica ion o he THO basis
o a sca e ing p oblem. We conside a one-dimensional com-
posi e objec , cha ac e ized by wo pa icles wi h masses
m1,m2, and coo dina es x1and x2. Thei educed mass is
␮
⫽m1m2/(m1⫹m2) and he o al mass is M⫽m1⫹m2. The
ela i e coo dina e is x⫽x1⫺x2, and he cen e o mass co-
o dina e is X⫽(m1x1⫹m2x2)/M. The co esponding Hamil-
onian is gi en by
h⫽⫺ ប2
2
␮
d2
dx2⫹ B共x兲,共2兲
whe e he xis he ela i e coo dina e and B(x) is he in e -
ac ion ha binds he pa icles. Ini ially, he composi e objec
is in i s g ound s a e
␺
B(x), which is an eigens a e o h
co esponding o an ene gy eB. This objec collides wi h a
massi e pa icle, o wall. The in e ac ion o he sys em wi h
he wall is gi en by a unc ion V(X,x), which depends on
he cen e o mass as well as on he in e nal coo dina e. I he
pa icles in e ac independen ly wi h he wall, hen V(X,x)
⫽V1(x1)⫹V2(x2), bu his will no be ue in gene al, i
he e a e pola iza ion e ec s. Thus, he comple e Hamil-
onian can be w i en as
H⫽⫺ ប2
2M
d2
dX2⫹V共X,x兲⫹h.共3兲
The xdependence o he in e ac ion V(X,x) can be ex-
panded in e ms o a amily o o hogonal polynomials
Pm(x), whe e m ep esen s he o de o he polynomial.
These polynomials a e o hogonal wi h espec o he weigh
unc ion gi en by
␺
B(x)2, so ha
冕
dx
␺
B共x兲2Pm共x兲Pn共x兲⫽
␦
共n,m兲.共4兲
Explici exp essions o he i s ew polynomials a e, in
e ms o he expec a ion alues o xnin he g ound s a e, and
assuming ha
␺
B(x)⫽
␺
B(⫺x),
P0共x兲⫽1, 共5兲
P1共x兲⫽x
冑
具
x2
典
,共6兲
P2共x兲⫽x2⫺
具
x2
典
冑
具
x4
典
⫺
具
x2
典
2.共7兲
Thus, he in e ac ion can be expanded as
V共X,x兲⫽兺
mVm共X兲Pm共x兲,共8兲
Vm共X兲⫽
冕
dx
␺
B共x兲2Pm共x兲V共X,x兲.共9兲
I should be no iced ha he i s e m in his expansion,
which is independen o he in e nal a iable x, co esponds
o he expec a ion alue o he in e ac ion V(X,x) in he
g ound s a e o he composi e objec , which is he olding
po en ial. This is gi en by
V 共X兲⫽V0共X兲⫽
冕
dx
␺
B共x兲2V共X,x兲.共10兲
The o he e ms gi e ise o he idal o ces, which can in-
duce he exci a ion o he composi e objec du ing he colli-
sion. Fo he pu pose o his pape , we will conside he case
in which he composi e objec consis s on wo iden ical pa -
icles. Then, he unc ion V(X,x) is e en in x, and only he
polynomials o e en o de con ibu e o he expansion.
Mo eo e , o he sake o simplici y, we will e ain only he
e ms up o m⫽2. Thus, we ha e
V共X,x兲⫽V 共X兲⫹V2共X兲P2共x兲.共11兲
In a semiclassical app oach 关14兴, he olding po en ial de e -
mines he ajec o y X( ) o he cen e o mass o he objec .
The ajec o y can be ob ained by sol ing he di e en ial
equa ion
M
2
冉
dX共 兲
d
冊
2
⫹V 共X兲⫽E⫺eB.共12兲
The u ning poin o he classical ajec o y X0occu s when
E⫺eB⫽V (X0). I he ime ⫽0 is aken when X( )⫽X0,
hen he ajec o y X( ) is an e en unc ion o he ime. The
idal po en ial, which is esponsible o he p ojec ile exci a-
ion, is gi en by
VT共X,x兲⫽V2共X兲P2共x兲.共13兲
In a semiclassical ea men , he ajec o y X( ) is used o
conside he idal po en ial VT„X( ),x…as a ime-dependen
ope a o ha ac s on he in e nal coo dina e x. Mo eo e , in
he case ha we a e conside ing, he dependence in he cen-
I. MARTEL e al. PHYSICAL REVIEW A 65 052708
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e o mass and ela i e coo dina es ac o ize, so ha he
e ec o he idal o ces a e desc ibed by he ope a o P2(x),
which ac s wi h an in ensi y F( )⫽V2„X( )…. The in e nal
s a e will e ol e sa is ying he equa ion
iបd
d
␺
共x, 兲⫽关h⫺eB⫹F共 兲P2共x兲兴
␺
共x, 兲,共14兲
wi h he bounda y condi ion ha o →⫺⬁, he wa e unc-
ion is ha o he g ound s a e
␺
B(x).
Recapi ula ing, Eq. 共14兲 ep esen s, wi hin some eason-
able app oxima ions, he ime e olu ion o he in e nal s a e
o a wo-pa icle sys em ha collides wi h a wall. This equa-
ion con ains de i a i es wi h espec o and x, and so i is
di icul o sol e i exac ly. Besides, as he eigens a es o h
con ain bo h bound and con inuum s a es, one canno jus
p ojec on he eigens a es o hand sol e he coupled equa-
ions.
We will show in he ollowing sec ion ha he THO
me hod p o ides a ini e basis o no malizable s a es, which
allow o ind an app oxima e solu ion o Eq. 共14兲. Besides, as
he numbe o THO s a es inc eases, he ele an sca e ing
magni udes con e ge.
We will also conside wo dynamical app oxima ions o
Eq. 共14兲. The adiaba ic app oxima ion a ises when he cha -
ac e is ic ime scale o he in e ac ion, gi en by he ime
ange o he unc ion F( ), is much longe han he ime
scale o he in e nal mo ion, ប/eB. In his limi , he ime-
dependen wa e unc ion can be app oxima ed by he exp es-
sion
␺
Ad共x, 兲⫽N共 兲
冋
␺
B共x兲⫺F共 兲1
h⫺eBP2共x兲
␺
B共x兲
册
⫻exp关⫺i
␾
共 兲兴,共15兲
whe e N( ) is a no maliza ion ac o , which a ies slowly,
and he phase
␾
( ) sa is ies he equa ion
␾
共 兲⫽⫺
␣
ប
冕
⫺⬁
d ⬘F共 ⬘兲2.共16兲
The pa ame e
␣
is he pola izabili y associa ed o he ope a-
o P2(x). I is gi en by he exp ession
␣
⫽
具
␺
B
兩
P2共x兲1
h⫺eBP2共x兲
兩
␺
B
典
.共17兲
I should be no iced ha , in he adiaba ic app oxima ion, he
objec always eme ges om he sca e ing p ocess in i s
g ound s a e. Indeed, he unc ion F( ), which is associa ed
o he couplings, anishes as →⬁. The only e ec ha
a ises om he coupling is a phase shi in he g ound-s a e
wa e unc ion. This phase shi is de e mined by he alue o
he pola izabili y
␣
. So, he adequacy o any app oxima e
ea men o con inuum disc e iza ion can be judged by com-
pa ing he alue o
␣
ob ained om he disc e iza ion wi h
he exac alue. We will make his compa ison in he ollow-
ing sec ion o he THO basis.
The sudden app oxima ion is opposi e o he adiaba ic
one. I a ises when he ime scale o he in e ac ion is much
sho e han ha o he in e nal mo ion. Thus, he in e nal
coo dina es a e e ec i ely ozen du ing he sca e ing. The
sudden app oxima ion is ob ained by igno ing he e m h
⫺eBin Eq. 共14兲. Tha allows o in eg a e wi h espec o he
ime a iable, o gi e
␺
Su共x, 兲⫽
␺
B共x兲exp关⫺i
␾
共 兲P2共x兲兴,共18兲
whe e he phase
␾
( ) is gi en by
␾
共 兲⫽1
ប
冕
⫺⬁
d ⬘F共 ⬘兲.共19兲
In he sudden app oxima ion he objec eme ges om he
sca e ing p ocess in a s a e whose densi y dis ibu ion is he
same as ha o he g ound s a e. Howe e , i can be qui e
di e en om he g ound s a e because he addi ional phase
depends on he a iable x. The p obabili y ampli ude o e-
maining in he g ound s a e is gi en by
具
␾
B
兩
A共⌽兲
兩
␾
B
典
⫽
冕
dx
␺
B共x兲2exp关⫺i⌽P2共x兲兴,共20兲
whe e ⌽⫽
␾
(⬁). I he con igu a ion space is es ic ed, by
means o some con inuum disc e iza ion p ocedu e, he ex-
p ession abo e will be modi ied. In he ollowing sec ion we
will e alua e he con e gence o he elas ic ampli udes in he
THO basis, as a unc ion o ⌽. No e ha he ⌽, which is
associa ed o he in eg al o he coupling po en ial along he
ajec o y, is a dimensionless pa ame e ha measu es he
impo ance o he coupling. Small alues o ⌽indica e ha
he elas ic sca e ing domina es, while la ge alues o ⌽im-
ply ha exci a ion domina es.
III. SCATTERING CALCULATIONS IN THE THO BASIS
We make use o he THO basis o expand he wa e unc-
ion
␺
(x, ). The THO basis 关9兴is ob ained om he g ound-
s a e wa e unc ion by he exp ession
␾
n
THO共x兲⫽Hn„s共x兲…
␺
B共x兲,共21兲
whe e Hn(s) is a p ope ly no malized He mi e polynomial,
and s(x) is gi en by Eq. 共1兲. Fo hese calcula ions, he
binding in e ac ion has been aken as a Po
¨schl-Telle po en-
ial 关15兴, gi en by
B共x兲⫽ 0/cosh2共
␤
x兲.共22兲
The po en ial dep h 0⫽⫺ប2
␤
2/
␮
is aken so ha i only
accommoda es one bound s a e, which has an ene gy eB⫽
⫺ប2
␤
2/2
␮
, and ha is gi en by he analy ic wa e unc ion
␺
B共x兲⫽N/cosh共
␤
x兲.共23兲
We can use he THO basis 共21兲, wi h n⫽0,...,N, o
diagonalize he Hamil onian h. This gi es ise o N⫹1
eigens a es
␺
j
h(x), whose co esponding eigen alues a e ej,
o j⫽0 oN. The s a e wi h j⫽0 is p ecisely he g ound
CONTINUUM COUPLING IN ONE-DIMENSIONAL... PHYSICAL REVIEW A 65 052708
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s a e
␺
B(x), which coincides wi h
␾
0
THO(x). The o he s a es
a e no malizable s a es, which ep esen he con inuum in he
THO basis.
In Fig. 1 we p esen he alues o he ene gies eiob ained
om he diagonaliza ion o he in e nal Hamil onian in he
THO basis as a unc ion o he numbe o s a es included in
he THO basis. The ene gy scale is in uni s o ប2
␤
2/
␮
,so
ha he bound s a e has eB⫽⫺1/2. In Fig. 2 we p esen he
en eigen unc ions o hcons uc ed om he THO basis wi h
N⫹1⫽10 共nine con inuum s a es plus he bound g ound
s a e兲. We only had o include he wa e unc ions wi h posi-
i e pa i y, which a e he ones connec ed by he in e ac ion.
The ma ix elemen s o he in e ac ion a e p opo ional o
he ma ix elemen s o he ope a o P2(x). These ma ix el-
emen s can be calcula ed in he THO basis as
具
␾
n
THO
兩
P2
兩
␾
m
THO
典
⫽
冕
dx
␺
B共x兲2Hn„s共x兲…P2共x兲Hm„s共x兲….
共24兲
F om his ma ix, one can also calcula e he ma ix elemen s
o P2in he basis o eigens a es o he Hamil onian
兩
␺
j
h
典
.
No e ha one can diagonalize he ope a o P2(x) in he THO
basis. Le us label he eigens a es by
兩
␺
k
P2
典
and he eigen al-
ues P2(k).
We will s udy he adequacy o he THO basis o desc ibe
he pola izabili y
␣
. As we ha e a gued in he p eceding
sec ion, his is ele an o he desc ip ion o he sca e ing
p ocess in he adiaba ic limi . In he THO basis, he exp es-
sion o he pola izabili y is
␣
共THO兲⫽兺
j⫽
”0
具
␺
B
兩
P2
兩
␺
j
h
典具
␺
j
h
兩
P2
兩
␺
B
典
ej⫺eB.共25兲
In Table I we p esen he con e gence o his magni ude,
exp essed in uni s o
兩
eB
兩
⫺1, as a unc ion o he numbe o
THO s a es. As we can see, he con e gence is e y as .
We will nex conside he THO basis o desc ibe he elas-
ic sca e ing ampli udes in he sudden app oxima ion. The
exp ession co esponding o Eq. 共20兲in he THO basis can
be o mula ed using he eigens a es o he ope a o P2(x)as
具
␾
B
兩
A共⌽兲
兩
␾
B
典
⫽兺
k
具
␺
B
兩
␺
k
P2
典
2exp关⫺i⌽P2共k兲兴.
共26兲
The p obabili y o emaining in he g ound s a e is gi en
by he squa e o his ampli ude. The esul s a e plo ed in
Fig. 3, as a unc ion o ⌽. They indica e ha he numbe o
THO s a es needed o ob ain he ull sudden calcula ion in-
c eases as he coupling s eng h ⌽inc eases.
We inally conside he gene al case, in which we do no
make use o he adiaba ic o sudden app oxima ions. Fo he
pu pose o he calcula ions, we assume ha he olding po-
en ial can be app oxima ed by an exponen ial o m, o dis-
ances beyond he u ning poin
VF共X兲⫽VF共X0兲exp关⫺共X⫺X0兲/a兴.共27兲
I is s aigh o wa d o ob ain he ajec o y in his case. In
e ms o he a iable y⫽(X⫺X0)/a, one ge s
anh共 /2a兲⫽⫾
冑
1⫺exp关⫺y共 兲兴,共28兲
whe e ⫽
冑
2(E⫺eB)/Mis he asymp o ic eloci y. In his
equa ion, ⫽0 co esponds o he dis ance o closes ap-
TABLE I. Con e gence o he pola izabili y
␣
, in uni s o
(2
␮
/ប2
␤
2), as a unc ion o he numbe o posi i e pa i y con-
inuum s a es in he THO basis.
N
␣
1 0.52155
2 0.70895
3 0.71720
4 0.71721
5 0.71721
Exac 0.71721
FIG. 1. Ene gy eigen alues o he Hamil onian o he composi e
objec as a unc ion o he numbe o s a es, N⫹1, in he THO
basis. Ene gies a e gi en in uni s o (ប2
␤
2/
␮
).
FIG. 2. Wa e unc ions o he composi e objec in he THO
basis wi h N⫹1⫽10, exp essed as a unc ion o
␤
x共dimension-
less兲.
I. MARTEL e al. PHYSICAL REVIEW A 65 052708
052708-4
p oach, y⫽0. I should be no iced ha he dep h o he po-
en ial VFonly de e mines he dis ance o closes app oach.
The ajec o y, measu ed wi h espec o X0, only depends on
he ange a.
We will also assume ha he coupling e m is also expo-
nen ial, wi h he same ange as he olding po en ial. Then,
V2共X兲⫽V2共X0兲exp关⫺共X⫺X0兲/a兴.共29兲
I his is exp essed in e ms o he ime, one ge s
F共 兲⫽V2„X共 兲…⫽V2共X0兲cosh⫺2共 /2a兲.共30兲
F om his exp ession, one ge s ha he cha ac e is ic ime o
he collision is gi en by Tc⫽a/ . This ime is o be com-
pa ed wi h a cha ac e is ic ime o he in e nal mo ion,
which is Ti⫽ប/
兩
eB
兩
. So, we de ine an adiaba ici y pa ame e
␰
⫽Tc/Ti. Small alues o
␰
co espond o he sudden limi ,
and la ge alues o he adiaba ic limi . Besides, we will de-
ine a dimensionless ime
␶
⫽ /Ti. The alue o ⌽ o F( )
is 4V2(X0)a/(ប ). Thus, we can w i e he in e ac ion, in
e ms o sui able dimensionless pa ame e s, as
F共
␶
兲⫽⌽
兩
eB
兩
4
␰
cosh⫺2关
␶
/共2
␰
兲兴.共31兲
Wi h his exp ession, he equa ion o he e olu ion can be
w i en as
id
d
␶␺
共x,
␶
兲⫽
冋
h⫺eB
兩
eB
兩
⫹⌽
4
␰
cosh⫺2关
␶
/共2
␰
兲兴P2共x兲
册
␺
共x,
␶
兲.
共32兲
We expand he unc ion in e ms o he eigens a es o hin a
THO basis. This gi es
␺
共x,
␶
兲⫽兺
j⫽0
N
cj共
␶
兲
␺
j
h共x兲exp关⫺i
␶
共ej⫺eB兲/
兩
eB
兩
兴.共33兲
Subs i u ing his expansion in he p e ious equa ion, and p o-
jec ing wi h
␺
i
h(x) one ge s,
id
d
␶
ci共
␶
兲⫽兺
j⫽0
N⌽
4
␰
cosh⫺2关
␶
/共2
␰
兲兴
具
␺
i
h
兩
P2
兩
␺
j
h
典
cj共
␶
兲
⫻exp关⫺i
␶
共ej⫺ei兲/
兩
eB
兩
兴,共34兲
wi h he bounda y condi ion ha , o
␶
→⫺⬁, only c0(
␶
)
⫽1 and he o he componen s anish.
The p obabili y o emaining in he g ound s a e, a e he
sca e ing, is gi en by
兩
c0(⬁)
兩
2. This alue depends on he
pa ame e ⌽, which measu es he coupling s eng h, and he
pa ame e
␰
, which measu es he deg ee o adiaba ici y. In
Fig. 4 we ep esen he alue o he g ound-s a e p obabili y
e sus
␰
, o a ixed alue o he coupling s eng h pa ame e
⌽⫽1, calcula ed in he THO basis. We see ha he esul s
con e ge apidly as he numbe o THO s a es inc eases.
Only when he adiaba ici y pa ame e is e y small (
␰
⬍0.1) he con e gence is no so as . We ha e pe o med
calcula ions o o he alues o he coupling s eng h, and we
ind ha he con e gence o he THO sca e ing calcula ions
is e y good excep o he cases in which bo h he coupling
s eng h is la ge and he adiaba ici y pa ame e is small, his
is, o s ong coupling e y close o he sudden limi .
We ha e e alua ed he a e age alue o he ene gy o he
b eakup s a es ha a e p oduced a e he sca e ing p ocess,
weigh ed by he co esponding exci a ion p obabili ies. The
esul s in Fig. 5 show ha he a e age exci a ion ene gy in-
c eases as one goes o he sudden limi . Tha indica es ha
one should be ca e ul when applying he sudden app oxima-
ion, which implies neglec ing he exci a ion ene gy, e en in
cases in which he adiaba ici y pa ame e is small. We ind
ha he con e gence o he THO calcula ion is sa is ac o y,
FIG. 3. G ound-s a e p obabili y in he sudden limi as a unc-
ion o he pa ame e ⌽共dimensionless兲. The hick ull line is he
ull sudden esul . The dashed lines co espond o he THO disc e i-
za ion, o se e al numbe s o s a es. The hin ull line is he sudden
calcula ion excluding con inuum- o-con inuum coupling.
FIG. 4. B eakup p obabili y as a unc ion o he adiaba ici y
pa ame e
␰
共dimensionless兲, o a ixed alue o he coupling pa-
ame e ⌽⫽1. Lines co esponding o hose ma ked in he legend
box as THO a e he ull THO esul s o di e en numbe s o s a es
included in he basis. The hin ull line is he con e ged esul
excluding con inuum- o-con inuum coupling.
CONTINUUM COUPLING IN ONE-DIMENSIONAL... PHYSICAL REVIEW A 65 052708
052708-5

excep o
␰
⬇0.5. The e he b eakup p obabili ies a e e y
small, and his induces unce ain ies in he e alua ion o he
a e age ene gy.
We ha e in es iga ed he b eakup p obabili y dis ibu ion
ob ained making use o he THO disc e iza ion. We ha e
calcula ed app oxima ely he di e en ial p obabili y o exci-
a ion as a unc ion o he ela i e momen um o he b eakup
agmen s. The e alua ion o his unc ion equi es o es i-
ma e he in e al ha co esponds, in he ue con inuum, o
each eigens a e o he in e nal Hamil onian in he THO basis.
The a e age momen um o ou con inuum s a es can be ob-
ained in e ms o he ene gy by means o pi
2/2
␮
⫽ei.A
simple in e pola ion p ocedu e yields he ange in e ms o
he momen a o he neighbo ing s a es,
⌬i⫽1
2pi⫹1⫺1
2pi⫺1,2⬍i⬍N,共35兲
⌬1⫽1
2p2,共36兲
⌬N⫽⫺2pN⫺1⫹3
2pN⫹1
2pN⫺2.共37兲
Thus, we can exp ess he di e en ial exci a ion p obabili y a
ene gies close o he THO eigen alues by
冉
dP
dp
冊
p⫽pj
⫽Pj
⌬j.共38兲
In Fig. 6 we p esen he b eakup p obabili y dis ibu ion ob-
ained in THO calcula ions using di e en numbe o s a es.
The coupling pa ame e is aken as ⌽⫽1 as in he p eceding
calcula ion. The adiaba ici y pa ame e is aken as
␰
⫽0.15,
which co espond o an in e media e si ua ion be ween he
adiaba ic and sudden limi s. We can see ha he e is a ea-
sonable con e gence o he calcula ions wi h di e en num-
be o s a es. We ha e also pe o med calcula ions wi h di -
e en alues o he coupling s eng h and he adiaba ici y
pa ame e . We ind ha la ge coupling s eng h inc eases he
b eakup, bu does no modi y he o m o he momen um
dis ibu ion o he agmen s. Howe e , la ge adiaba ici y
pa ame e s 共co esponding o slowe collisions兲mo e he
momen um dis ibu ion o smalle alues. The s uc u e o
he b eakup dis ibu ion shows a single maximum a ce ain
momen um dis ibu ion, wi h he excep ion o ce ain alues
o he adiaba ici y pa ame e o each coupling s eng h, o
which he b eakup p obabili y is e y small, and he dis i-
bu ions shows wo maxima.
IV. EFFECT OF CONTINUUM-TO-CONTINUUM
COUPLING IN THE THO BASIS
Ha ing es ablished ha he THO basis is an adequa e
me hod o desc ibe he con inuum o b eakup s a es, we wan
o assess he ques ion o whe he he con inuum- o-
con inuum coupling is impo an . Fo ha , we ha e pe -
o med calcula ions in he THO basis whe e we ha e igno ed
con inuum- o-con inuum coupling, bo h diagonal and nondi-
agonal. Ou i s obse a ion is ha he calcula ion in he
THO basis con e ges much as e in he cases in which
con inuum- o-con inuum coupling is neglec ed. Usually, i is
enough o in oduce 3 o 4 con inuum s a es in he THO
basis o ge con e gence. In Fig. 3 we ha e p esen ed he
exac sudden esul excluding con inuum- o-con inuum cou-
pling. We ha e p esen ed in Figs. 4, 5, and 6 he calcula ions
wi h nine con inuum s a es in he THO basis, which p ac i-
cally coincide wi h he calcula ions wi h 7 o 8 con inuum
s a es.
I should also be no iced ha , compa ing he calcula ions
in Fig. 3, he b eakup p obabili y is conside ably enhanced
when con inuum- o-con inuum coupling is neglec ed. This is
due, in ou calcula ions, o he p ope ies o he ope a o
FIG. 5. A e age o he ene gy o he b eakup s a es, in uni s o
(ប2
␤
2/
␮
), as a unc ion o he adiaba ici y pa ame e
␰
共dimension-
less兲, o a ixed alue o he coupling pa ame e ⌽⫽1. The lines
co espond o he THO disc e iza ion, o se e al numbe s o s a es.
The hin line is he con e ged esul excluding con inuum- o-
con inuum coupling.
FIG. 6. P obabili y o exci a ion o he con inuum, in uni s o
(ប
␤
)⫺1, as a unc ion o he b eakup momen um, in uni s o (ប
␤
),
o di e en numbe s o s a es in he THO basis. The coupling
s eng h is ⌽⫽1 and he adiaba ici y pa ame e is
␰
⫽0.15. The hin
line is he con e ged esul excluding con inuum- o-con inuum cou-
pling.
I. MARTEL e al. PHYSICAL REVIEW A 65 052708
052708-6
P2(x) in he idal po en ial. The expec a ion alue o his
ope a o anishes o he g ound s a e. Howe e , o con-
inuum s a es, wi h a la ge spa ial ex ension, he expec a ion
alue is posi i e, and hence he idal po en ial has a epulsi e
e ec in hese s a es, and his dec eases he p obabili y o
b eakup.
We ha e obse ed ha his e ec does no only occu in
he sudden limi . Fo ini e alues o he adiaba ici y pa am-
e e , he calcula ions ha igno e con inuum- o-con inuum
coupling gi e in gene al la ge b eakup p obabili ies. This is
shown in Fig. 4. This e ec is mo e acu e as he coupling
s eng h is la ge . Fo weak coupling s eng h ⌽Ⰶ1, he e -
ec o con inuum- o-con inuum coupling ge s smalle .
We ha e also in es iga ed he ene gy dis ibu ion o he
b eakup s a es ob ained in calcula ions ha igno e
con inuum- o-con inuum coupling. As i is shown in Fig. 5,
he a e age ene gy o he b eakup s a es is lowe in he cal-
cula ions ha neglec con inuum- o-con inuum coupling. Be-
sides, as he collision is as e 共adiaba ici y pa ame e
smalle 兲 he inc ease in he exci a ion ene gy is smalle han
when con inuum- o-con inuum coupling is conside ed.
The momen um dis ibu ion o he b eakup agmen s is
also e y di e en in he calcula ions neglec ing con inuum-
o-con inuum coupling, as i is shown in Fig. 6. These calcu-
la ions gi e much la ge b eakup p obabili ies, which a e
concen a ed on small alues o he agmen momen a.
V. SUMMARY AND CONCLUSIONS
In his wo k we ha e made use o a ecen ly p oposed
me hod o disc e ize he con inuum o b eakup s a es o
weakly bound sys ems. The me hod, named THO, p o ides a
basis o no malizable wa e unc ions, which a e gene a ed
by mul iplying he g ound s a e o he composi e objec by a
numbe N⫹1 o He mi e polynomials on a a iable s(x).
This a iable is ob ained as a local scale ans o ma ion om
he physical a iable x. As he numbe Ninc eases, he basis
app oaches comple eness, and he ue con inuum o b eakup
s a es should be accu a ely desc ibed in e ms o he THO
basis.
We ha e in es iga ed he adequacy o he THO basis o
desc ibe he e ec o he coupling o b eakup s a es on he
sca e ing o a composi e objec . We ha e conside ed a one-
dimensional p oblem in which a composi e objec , made up
o wo s uc u eless agmen s ha a e ini ially bound, col-
lides wi h a epulsi e po en ial ha depends bo h on he cen-
e o mass coo dina e and on he ela i e coo dina e o he
agmen s. This po en ial is app oxima ed as he sum o a
olding po en ial, which ac s on he cen e o mass coo di-
na e, and de e mines he classical ajec o y, and a idal po-
en ial, which is w i en as he p oduc o a coupling o m-
ac o imes an ope a o ac ing on he ela i e coo dina e. As
a esul o he collision, he composi e objec can b eakup
p oducing agmen s wi h a ce ain ene gy dis ibu ion. We
use a semiclassical app oxima ion, by which he cen e o
mass desc ibes a ajec o y ha is de e mined by he olding
po en ial. The e olu ion o he in e nal s a e o he composi e
sys em is de e mined by a ime-dependen Sch o
¨dinge equa-
ion, which is p ojec ed on a ini e THO basis.
We ind ha he dynamics o he collision can be cha ac-
e ized wi hin a semiclassical app oxima ion in e ms o wo
dimensionless pa ame e s. One is he coupling s eng h,
which is de ined as he ime in eg al o he coupling o m-
ac o along he ajec o y and he o he is he adiaba ici y
pa ame e , which is he a io o he collision ime and he
cha ac e is ic ime o he in e nal mo ion.
When he collision is slow, he adiaba ici y pa ame e is
la ge and one is in he adiaba ic limi . In his case, he e ec
o coupling o he con inuum is jus o induce a phase change
in he elas ic wa e unc ion. The calcula ions in he THO
basis con e ge e y quickly o he exac esul s in his case.
When he collision is as , he adiaba ici y pa ame e is
small and one is in he sudden limi . In his case, he numbe
o THO s a es needed o ob ain con e gence in he elas ic
p obabili y depends on he alue o he coupling s eng h.
We ha e pe o med calcula ions in in e media e si ua-
ions, o di e en alues o he coupling s eng h and he
adiaba ici y pa ame e s. We ind ha he con e gence in he
THO basis is sa is ac o y, excep in si ua ions in which bo h
he coupling s eng h is la ge and he adiaba ici y pa ame e
small, his is, in si ua ions o s ong coupling close o he
sudden limi .
We ha e used he THO basis o e alua e di e en sca e -
ing magni udes in ou model p oblem. We ind ha he p ob-
abili y o b eakup inc eases, in gene al, as he coupling
s eng h inc eases, and as he adiaba ici y pa ame e de-
c eases. Howe e , he e a e ce ain alues o he adiaba ici y
pa ame e , o each coupling s eng h, o which he b eakup
p obabili y ge s e y small. We ind ha he THO calcula-
ions, wi h di e en numbe s o s a es, p esen consis en ly
his ea u e.
We ha e used he THO basis o e alua e he ene gy dis-
ibu ion o he b eakup s a es. We ind ha he a e age ex-
ci a ion ene gy o he b eakup s a es does no depend e y
much on he coupling s eng h, and inc eases signi ican ly as
he adiaba ici y pa ame e dec eases. We ha e also e alua ed
he ene gy dis ibu ion o he b eakup s a es, inding ha he
dis ibu ion is wide o he lowe adiaba ici y pa ame e .
We ha e in es iga ed he ole o con inuum- o-con inuum
coupling in his p oblem. We ind ha he THO me hod con-
e ges e y as when con inuum- o-con inuum coupling is
neglec ed, e en in he sudden limi . We ind ha , when
con inuum- o-con inuum coupling is neglec ed, he b eakup
p obabili ies a e, in gene al, o e es ima ed. Also, he ene gy
dis ibu ion o he b eakup s a es becomes na owe . We in-
e p e hese esul s as a consequence ha , due o he o m o
he P2(x) ope a o ha gene a es he coupling, he e ec o
con inuum- o-con inuum coupling is epulsi e o he
b eakup s a es, and his educes he e ec o he coupling in
he ull calcula ions. We also ind ha , i con inuum- o-
con inuum coupling is neglec ed, we do no ge he special
alues o he adiaba ici y pa ame e o which b eakup is
e y small. So, we can conclude ha , a leas in he sca e -
ing p oblem unde discussion, con inuum- o-con inuum cou-
pling is e y impo an , and ha he THO basis is a use ul
me hod o s udy i s e ec . I should be no iced ha bo h
calcula ions, wi h con inuum- o-con inuum coupling and
wi hou i , con e ge sa is ac o ily in he THO basis.
CONTINUUM COUPLING IN ONE-DIMENSIONAL... PHYSICAL REVIEW A 65 052708
052708-7
We conclude ha he THO basis is adequa e o desc ibe
he coupling o b eakup s a es in sca e ing p oblems ha can
be ea ed wi hin he semiclassical app oxima ion. Fo a ull
quan um-mechanical calcula ion he THO basis p o ides
wi h a ini e se o no malized s a es ha ep esen he con-
inuum o b eakup s a es. These wa e unc ions a e used o
e alua e diagonal and ansi ion po en ials, which en e in a
s anda d coupled channel calcula ion. We ha e al eady ap-
plied his me hod o desc ibe he sca e ing o deu e ons on
hea y a ge s wi h sa is ac o y esul s 关16兴.
ACKNOWLEDGMENTS
This wo k was suppo ed in pa by he Spanish DGICYT
unde P ojec Nos. PB98-1111 and FPA2000-1592-C03-02.
We acknowledge use ul discussions wi h R. Lio a, C.H.
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