Revista Brasileira de Flsica, Vol. 13, n? 4 , 1983
Quaiitative Features of the Spectrum of One-Dimensional Three-Body Systems
C.P. MALTA,
F.A.B.COUTINHO and J. FERNANDO PEREZ
Instituto de Física, Universidade de São Paulo, Caixa Postal 20516, São Paulo, 01000, SP, Brasil
Recebido em 31 de janeiro de 1984
Atwact
General q u a l i t a t i v e f e a t u r e s o f t h e bound-spectrum o f a one-dimensional system of t h r e e t d e n t i c a l p a r t i c l e s a r e d e d u c e d i n t h e
case of Bose-Einstein, Fermi - D i rac and Bol tzmann s t a t i s t i c s . Using t h e
symmetry p r o p e r t i e s o f o n e - d i m e n s i o n a l h y p e r s p h e r i c a l h a r m o n i c s
components, i t i s shown how t o c o n s t r u c t e i g e n f u n c t i o n s h a v i n g d e f i n i t e
p a r i t y and d e f i n i t e t r a n s f o r m a t i o n p r o p e r t i e s under permutation o f any
particle pair.
1. INTRODUCTION
The N-body problem i s w e l l known t o
c l a s s i c a l and quantum theory.
be u n s o l u b l e i n b o t h
I n p a r t i c u l a r , the t h r e e - b o d y
a1 ready possesses the i n t r a c t a b i 1 i t y o f the general H- b o d y
problem
p r o b l em.
T h i s i n t r a c t a b i l i t y however shouldbe c a r e f u l l y q u a l i f i e d . When
a mathematician says t h a t t h e c l a s s i c a l two-body problem can be solved
he means t h a t aZl p o s s i b l e motions o f a generic two-body
known'.
problem
are
When he says t h a t the c l a s s i c a l p o t e n t l a l system w i t h two de-
grees o f freedom i s "hors deç poss i b i l i t 6 s de l a science contempora ine"
(reference 1 , pg.28) he means t h a t the cannot d e s c r i b e
motions of such a system.
Bruns'
a l l p o s s i b 1e
In the three-body problem t h i s i s so because
theorem 2 says t h a t t h e r e a r e n o t s u f f i c i e n t f i r s t
integrais o f
motion i n i n v o l u t i o n .
Nevertheless, f o r a s p e c i f i c c l a s s i c a l three-body system,
a
computer code can be w r i t t e n t o i n t e g r a t e t h e equations o f motion
determine t h e p o s i t i o n o f t h e system known a t any time. It
is
and
just a
question o f patience and money t o pay t o the computer owner.
What about quantum systems? I n t h i s paper we s e l e c t t h e simp l e s t example o f a many-body system: a quantum system w i t h two degrees
o f freedom, namely, t h r e e i d e n t i c a l i n t e r a c t i n g p a r t i c l e s moving on
a
l i n e . We concentrate on the bound-state spectrum o f the system andshow
t h a t computer I n t e g r a t i o n o f the time-independent schr;dlnger
cannot be attempted b l i n d l y : i t i s necessary t o
equation
separate f i r s t
the
e i g e n f u n c t l o n s according t o the symmetry invariances o f the system. We
thus i l l u s t r a t e , i n a simple c o n t e x t , how symmetry techniques
can be
a p p l i e d and completely worked o u t w i t h moderate ease.
For s o l v i n g the problem we have chosen t h e hyperspherical h a r monics ( o r K -harmonics) ~ e t h o d ~ which
'~,
has been e x t e n s l v e l y ~ s e d ~ ' ~ ,
as i t a l l o w s , i n p r i n c i p l e , f o r a systematic treatment o f the few body
problem. Hcwever,
i t has t h e drawback t h a t t h e h i g h d i m e n s i o n a l i t y
of
the hyperangle makes i t s v i s u a l i s a t i o n r a t h e r d i f f i c u l t . I n o r d e r t o
g a i n a b e t t e r i n s i g h t of the method, Amado and coelho e considered t h e
case o f one-dimensional three-body system, i n t e r a c t i n g
v i a a general
two-body p o t e n t i a l , and obtained an i n f i n i t e s e t o f coupled
d i f f e r e n t i a l equations (see equations
(9) and
ordinary
8
(10) ofArnadoand coelho )
which were then solved i n the p a r t i c u l a r case o f t h r e e i d e n t i c a l
par-
t i c l e s I n t e r a c t i n g v i a an a t t r a c t i v e 6 - p o t e n t i a l .
I n t h i s paper we i n v e s t i g a t e t h e symmetry p r o p e r t i e s ,
under
the permutation group, o f the K-harmonics (bound- state) s o l u t i o n s
the one-diniensional system o f i d e n t i c a l p a r t i c l e s i n t e r a c t i n g
a t t r a c t i v e two-body p o t e n t i a l V(lxi-x
lem o f
IV
via
for
an
.I).
As the one-dimensional prob3
i c l e n t i c a l p a r t i c l e s i n t e r a c t i n g v i a an a t t r a c t i v e 6 - p o t e n t i a l
has a s i n g l e 3 bound- state s o l u t i o n which i s t o t a l 1 y symrnetric under t h e
permutation o f any p a i r o f p a r t i c l e s , t h i s p a r t i c u l a r case w i l l n o t be
considered here. The three- dimensional problem of t h r e e
i n t e r a c t i ng
and
~ ~we~ bel ieve t h a t the s o l p a r t i c l e s i s considerably more i n v o ~ v e d
u t i o n g i v e n here i s o f considerable pedagogical value.
I n Section 2 we make a b r i e f e x p o s i t i o n o f t h e hyperspherical
harmonics niethod f o r the case o f t h r e e i d e n t i c a l p a r t i c l e s ;
independent: schr;dinger
equation i s reduced t o
coupled orclinary d i f f e r e n t l a l equations.
an
I n Section
e f f e c t o f t:he a c t i o n o f the permutation group
on
i n f i n it e
I n Section
set
3 we discuss
the
time
of
the
K- harmon i c s
components and c o n s t r u c t f u n c t i o n s , o f d e f i n i t e p a r i t y , which
a b a s l s for. i r r e d u c i b l e r e p r e s e n t a t i o n s l O ' "
the
provide
o f t h e S 3symmetry group.
4 these f u n c t i o n s a r e used f o r c o n s t r u c t i n g e i g e n f u n c t i o n s o f
d e f i n i t e p a r l t y and permutatlon p r o p e r t i e s . Concluding t h e paper,
we
present some v e r y -g e n e r a l p r o p e r t i e s e x h i b i t e d by t h e bound-statespec-
trum of a one-dimensional system of three identical
interacting
par-
ticles obeying Bose-Einstein, Fermi-Dirac and Boltzmann statistics.
2. THE HYPERSPHERICAL METHOD FOR ONE-DIMENSIONAL THREE-BODY SYSTEM
The ~chrgdin~er
equation for a system of three identical particles moving in one dimension, interacttng via an attractive two-body
potential V(1x.-x.l), is
-L
where E' = E
+
3
ECM and
In terms of the Jacobi coordinates
equation (I), wi th the center of mass rernoved, can
the ~chrgdin~er
wr i tten as
7
be
-
where
Introducing the "hyperspherical coordinates", the hyper-radius
the hyperangle 13
schr8dinger equation (4) can be wri tten as
P
and
where
V ( P , ~ ) = v(J2 P I cos0I)
'
+
v(J2 p IT Jj-
+
sine
~ ( fp i
-
1
Jj- s i n o + 7
1 cose[)
cosej)
(8
1
The hyperspherical ( o r K -harmonics) method f o r s o l v i n g t h e equation (7)
c o n s i s t s ol: expanding Jl(p,B)
i n terms o f a complete
eigenfunctions.
Following
Amado
i K0
e
/(2r)li2
, K I n t e g e r , so t h a t
and
set
8
coelho ,
of
angular
we use t h e s e t
S u b s t i t u t l o n o f t h e e x p a n s i o n given by equation (9)
in
the
time independent schr8dlnger equat ion ( 7 ) , leads t o t h e f o l lowing i n f i n i t e s e t o f coupled o r d i n a r y d i f f e r e n t i a l equations
where
One may now f e e l tempted t o e n t e r eq.(10)
i n a computer, trunc-
a t i n g t h e expansion a t some K
12. However, simply t r u n c a t i n g eq.(10)
ma x
K miix w i l l n o t work, because, as we s h a l l see i n the next sect i o n , e i g e n f u n c t i o n s o f d i f f e r e n t symmetry a r e r h i x e d i n
e q . ( I O),
a t some
although
i11
f a c t uncoupled. The separation o f e i g e n f u n c t i o n s o f d i f f e -
r e n t symmetry w i l l be done i n an elementary way i n t h e n e x t
The method we use i s simple but cannot be e a s i l y extended
dimension o r l a r g e r number o f p a r t i c l e s . The reader may
~ t h e separation i s done
s e r i e s o f i i r t i c l e s by ~ f r o s 'how
mensions f o r an a r b i t r a r y number o f p a r t i c l e s .
intend
I n another
t o p r e s e n t a pedagogical i n t r o d u c t i o n t o these
section.
to
higher
f ind i n
the
i n three- dipaper
we
methods f o r
the case of four (and N) particles moving in one-dimension".
Finally we should mention that in the three-dimensional case,
even after the separation of the different symmetry functions has been
achieved, there still remain formidable computational problems. These
are agravated if three-body forces are included. A way of dealing with
the situation is presented in reference 6.
3. SYMMETRY PROPERTIES OF THE K-HARMONICS SOLUTIONS
As is well kn~wn'~'", the invariance of the hamiltonian under
any symmetry group irnplies the existence of eigenfunctions exhibiting
the group transformation properties. We shall then explorethesymmetry
invariances of the three-body hamiltonian H for obtaining elgenfunctlons with def inite symmetry properties label led
by the K-harrnon ics
components (eq. (9)). Under the actlon of the parlty operator,
(x,,x2,
x3) go onto (-x,,-x2,-x ) , (<,TI)go onto (-<,-ri) (see eqs. (3a) and (3b))
and (p,B) go onto (p;rr+8) (see eq. (6)) so that the potent ia1 V of the
,
three-body system i s invar iant under par i ty transformation (see
eqs.
(21, (5) and ( 8 ) ) . Thus, there exlst eigenfunctions of H that have def i nlte parity and we shall now determine the set of K-values that enter
the hyperspherical harmonic expansion (eq. (9)) for an e i genfunct ion
with definite parity.
Due to parity invariance of the potential, the matrix elements
GKIVIK'>
,
given in eq. (1 I), have the fol lowing properties
Selection rule (12a) decouples K-even and K-odd equations in
the system (10) and we are left with two infinite sets of
coupled
ordinary differentlal equat ions: one for even K and one for odd K. So,
the eigenfunctions of H hav Ing positive (negative) parity wíll contain
onl y even (odd) K components when expanded in terms of the hyperspherical harmnics (eq. (9)).
Now, uslng property (12b), the differentlal equa t i ons
components R- (p) and R~(P)(eq. (10)) are
K
for
so t h a t ,
Tha: i n v a r i a n c e o f the p o t e n t i a l V under t h e group
S 3
(group
(2).
o f a11 permiitations o f t h r e e o b j e c t s ) i s e v i d e n t from expression
We s h a l l now o b t a i n the t r a n s f o r m a t i o n p r o p e r t i e s o f v a r i a b l e s
and (S,ri)
(p , 0 )
under the group S,.
Th6: perrnutation o f any p a i r ( i , j ) of p a r t i c l e s leaves
P
in-
v a r l a n t . As f o r the angle 0, under t b 2 a c t i o n o f the permutation operator
P. ., 11: transforms as f o l l o w s
23
P12
:
P13
'23
Using eqs.
4
w e l l known
and
(14a),
'
@ + r - e
8
+
5n/3
6
+
r/3
(14a)
- e
-
8
(14b) and (14c) i n eq.
transformation properties o f
5
(14b)
and ri
(14~)
(6)
we
obtain
the
In terms of the variabies p and 8, the invariance of the
po-
tential V under the action of the permutation
(Relation (16) is obtained by using eqs. (14a), (14b) and (14c) in expression (8) for V(p,8)).
Using the invariance property (16), the
matrix
elements
<KIvIK'>, given by eq.(ll), can be rewritten as
from which it follows that (remember that parity invariance requ i res
K -K even)
f
otherwise
1 0
Therefore each of the two infinite sets (K even and K odd) of
coupied
ordinary differential equations (10) splits into three infinite
sets
of K val ues
6n ,
ii even
:
K odd
: 1+6n,
with n, integer, running from
2+6n ,
4+6n
,
5+6n
3+6n
to +m.
Notice that
2
+
6n E
-
4 + 6(n+l)
-
5
and
(18a)
1 + 6n
+
6(n+l)
i.e., the elernents belonging to the sets (4+6n) and (5+6n) are minus
the elements of the sets (2+6n) and (1+6n) respectively.
Thti i n v a r i a n c e o f the p o t e n t i a l V under
the
action
of
the
permutation group S, i m p l i e s the i n v a r i a n c e o f t h e h a m i l t o n i a n H under
S, so t h a t t h e r e a r e e i g e n f u n c t l o n s o f H t h a t have d e f i n i t e transform-
a t i o n p r o p e r t i e s under permutatlon o f p a r t i c l e s , besides having
defi-
n i t e p a r i t y , We s h a l l show t h a t these e i g e n f u n c t i o n s can b e o b t a i n e d by
including
ir1
t h e K-harmonics expansion (1 1 ) t h e a p p r o p r i a t e s e t
o f K-
values ( e q . ( i 8 ) ) .
Eq!;.
(15a),
(15b) and (15c) show t h a t
the vectors
p r o v i d e a b i i s i s f o r the two dlmensional i r r e d u c i b l e (mixed)
t a t i o n o f t h e permutation group S,.
5
and
q
represen-
An o b j e c t t h a t transforms
5
as
w i l l c a r r y ahe index MS (mixed symmetric) as i t i s symmetric under t h e
permutation o f p a r t i c l e s 1 and 2 (see eq. ( l 5 a ) ) andmixes w i t h the c o r r e sponding V under permutation o f p a r t i c l e s 1 and 3 o r 2 and
3 (see eqs.
(15b) and ( 1 5 ~ ) ) .Analogously, an o b j e c t t h a t transforms l i k e
will
V
c a r r y t h e iiidex MA (mixed antisymmetric) as i t i s antisymmetric
under
the perrnutation o f p a r t i c l e s 1 and 2 (see eq. ( 15a)) and rnixes w i t h t h e
correspondiiig
eqs.
5
under permutation o f any o t h e r p a i r o f p a r t i c l e s ( see
(15b) and ( 1 5 ~ ) ) .An o b j e c t t o t a l l y symme t r i c ( a n t i ~ ~ m m e t r i wi11
c)
c a r r y the index
S(A).
Using eqs.
ties,under
(lha),
(14b) and ( 1 4 ~ 1 , t h e t r a n s f o r m a t i o n proper-
S 3 , o f cos(K0) and sin(K8) a r e e a s i l y obtained.
We f i n d
t h a t f o r each s e t o f K - v a l u e s g i v e n i n e q . 0 8 ) the f u n c t l o n s
cos(K0)
and sin(K0) e x h i b i t d e f i n i t e symmetry p r o p e r t i e s under permutations o f
p a r t i c l e p a i r s . Thus, cos(6n8) ( s i n ( 6 n 8 ) ) i s t o t a i i y symmetric
(anti-
symmetric), p r o v i d i n g a one-dimensional symmetric (antisymmetric)
reducible representation o f
S,,
with positive parity.
cos(I(2+6n)0] i s MS and sinC(2+6n)8]
ir-
The f u n c t i o n
i s MA thus p r o v i d i n g a
basis
for
the two-dimensional mixed i r r e d u c i b l e r e p r e s e n t a t i o n o f S 3 ( a l s o
with
the p o s i t i v e p a r i t y ) . Another p o s i t i v e p a r i t y mixed i r r e d u c i b l e
rep-
r e s e n t a t i o n i s provided by cos lI(4+6n)0]
(MS) and s i n C(4+6n)~] ( MA). As
f o r t h e n e g a t i v e p a r i t y i r r e d u c i b l e representations,
(cos 1]3+6n)8]
sin[(3+6n)8]
) p r o v i d e t h e me-dimensional t o t a l ly symmetric
(ant i
-
synnnetric) i r r e d u c i b l e r e p r e s e n t a t i o n . The odd p a r i t y two-d i m e n s i o n a 1
mixed i r r e d u c i b l e r e p r e s e n t a t i o n s o f S , a r e provlded by
(MS) and cos [ l i + 6 n ) ~ ] (MA) and a1 so by s i n ((5+6n)0]
sin
E1 +6n)6]
(MS) and cos [(5+6njj
(MA). These p r o p e r t i e s o f sin(K8) and c o s ( ~ 6 ) wi li be used i n
the n e x t
s e c t i o n f o r c o n s t r u c t l n g e l g e n f u n c t i o n s o f p o s i t i v e and n e g a t i v e p a r i t y
t h a t a r e symmetric, antisymmetric and mixed symmetrlc under permutation
o f partlcles.
We conclude t h i s s e c t i o n w i t h a comment about the
usefulness
o f mixed symmetry e i g e n f u n c t i o n s . Systems o f i d e n t i c a l fermions (bosons)
must be d e s c r i bed by t o t a l 1 y antisymmetric (symmetric)
wavefunctions.
I f t h e p a r t i c l e s have o n l y one degree o f freedom then o n l y t h e t o t a l l y
symmetric ( f o r bosons) o r t o t a l 1 y antisymmetric ( f o r fermions)
f u n c t l o n s a r e admissible. However,
eigen-
i f the p a r t i c l e s have anextradegree
o f freedom, f o r instance, spin, u s i n g the w e l l k n ~ w n ' l " ~p r e s c r i p t i o n
o f mul t i p l y i n g mixed r e p r e s e n t a t i o n s , we can o b t a i n t o t a l
ei genfunc-
t i o n s t h a t a r e t o t a l l y symmetric o r antisymmetric.
4. CONSTRUCTION OF EIGENFUNCTIONS WITH DEFINITE PARITY AND PERMUTATION
PROPERTIES
We s h a l l f i r s t c o n ç t r u c t the e i g e n f u n c t i o n s t h a t a r e
totally
eigenfunc-
symmetric o r t o t a l l y antisymmetric. As we have seen, these
t i o n s a r e associated w i t h t h e s e t s ~ = 6 n( c o s ( 6 n ~ ) )and sen(6ne)),
*3+6n
( s i n [73+6n)87 and cos [T3+6n)e]
and
) (posi t i v e and n e g a t i v e p a r i t y
r e s p e c t i v e l y ) . For these K sets, expansion (10) can be wri.tten as
and
respectively.
I n t r o d u c i n g the n o t a t i o n
o
R6n ( P )
the e i g e n f u n c t i o n s
from eq. (19)
if
= R6,(P)
S
(p,O)
and
~
~
~ = (- R0- 6 n) ( ~ )
(p,B) a r e immediately
,
n#O
o b t a i ned
A
2i
=
(271)
,I
m
(p) s i n ( 6 n e )i t - x
6n
(271)
1
m
$n(e)
sin(6ne)
Y1=-
(23)
E
Analogousl y, i n t r o d u c i ng t h e e igenvectors R3+6n(~) and R3+6n
O (P)
t h e e i g e n f u n c t i o n s O , $ - ) ( P , ~ ) and $ j - ) ( p , e )
f rom eq.
are
i r n m e d i a t e l y obtained
(;!O)
Ar; we have seen, t h e mixed symmetry e i g e n f u n c t i o n s
s o c i a t e d w i t h K=2+6n and K=4+6n sets, f o r p o s i t i v e p a r i t y ,
are
as-
and
with
&1+6n and K=5+6n sets, f o r n e g a t i v e p a r i t y . Expansion (11) f o r
s e t s are,
and
respectively
the K
Using eqs.(l3)
and (18a) i t can be e a s i l y v e r i f i e d t h a t
c)
t h e r e f o r e we s h a l l conslder o n l y t h e e i g e n f u n c t l o n s
(p,0).
l n o r d e r t o e x h i b i t t h e mixed symmetry c h a r a c t e r o f t h e e i g e n f u n t i q n s (27) t o (28a), the s imple and elegant
rnethod
w i l i be used. The f o l l o w l n g complex conjugate
Simonov'
d e v i sed
vectors
by
are
introduced
Z
= sino+
i
cose
T h e i r transformation p r o p e r t i e s under
of
5
and r~ (eqs.
(lSa),
(15b),
P.. can be obtained from those
Z = (E+iq)/p ( e q . ( 6 ) ) . The
23
(15c) s i n c e
transformat ions a r e
Likewise, f o r any given mixed symmetry b a s i s (MS, MA ) , complex
conju-
gate v e c t o r s o f t h e form i n eq. (30)
MZ = MS
+
{MA
and
MZ, = MS
can be introduced ( t h e y transform, o f course,
i ndexes)
.
As
(sin[(1+6n)8])
d i scussed
in
the
previous
i s a MS f u n c t i o n and sin[(2+6n)0]
-
t ike Z
sect ion
where
692
thus the
cos[(2+6n)0]
(cosC(l+6n)0])
MA f u n c t i o n o f p o s i t i v e (negative) p a r i t y . Therefore,
(27) can be w r i t t e n as
and Z*,
$$)(p,B)
is
a
i n eq.
and
Analogously, $;-)(P,B)
i n eq.(28) can be w r i t t e n as
where
and
Thus,
$2)
(p.8)
form under S , as Z and Z*,
the eigenfiinct ion
i n eq. (27), and i t s , complex conjugate,
r e s p e c t i v e l y (eqs.
.
-i $A-) ( p , ~ ) (eq (28)) , and i t s complex
they t r a n s f o r m 1 i k e Z* and Z r e s p e c t i v e l y (eqs.
We now show how t h e e i g e n f u n c t i o n s
trans-
(31a) and (31b). As
for
conjugate,
(31b) and ( 3 l a ) ) .
$it)
( p , ~ ) and
($i(p,~))*
can be used t o o b t a i n t o t a l l y symmetric o r t o t a l l y antisymmetric funct l o n s . As n~entioned i n t h e end o f s e c t i o n 3, t h e p a r t i c l e s
must h a v e
an e x t r a degree o f freedom. We s h a l l assume t h a t t h e p a r t i c l e s
s p l n 1/2 arid t h e eigenvector correspondlng t o s p i n up (down)
have
wi 1 1
be
denoted by u ( d ) . Spin e i g e n f u n c t i o n s o f t h e t h r e e - p a r t i c l e system t h a t
under t h e permutation group S 3
transform as Z and ZJ w i l l be o b t a i n e d
through t h a t procedure o f combining t h e MS and MA s p i n f u n c t i o n s as i n
eq.(30a).
The t h r e e - p a r t i c l e mixed symmetry f u n c t i o n s 1 5 have t o t a l s p i n
l / 2 and s u p e r s c r i p t I 1 / 2 w i l l be used t o i n d i c a t e t h e Z-component
the t o t a l s p i n . The s p i n up f u n c t i o n s a r e
of
and t h e s p i n down a r e
The c o r r e s p o n d i n g complex c o n j u g a t e v e c t o r s , w h i c h t r a n s f o r m as Z and
Z* ( e q s . ( 3 l a ) ,
(31b)), a r e
and
The t o t a l l y symmetric and t o t a l l y a n t i s y m m e t r i c w a v e f u n c t i o n s
(eqs.
a r e now o b t a i n e d f r o m t h e a p p r o p r i a t e c o m b i n a t i o n o f $ 2 ) (p,0)
(27) and ( 2 8 ) ) , and t h e i r complex c o n j u g a t e s ,
t h e l r complex c o n j u g a t e s ,
with
(eqs. (36) and ( 3 7 ) ) . The
$i112(p,0),
positive
and
parity
functlons are
and t n e n e g a t i v e p a r i t y a r e
O f course,
i f t h e t h r e e - f e r m i o n s ystem has no o t h e r degree o f
freedom b e s i d e s s p i n , o n l y t h e above t o t a l l y a n t i s y m m e t r i c
t tons )O
';
(p,0;
i
(eqs.
(39) and (41 1) a r e a d m i s s i b l e .
eigenfunc-
5. CONCLUSIONS
We now a n a l y s e some q u a l i t a t i v e f e a t u r e s
spectrum o f one- dirnensional t h r e e - p a r t i c l e
of
t h e bound-state
system d e s c r i bed
a
by
h a m i l t o n i a n i n v a r i a n t under p a r i t y and under t h e g r o u p S,.
We s h a l l
c o n s i d e r t h e cases o f bosons o f s p i n zero,
1/2
fermions o f s p i n
a l s o t h e case o f d i s t i n g u i s h a b l e p a r t i c l e s ( ~ o l t z m a n ns t a t i s t i c s )
For a s y s t e m o f b o s o n s o f z e r o s p i n ,
the
only
(eqs.
Due t o t h e absence o f c e n t r i f u g a l b a r r i e r f o r K=O,
(25)).
$i-)
d e s c r i bed by
QS(-1
t h e ground-
X
( p , ~ ) , f o r which t h e l o w e s t component i a
+Sf)
t h e r e rnay be a whole f a m i l y o f s t a t e s o f t y p e
(p,B),
6- potent ia1
(22) and
There may be an e x c i t e d s t a t e
h a v i n g a c o n s i d e r a b l e c e n t r i f u g a l b a r r i e r . Depending on t h e
depth,
.
admissible
e i g e n f u n c t i o n s a r e t h e t o t a l l y symrnetric ones, $ L f ) ( p , 8 )
- s t a t e e i g e n f u n c t i o n s w i l l be ) p ) ( p , B ) .
and
= 3, thur
potentiai
( p , 0)
b u t no mixed symmetry t y p e i s a l l o w e d . I n t h e case
(+)
t h e r e i s o n l y one- bound- state, qS
(p,8).
and
of
the
,
Noiu f o r a system o f s p i n 1 / 2 f e r m i o n s ,
spin- space wi
the e i g e n f u n c t i o n i n
l l have mixed symmetry c h a r a c t e r due t o Paul i p r i n c i p I e
(two s t a t e s f o r accornodating t h r e e p a r t i c l e s ) . Thus t h e s p a t i a l e i g e n f u n c t i o n w i l i be e i t h e r $ k ) ( p , 8 )
o r $ L - ) ( P , ~ ) (eqs.
(27) and
+)
o r $i-)(p,0;f
(eqs.
3
(o,%;+)
n d ( 4 1 ) ) . Due t o t h e c e n t r i f u g a l
barrier
Y
= -1 ,5)
we e x p e c t t h e e 4 g e n f u n c t i o o s $ L - ) ( p , 0 )
(which
containi
eigenfunc-
components) t o have an e i g e n v a l u e smal l e r t h a n t h a t o f t h e
tion $F)(p,8)
,
(28))
6)
t h e c o r r e s p o n d i ng t o t a 1 e i g e n f u n c t i o n b e i ng, r e s p e c t i v e l y,
(which c o n t a i n s b = 2, -4 components).
F i i i a l l y , f o r a system o f d i s t i n g u i s h a b l e p a r t i c l e s , ( ~ o l t z m a n n
s t a t i s t i c s ) , a l l symmetry t y p e s t a t e s a r e a l l o w e d . The
ground- state
wi I 1 be d e s c r i bed by
$f)
(p, 0 )
(eq (22) ) due t o t h e absence
of
t r i f u g a l bai-rier f o r
K=O. The o t h e r symmetry s t a t e s , due t o
the
.
t r i f u g a l b a i - r i e r , s h o u l d o c c u r i n t h e f o l l o w i n g o r d e r : 'ii(-)(p,8)
(2811,
( ~ ~ (eq.
0 ) (27)
(2.5).
1,
( 2 6 ) ) and $ 1 + ) ( ~ , 8 ) (eq.
The degeneracy o f
O
eq.
cast
(25),
i n t o 1:he f o r m
For
(eq S.
~ ~ i8s ) shown t o be f a l s e by examininq
-
(10) f o r t h e r a d i a 1 v e c t o r s
eq. (24) w h i c h e n t e r i n
rf:spectively.
( a p p a r e n t l y degenerate)
cen-
( eq.
(23)).
$I')S, A (
t h e systern o f d i f f e r e n t i a 1 eqs.
and R3+6n(p)
M
(-1
$S,Aí~,B)
cen-
<+6n(p)
(p,8)
eq . (261,
t h e system
and
<+6n(~)
$h')
( Pie
1,
-3-6nrq $6n1(~)
i
E $+6n,(~)
i t becomes
For an interparticle potential monotonically increasing in the
o
particle distance, the potential is more attractive for R3+6n(p)
for
Rf+6n(~)
and the state $~-)(P,B) wil l be lower
than
inter-
Therefore, assuming that the potential is attractive enough to
a11 the states, the band of states with parity and
than
$l-)(p,B).
bind
perrnutat ion sym-
metry will occur in the order
(of course
and
($L-)
)
B
,
p
(
:
$
+A-)
(p,e)
( O , @ ) )* and
,
(-1 (p,w ,
($k)(P,e)
(+I
(P,B)
) * are degenerate wi th
$h-)
(&O)
respectively).
Depending on the potential depth, there rnay be whole farnilies
o f excited states f o r each type of symmetry state and the above order
of occurence of states refers to the lowest lying state of
each
sym-
metry family.
F.A.B.
Coutinho and J. Fernando Perez gratefully acknowledge
partia1 support from CNPq (~rasi
I).
We thank Dr. B.V. Carlson for his useful comments.
REFERENCES
1 . Arnol d,V., Méthodes Mathematiques de Za Mécanique CZassique (Ed i tion
Mir ~oscow)1976.
2. Wh 1t talcer, E. T. , A Treatise on the AnaZyticaZ ü y m i c s of ParticZes
and RZgid BodZes (New York Dover Publ i c a t i o n s ) 1944.
3. Galbrai th,H.W.,
J.Math.
Phys. 12, 2382 (1971).
4. SImonov,Yu A., Sov. J. o f Nucl. Phys. 3 , 46 (1966).
5.
B a l l o t , J.L.
6. Das,T.F:.,
and R i ~ e l 1 e ~ M . Fde
. La, Ann. o f Physics 127, 62 (1580).
Coelh0,H.T.
and Ripelle,M.F.de
La, Phys.Rev.
2288
C 26,
(1982).
7. Ballot,J.L.,
R i p e l l e , M.F. de La and Levinger,J.S.,
Phys. Rev. C26,
2301 (1982).
8. Amad0,R.D.
and Coelho,H.T.,
9. McGuíre, J.B.,
10. Hamermesh
J.Math.Phys.
Am. J . o f Physics 46, 1057 (1978).
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GroupTheory ( ~ d d i s o n - ~ e s l ePyu b l i s h i n g Company,
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Inc.
.
1 1 . E1 1 i o t t , J. P. and Dawber, P.G ., Symetry i n Physics, v01 1 ( ~ h eMacM i l l a n Press LTD London and Basingstoke) 1979.
12. One should mentlon t h a t f o r making a computer c a t c u l a t i o n
b e t t e r t o d e f i n e RK = $JK(~)/pl" o b t a i n i n g from eq.(10)
rhe
it
is
following
which i s somewhat e a s i e r t o s o l v e nurnerically due t o the a b s e n c e o f t h e
f i r s t derivative.
13. Fomin, B.A.
and Ef ros, V.D.,
S o v i e t J. Nucl
.
Phys. 34, 327 (1982) and
references t h e r e i n .
14. Conceição,E.M.F.
da, Malta,C.P.
and Coutinho F.A.B.
-
In
prep-
aration.
15. Close, F.E.,
An Introduction t o &uarks and Partons (Academic Press
London New York and San Francisco) 1979.
Resumo
Aspectos q u a l i t a t i v o s do espectro pontual de um sistema de
t r ê s p a r t í c u l a s i d ê n t i c a s movendo-se em uma dimensão s ã o ana 1 i 5ados.
Usando-se o método dos harmônicos-K mostra-se como c o n s t r u i r funçoesde
onda com paridade e propriedades de transformaçao bem d e f i n i d a s sob
permutação de qualquer par de p a r t i c u l a s .