Acciden al degene acy in a simple quan um sys em: A new symme y
g oup o a pa icle in an impene able squa e-well po en ial
F. Ley az
Ins i u o de Fı
´sica, Labo a o io de Cue na aca, UNAM, Apdo. Pos al 139-B, Cue na aca, Mo elos, Mexico
A. F ank
Ins i u o de Fı
´sica, Labo a o io de Cue na aca, UNAM, Apdo. Pos al 139-B, Cue na aca, Mo elos, Mexico
and Ins i u o de Ciencias Nuclea es, UNAM, Apdo. Pos al 70-543, Ci cui o Ex e io , C. U. 04510
Mexico, D. F., Mexico
R. Lemus
Ins i u o de Ciencias Nuclea es, UNAM, Apdo. Pos al 70-543, Ci cui o Ex e io , C. U. 04510 Mexico,
D. F., Mexico
M. V. And e
´s
Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea Facul ad de Fı
´sica, Uni e sidad de Se illa,
Apdo. 1065, 41080 Se illa, Spain
~Recei ed 2 Decembe 1996; accep ed 30 Ap il 1997!
The wo-dimensional squa e-well po en ial is one o he simples quan um-mechanical sys ems ha
exhibi s acciden al degene acy. We show ha he double degene acy p esen is a consequence o a
dynamical symme y and de i e a new symme y g oup associa ed wi h he sys em. © 1997
Ame ican Associa ion o Physics Teache s.
I. INTRODUCTION
Acciden al degene acy in quan um sys ems usually signals
he p esence o a hidden symme y g oup o which he de-
gene acy is ende ed no mal, ha is, in e ms o which he
obse ed s a e degene acies co espond o he dimensions o
he i educible ep esen a ions o he g oup.1This is he case
o he non ela i is ic hyd ogenic a om, whe e he SO(4)
symme y explains he angula momen um independence o
he ene gy spec um.2Many o he physical sys ems display-
ing la ge degene acies han an icipa ed ha e been s udied,
and in mos ins ances he co esponding hidden symme ies
ha e been iden i ied. In mos cases, howe e , his equi es a
e y ca e ul and de ailed analysis, which has p omp ed he
au ho s o Re . 3 o compa e i mo e wi h a han wi h
science. Mo eo e , hese symme ies usually co espond o
con inuous ans o ma ions and he co esponding symme y
g oups a e hus Lie g oups o supe g oups.4I may come as
a su p ise ha one o he simples and mos s udied sys ems
in in oduc o y quan um mechanics, ha o he ee pa icle
in an impene able squa e box, exhibi s acciden al degen-
e acy and ha , o ou knowledge, no g oup- heo e ical expla-
na ion has been p o ided in he li e a u e. The p oblem is
discussed in many quan um mechanics ex books. Fo ex-
ample, Libo assigns he degene acy o he x↔ysymme y
ans o ma ion be ween wa e unc ions, bu does no discuss
ei he he symme y classi ica ion o he s a es o he dis inc-
ion be ween no mal and acciden al degene acies in he
sys em.5O he books do analyze he geome ical symme ies
in he p oblem, bu ail o explain he p esence o acciden al
degene acy.6In con as o o he hidden symme y analyses
ha ha e been ca ied ou , a dis inguishing ea u e o his
sys em ~and ha o i s h ee-dimensional ex ension!is ha
he pa en geome ical symme y o he box co esponds o a
poin g oup and no o a con inuous se o ans o ma ions.
Pe haps o his eason he p oblem has no been s udied
using he adi ional me hods,3since i s analysis equi es a
combina ion o disc e e and con inuous g oup echniques. In
his a icle we e-examine he wo-dimensional pa icle in a
box and de i e, using g oup- heo e ical a gumen s, he dy-
namical symme y esponsible o he obse ed double de-
gene acy. We also cons uc a new symme y g oup which
a ises om he combina ion o his ~con inuous!symme y
and he eigh disc e e ope a ions o he squa e-well geome -
ic symme y g oup. The new g oup is simila o he kind o
space g oups common in solid s a e physics. We hen p o-
ceed o show explici ly ha he g oup explains he obse ed
double degene acy o he sys em.
II. THE SQUARE-WELL POTENTIAL
A ee pa icle enclosed by an impene able wo-
dimensional squa e box o side Lhas he eigens a es
c
n1n2~x,y!52
Lsin
S
n1
p
x
L
D
sin
S
n2
p
y
L
D
,~1!
whe e n1,n2a e posi i e in ege s. These s a es sa is y he
condi ion
c
n1n250 a he bounda ies o he box. In Fig. 1 we
indica e he coo dina e sys em selec ed o he s a es ~1!.I is
in his sys em ha he eigen unc ions ake he simple o m
~1!. The co esponding eigen alues a e gi en by
En1,n25 2
p
2
2
m
L2~n1
21n2
2!.~2!
We can eadily see ha he e a e wo kinds o degene acies
p esen . The i s kind, which is he one we shall conside in
his pape , is a double degene acy ha occu s whene e n1
Þn2, since by in e changing he n1and n2labels in ~1!,we
ind
En1,n25En2,n1.~3!
The second kind o degene acy is mo e sub le and occu s
when we ha e he ollowing kind o ‘‘Py hago ean’’ ela-
ions:
n1
21n2
25n3
21n4
2wi h niÞnj o all i,j,~4!
1087 1087Am. J. Phys. 65 ~11!, No embe 1997 © 1997 Ame ican Associa ion o Physics Teache s
and highe o de ela ions o his so , i.e., he equali y o
h ee o mo e o hese sums o squa es. The ull degene acy
o he p oblem, including he ones in ~4!, has been discussed
by Wai-Kee Li, gi ing ise o a o mula which a ises om
he Gaussian ac o iza ion heo em. This esul is use ul o
e alua e he deg ee o degene acy as he quan um numbe s
g ow.7We shall no a emp o explain he exis ence o he
‘‘Py hago ean’’ degene acy ~4!in e ms o a la ge symme-
y g oup in his pape , so om ou poin o iew hey will
emain ‘‘acciden al.’’ I is, o cou se, an in e es ing ques ion
whe he hey can also be unde s ood in his way, a ma e
which p obably in ol es he es ablishmen o some kind o
new connec ion be ween g oup- heo e ical me hods and
numbe heo y. The analysis o he double degene acies ~3!,
howe e , may shed some ligh on his ques ion.
Re u ning o Eq. ~3!, we shall now s a by showing ha
he appa en symme y o he sys em is unable o accoun o
his beha io . Wha is he explici symme y g oup associ-
a ed wi h he pa icle in he squa e box? The sys em is
clea ly in a ian unde all ope a ions ha ans o m he
squa e box on o i sel , i.e., he C4 poin g oup o powe s o
p
/4 o a ions and e lec ions.8These ope a ions a e indica ed
in Fig. 2, while in Table I we w i e down he C4 -cha ac e
able. A he igh o Table I we indica e he way ha he
coo dina es and angula momen um ope a o s ans o m un-
de he g oup. We see om Table I ha his g oup has i e
kinds o i educible ep esen a ions ~I.R.!, ou o hem one
dimensional and a single wo-dimensional one. In o de o
his g oup o explain he degene acy ~3!o he sys em, he
se o doubly degene a e s a es
c
n1n2and
c
n2,n1, wi h n1Þn2, should ans o m as he wo-
dimensional i educible ep esen a ion, E. We shall p o e,
howe e , ha his is no he case. To his end, we de ine he
linea combina ions
n1n2
651
A
2~11
d
n1n2!~
c
n1n26
c
n2n1!,~5!
which ha e he same ene gy spec um as he s a es ~1!and
a e mo e app op ia e, since hey ca y he I.R. o C4 ,aswe
now show by applying o hem he ope a o s o he g oup. I
is enough o conside a single ope a o in each C4 class,
om which we ind
E
ˆ
n1n2
65
n1n2
6,~6a!
C
ˆ2
n1n2
65~2!n11n2
n1n2
6,~6b!
C
ˆ4
n1n2
65
a
n1n2
6
n1n2
11
b
n1n2
6
n1n2
2,~6c!
s
a
n1n2
65
a
n1n2
6
n1n2
12
b
n1n2
6
n1n2
2,~6d!
s
d
a
n1n2
656
n1n2
6,~6e!
whe e
a
n1n2
651
2~~2!n2116~2!n111!,~7a!
b
n1n2
651
2~~2!n26~2!n111!.~7b!
The ope a ions conside ed in ~6!a e indica ed in Fig. 2 and
he esul can be simply deduced om he ac ion o he g oup
ope a o s on he coo dina es (x,y) and subs i u ing in he
c
’s
in ~1!and he
’s in ~5!. Fo example, C
ˆ2x5L2x,C
ˆ2y
5L2y, and he esul ~6b!can be ound by di ec subs i u-
ion in he wa e unc ions.
I is impo an o emphasize ha in o de o ca y ou he
symme y analysis o he sys em i is necessa y o use he
s a es ~5!and no he wa e unc ions ~1!, since he la e , in
gene al, do no ans o m i educibly unde he C4 g oup,
bu a he as a linea combina ion o wo di e en ep esen-
a ions. This is an impo an poin , which is o en a sou ce o
con usion in quan um mechanics ex books. Fo example, he
s
ˆd
aope a o in ~6e!, while clea ly connec ing he s a es
c
n1n2
and
c
n2n1, does no mix he s a es ~5!among hemsel es.
The mixing o he
c
’s, while implying a degene acy, does
no p o ide a eason o i s occu ence.
Fig. 1. Selec ed ~unp imed!coo dina e sys em o he squa e-well po en ial,
wi h o igin a a co ne o he box.
Fig. 2. P imed coo dina e sys em o symme y elemen s o he g oup C4 ,
wi h o igin a he cen e o he box.
Table I. The C4 cha ac e able. A he igh we indica e he ans o ma ion
p ope ies o he coo dina es x8,y8,z8and he angula momen um ope a o s
Rx,Ry,Rzunde he ac ion o he g oup, wi h espec o he e e ence ame
o Fig. 2.
C4 EC
22C
42
s
2
s
d
A
111111z
8
,x
8
2
1y
8
2
A
21112121R
z
8
B
11121121x
8
2
2y
8
2
B
21121211x
8
y
8
E222000(x
8
,y
8
), (Rx8,Ry8)
1088 1088Am. J. Phys., Vol. 65, No. 11, No embe 1997 Ley az e al.
We now e u n o ou discussion. F om ela ions ~6!and
he cha ac e Table I we can iden i y he symme y cha ac e
o he
n1,n2
6acco ding o he pa i y o he labels. Fo ex-
ample, i n152k,n252l~k,lin ege s!,we ind
a
2k,2l
1521,
a
2k,2l
250,
b
2k,2l
150,
b
2k,2l
251~8a!
and om ~6!:
E
ˆ
2k,2l
65
2k,2l
6,C
ˆ2
2k,2l
65
2k,2l
6,C
ˆ4
2k,2l
657
2k,2l
6,
~8b!
s
ˆ
a
2k,2l
652
2k,2l
6,
s
ˆd
a
2k,2l
65
2k,2l
6.
We see ha
2k,2l
1and
2k,2l
2a e no mixed by he g oup
ope a o s and, u he mo e, by compa ing ~8b!wi h Table I
we ind ha
2k,2l
1→B2,
2k,2l
2→A2~lÞk!~9!
is hei symme y cha ac e . No e ha due o ~5!
n,n
250 and
only
2k,2k
1su i es, bu s ill ans o ms as B2. Doing he
same analysis o n152k11, n252l11, we ind
2k11,2l11
1→A1,
2k11,2l11
2→B1~lÞk!.~10!
Again, o l5k he A1s a e
2k11,2k11
1is nondegene a e.
Finally, o (2)n1Þ(2)n2, we ind ha
n1n2
1and
n1n2
2do
mix, since, e.g., om ~6!and ~7!
C
ˆ4
2k,2l11
15
2k,2l11
2.~11!
We conclude ha he pai o s a es
n1,n2
6,n1Þn2, ca y he
double- alued I.R. Eonly when (2)n1Þ(2)n2. Fo his
case he degene acy ~3!is indeed explained by C4 . How-
e e , in he case when n1Þn2and he pa i ies a e he same,
o bo h e en and odd alues o he quan um numbe s he se
o degene a e s a es
n1n2
1and
n1n2
2ca y di e en one-
dimensional I.R. o C4 and a e hus ‘‘acciden ally’’ degen-
e a e unde he ac ion o his g oup. F om he poin o iew
o C4 he e is no eason o (B2,A2)o (A
1,B
1
) s a es o be
degene a e. In Figs. 3 and 4 we display pa icula
n1,n2
6
wa e unc ions associa ed wi h Eand (B2,A2) symme ies,
espec i ely. No e ha no ob ious geome ic ope a ion can
ans o m he
1in o he
2 o he la e case.
III. HIDDEN SYMMETRY
To a emp an explana ion o his beha io we now sea ch
o addi ional ope a o s ha can mix he s a es ~9!and ~10!
among hemsel es and hus should lie ou side C4 . To his
end, we emind he eade ha in he case o he non ela i -
is ic hyd ogenic a om a simila si ua ion occu s, and he so-
lu ion s ems om he exis ence o a nongeome ical symme-
y a ising om he pa icula ~Coulomb!po en ial in ol ed.2
This is a so-called ‘‘dynamical symme y,’’ exp essed
h ough he conse a ion o he Runge–Lenz ec o Aand
h ough he closu e ~a ixed ene gy!o he commu a ion
ela ions in ol ing Aand he angula momen um L, co e-
sponding o an SO(4) symme y.2In he squa e-box sys em,
howe e , he e is no po en ial inside he box while he e is an
in ini e po en ial a he bounda y, and i is he shape o he
box ha en i ely imposes he C4 symme y. Since he
eigens a es ~5!sa is y he app op ia e bounda y condi ions,
he symme y g oup o he sys em can be mo e simply de-
ined h ough he ollowing equi emen s. The symme y op-
e a ions g
ˆishould commu e wi h he pa icle’s Hamil onian
inside he box:
H
ˆ52 2
2
m
S
]
2
]
x21
]
2
]
y2
D
,~12!
i.e.,
@g
ˆi,H
ˆ#50, ~13!
and, in addi ion, when ac ing on he eigens a es ~5! he g
ˆi
should lead o s a es wi h he same bounda y condi ions. An
a bi a y O(2) ans o ma ion ~ wo-dimensional o a ions
and e lec ions!sa is ies ~13!, bu only i s C4 subg oup op-
e a ions p ese e he bounda y condi ions. By imposing he
Fig. 3. Con ou plo o he wa e unc ion ~a!
23
1and ~b!
23
2.
1089 1089Am. J. Phys., Vol. 65, No. 11, No embe 1997 Ley az e al.
abo e equi emen s we can a oid dealing wi h he awkwa d
o m o he po en ial when sea ching o o he symme ies. I
a dynamical symme y ope a o D
ˆexis s o he pa icle in
he box, i should commu e wi h H
ˆbu no wi h he C4
ans o ma ions, since we expec ha D
ˆwill mix he A2and
B2s a es ~9!as well as he A1and B1eigen unc ions ~10!,
while p ese ing he bounda y condi ions sa is ied by he
n1,n2
6. These equi emen s a e only possible i D
ˆ ans-
o ms one o he wo s a es in ~9!o ~10!in o a linea com-
bina ion o he same wo s a es. In pa icula , we shall sea ch
o an ope a o ha ans o ms one o he s a es in o he
o he . F om he poin o iew o symme y, his implies ha
i should ha e de ini e enso ial p ope ies unde C4 ,D(G),
whe e Gis one o he I.R. in Table I, such ha
G^A15B1,G^B15A1,G^A25B2,G^B25A2.
~14!
F om Table I we eadily ind ha G5B1, since he en ies in
he able coincide wi h he g oup elemen s o one-
dimensional ep esen a ions. Fu he mo e, om Table I we
also see ha B1^E5E, and hus he degene a e se s wi h
(2)n1Þ(2)n2a e le unal e ed by a B1 enso . F om he
igh -hand side o Table I we see ha in he e e ence sys em
o Fig. 2 he combina ion x82–y82 ans o ms as he B1I.R.
unde he g oup ope a ions, al hough his is no ue in he
e e ence ame chosen o he wa e unc ions ~1!. The same
linea combina ion o squa ed momen a, howe e , ans-
o ms acco ding o he B1I.R. in bo h e e ence sys ems. We
conclude ha he ope a o
D
ˆ~B1!5
]
2
]
x22
]
2
]
y2,~15!
sa is ies he equi emen s we se up om he ou se . As a
check, by applying ~15! o he s a es ~5!we ind
D
ˆ~B1!
n1n2
65
p
2
L2~n2
22n1
2!
n1n2
7,~16!
as equi ed. This ope a o cons i u es he ‘‘dynamical sym-
me y’’ ope a o ~analogous o he Runge–Lenz ec o in he
Coulomb sys em!, which does no a ise om geome ical
conside a ions bu om he speci ic Hamil onian ~12! o-
ge he wi h he bounda y condi ions which impose he C4
symme y. We again s ess ha in his case he commu a ion
wi h he Hamil onian ~12!does no au oma ically gua an ee
ha he bounda y condi ions a e p ese ed. The combina ion
~15!is he only ope a o ha does so and has he app op ia e
enso p ope ies unde C4 . Be o e conside ing he combi-
na ion o his ope a o wi h he C4 ans o ma ions, we e-
ma k ha i is possible o in e p e he la ge symme y in
e ms o supe symme ic quan um mechanics, in he ollow-
ing sense.9We may iden i y D
ˆ(B1)in ~15!as a ‘‘supe sym-
me ic cha ge’’ Q
ˆ[D
ˆ(B1), since o he single s a es
Q
ˆ
n,n
15
S
]
2
]
x22
]
2
]
y2
D
n,n
150, ~17!
and hus each
n,n
1can be conside ed o be a acuum s a e in
he language o supe symme ic quan um mechanics
~SSQM!.8In his app oach he
n1n2
6play he ole o supe -
symme ic pa ne s and one may de ine a supe symme ic
Hamil onian ~wi h Q
ˆ†5Q
ˆ!:
H
ˆs51
2
$
Q,Q†
%
5
S
]
2
]
x22
]
2
]
y2
D
2
.~18!
Since B1^B15A1,H
ˆscommu es wi h C4 , and also wi h
bo h H
ˆand D
ˆ(B1)5Q
ˆ. No e ha he whole spec um can be
o ganized as a se ies o ‘‘ owe s’’ wi h he single
n,n
1,
ollowed by supe symme ic double s
n
¯
,n
6, wi h n
¯
.n. This
is an in iguing in e p e a ion, bu he desc ip ion is no com-
ple e, since he ac ual ela ion be ween he C4 symme ies
and D
ˆ(B1)is no speci ied in his amewo k. In Sec. IV we
de i e a new symme y g oup o he sys em.
Fig. 4. Con ou plo o he wa e unc ion ~a!
24
1and ~b!
24
2.
1090 1090Am. J. Phys., Vol. 65, No. 11, No embe 1997 Ley az e al.
IV. A NEW SYMMETRY GROUP FOR THE
SQUARE WELL
When a emp ing o de ine a la ge symme y g oup o
he sys em which con ains he geome ical C4 g oup as a
subg oup, we shall need o combine hese ans o ma ions
wi h he con inuous ones gene a ed by D
ˆ(B1). Fini e ans o -
ma ions associa ed wi h D
ˆ(B1)can be ob ained by exponen-
ia ion:
U
ˆ~
a
!5exp~i
a
D
ˆ~B1!!5exp
F
i
a
S
]
2
]
x22
]
2
]
y2
D
G
,~19!
whe e
a
is a coo dina e measu ing he ampli ude o he
ans o ma ion. Abs ac ly, he U
ˆ(
a
) con o m o an Abelian
one-pa ame e con inuous g oup,8bu no in o ma ion ega d-
ing he ange o
a
can be deduced om ~19!. When ac ing
wi h ~19!on he squa e-well wa e unc ions ~1!o ~5!, how-
e e , one can see ha
a
is bounded and hus ha he g oup is
compac . Using ~16!we eadily ind
U
ˆ~
a
!
c
n1n2~x,y!5ei
a
kn1n2
c
n1n2~x,y!,~20!
whe e
kn1n25
p
2
L2~n2
22n1
2!.~21!
As (n2
22n1
2) is an in ege numbe , we conclude ha
a
p
2/L2is a pe iodic a iable, whe e he pe iod depends on
he ela i e pa i ies o n1and n2. Thus, o (2)n15(2)n2,
n2
22n1
254l,lin ege ,
which implies ha we may es ic
a
o he ange
0<
a
p
2
L2l,
p
2o 0<
a
,L2
2
p
l.~22!
On he o he hand, o (2)n1Þ(2)n2,
n2
22n1
252l11, lin ege ,
implying ha
0<
a
p
2
L2~2l11!,2
p
,o 0<
a
,2L
2
p
~
2l11
!
.~23!
We see om ~20!–~23! ha his one-pa ame e g oup can
be labeled @up o possible addi ional degene acies o he
‘‘Py hago ean’’ ype ~4!# by he disc e e se o numbe s kn1n2
o , al e na i ely, by he in ege numbe s
k
¯
n1n25L2
p
2kn1n25n2
22n1
2.~24!
No e ha he double-degene acy ~3!implies ha kn1n2and
kn2n152kn1n2a e associa ed wi h he same ene gy eigen-
alue. The single s a es co espond o knn[0.
We a e now in a posi ion o s udy he ull symme y op-
e a ions combining U
ˆ(
a
) and C4 . The la e g oup can be
con enien ly de ined by he cose expansion ~see Fig. 2!
C4 5C2 1
s
d
aC2 ,~25!
whe e C2 is he no mal ~Abelian!subg oup
C2 5
$
g
ˆ
%
5
$
E,
s
a,
s
b,
s
a
s
b5C2
%
.~26!
The meaning o ~25!is ha he ull C4 g oup may be gen-
e a ed by conside ing he subg oup ~26!plus he elemen s
a ising om he ac ion o
s
d
aon he C2 ope a ions. Since
he ope a ions g
ˆPC2 can only change he signs o
]
xand
]
y~see Fig. 2!, i is clea om ~19! ha
g
ˆU
ˆ~
a
!5U
ˆ~
a
!g
ˆ,~27!
while
s
ˆd
a~which exchanges
]
xand
]
y!gi es
s
ˆd
aU
ˆ~
a
!5U
ˆ~2
a
!
s
ˆd
a.~28!
We also no e ha he in a iance o C2 implies
s
ˆd
ag
ˆ5g
ˆ*
s
ˆd
awhe e g
ˆ,g
ˆ*PC2 ,~29!
o explici ly
E*5E,
s
a*5
s
b,
s
b*5
s
a,C2
*5C2.~30!
Using ~27!–~29!we ind ha in he new symme y g oup
he e a e wo kinds o elemen s, which we deno e by U
ˆ(
a
)g
ˆ
and U
ˆ(
a
)
s
ˆd
ag
ˆ, wi h mul iplica ion able
U
ˆ~
a
!g
ˆ1U
ˆ~
b
!g
ˆ25U
ˆ~
a
1
b
!g
ˆ1g
ˆ2,~31a!
U
ˆ~
a
!g
ˆ1U
ˆ~
b
!
s
ˆd
ag
ˆ25U
ˆ~
a
1
b
!
s
ˆd
ag
ˆ1
*g
ˆ2,~31b!
U
ˆ~
a
!
s
ˆd
ag
ˆ1U
ˆ~
b
!g
ˆ25U
ˆ~
a
2
b
!
s
ˆd
ag
ˆ1g
ˆ2,~31c!
U
ˆ~
a
!
s
ˆd
ag1U~
b
!
s
ˆd
ag
ˆ25U
ˆ~
a
2
b
!g
ˆ1
*g
ˆ2.~31d!
Deno ing he one pa ame e g oup o U
ˆ(
a
)byD(1), i is
also clea om ~27!and ~28! ha o RPC4
R
ˆU
ˆ~
a
!R
ˆ215U
ˆ~
ea
!,~32!
whe e
e
511i R5g
ˆ
,
e
521i R
ˆ
5
s
ˆ
d
a
g
ˆ
.~33!
Equa ions ~32!and ~33!imply ha unde he ac ion o he
C4 elemen s, he D(1) se o ope a o s ans o m among
hemsel es. This is he de ini ion o an in a ian subg oup
and his indica es ha we may deno e he symme y g oup o
~31!in he semidi ec p oduc o m10
G5D~1!∧C4 .~34!
In he nex sec ion we cons uc he I.R. o his g oup and
p oceed o p o e ha unde i s ac ion he acciden al degen-
e acy ~3!is ende ed no mal.
V. THE IRREDUCIBLE REPRESENTATIONS OF G
Gi en he semidi ec p oduc o m o G, i s ep esen a-
ions can be cons uc ed using an induc ion p ocedu e10 om
D(1) o G. This is a somewha echnical endea o which is
di icul o he nonspecialis , so in his sec ion we ollow a
simple bu equi alen ou e which does no explici ly e-
qui e he induc ion concep s. Since we a e in e es ed in he
pa icula I.R. o Gspanned by he squa e-well’s wa e unc-
ions ~1!@o equi alen ly ~5!#, we may use he D(1) ep e-
sen a ions ~20!and ~21!. We s a by conside ing he I.R. o
he subg oup o ~34!gi en by
D~1!^C2 ,~35!
1091 1091Am. J. Phys., Vol. 65, No. 11, No embe 1997 Ley az e al.

whe e he di ec p oduc sign is due o ~27!. This is an Abe-
lian subg oup whose I.R. a e hus one dimensional.8Deno -
ing he ope a o s o his g oup by U
ˆ(
a
)g
ˆ, we ind
U
ˆ~
a
!g
ˆ
c
n1n2~x!5ei
a
kn1n2g
ˆ
c
n1n2~x!
5ei
a
kn1n2
x
~g
ˆ!
c
n1n2~x!,~36!
whe e
x
(g) a e elemen s o he C2 cha ac e able ~see
Table II!. I is easy o check di ec ly o he
c
n1n2(x) s a es
ha
x
~E!51,
x
~
s
a!5~2!n211,
x
~
s
b!5~2!n111,~37!
x
~C2!5~2!n11n2.
Equa ions ~36!and ~37!p o ide a comple e classi ica ion o
he I.R. o he subg oup ~35!spanned by he s a es
c
n1n2(x).
Compa ing ~37!wi h Table II we eadily iden i y he co e-
sponding C2 I.R. In o de o gene a e he co esponding
ep esen a ions o G, we should now include he ac ion o
s
ˆd
aas implied by he cose expansion ~25!and he g oup
able ~31!. Since
s
ˆd
a
c
n1n2~x!5
c
n2n1~x!,~38!
he ac ion o he C4 ope a ions ha a e no in C2 span a
wo-dimensional space whene e n1Þn2. Fo he case
whe e n15n2, howe e , he I.R. emains unidimensional:
U
ˆ~
a
!R
ˆ
c
nn~x!5ei
a
knn
x
~R!
c
nn~x!5
x
~R!
c
nn~x!,~39!
o all RPC4 , since knn50. The
x
(R)in~39!a e he ele-
men s o he C4 cha ac e able, and we ha e shown in Eqs.
~9!and ~10! ha he co esponding I.R. a e ei he A1~ o
e en n!o B2~ o odd n!. The knn50 I.R. hus coincide
wi h he C4 ones. Re u ning o he n1Þn2case, we ac on
he wo-dimensional space spanned by
c
n1n2(x) and
s
ˆd
a
c
n1n2(x)5
c
n2n1(x),
U
ˆ~
a
!g
ˆ
S
c
n1n2
c
n2n1
D
5
S
eik
a
x
~g!
c
n1n2
U
ˆ~
a
!g
ˆ
s
ˆd
a
c
n1n2
D
5
S
eik
a
x
~g!
c
n1n2
s
ˆd
aU
ˆ~2
a
!g
ˆ*
c
n1n2
D
5
S
eik
a
x
~g!
c
n1n2
e2ik
a
x
~g*!
s
ˆd
a
c
n1n2
D
5
S
eik
a
x
~g!0
0e2ik
a
x
~g*!
D
S
c
n1n2
c
n2n1
D
,~40!
whe e we used ~27!,~28!, and ~29!and ha e w i en kn1n2as
k.
Doing he same o U
ˆ(
a
)
s
ˆd
ag
ˆwe ind
U
ˆ~
a
!
s
ˆd
ag
ˆ
S
c
n1n2
c
n2n1
D
5
S
0e2ik
a
x
~g!
eik
a
x
~g*!0
D
S
c
n1n2
c
n2n1
D
.
~41!
In bo h ~40!and ~41!,
x
(g)@o
x
(g*)#a e gi en by ~37!
@ ecalling Eqs. ~30!#. As a las poin in he cons uc ion o he
I.R., i is necessa y o ind he ep esen a ion o U
ˆ(
a
)R
ˆin
he basis ~5!, which as discussed be o e is he one ha spans
he C4 I.R. This is achie ed by means o he o hogonal
ma ix
S51
&
S
11
211
D
,~42!
which ans o ms ~40!and ~41!in o he sligh ly mo e com-
plica ed o m
D~U~
a
!g!51
2
F
eik
a
x
~g!1e2ik
a
x
~g*!e2ik
a
x
~g*!2eik
a
x
~g!
e2ik
a
x
~g!2eik
a
x
~g*!eik
a
x
~g!1e2ik
a
x
~g*!
G
,~43a!
D~U~
a
!
s
d
ag!51
2
F
eik
a
x
~g*!1e2ik
a
x
~g!eik
a
x
~g!2eik
a
x
~g*!
eik
a
x
~g*!2e2ik
a
x
~g!2eik
a
x
~g*!2e2ik
a
x
~g!
G
,~43b!
whe e Dco esponds o he ep esen a ion ma ix.
Using ~43! oge he wi h ~30!and ~37!we can eadily ind
he explici o m o he ep esen a ion o e e y C4 elemen ,
in pa icula , o he gene a o s
s
ˆ
aand
s
ˆd
a. We can hen
s udy he ep esen a ions o he C4 subg oup con ained in
D, a p ocedu e known as he subduc ion G↓C4 by aking
a
50in~43!:
D~
s
a!52 1
2
F
~2!n11~2!n2~2!n12~2!n2
~2!n12~2!n2~2!n11~2!n2
G
~44!
and
D~
s
d
a!5
F
10
021
G
.~45!
Table II. The C2 cha ac e able.
C2 EC
2
s
a
s
b
A
11 111
A
2112121
B
1121121
B
2121211
1092 1092Am. J. Phys., Vol. 65, No. 11, No embe 1997 Ley az e al.
F om ~44!we immedia ely conclude ha when
(2)n15(2)n2 he ep esen a ions a e educible, while o
(2)n1Þ(2)n2 hey emain i educible, as we had al eady
ound in Sec. III.
We ha e hus succeeded in de i ing he explici o m o
he I.R. o Ggi en in ~43!, spanned by he squa e-well s a es
~5!and which a e labeled by a pai o quan um numbe s
(kn1n2,G). We should dis inguish be ween he n15n2and
n1Þn2cases.
~a!Fo n15n2, all he I.R. a e one dimensional, knn50.
The Gis a C4 label, ei he A1o B2 o e en and odd
n, espec i ely.
~b!Fo n1Þn2, he I.R. o Ga e bidimensional and la-
beled by (kn1n2,G), whe e Gis a C2 I.R. By aking
a
50in~43!we a i e a he subduc ion o hese ep-
esen a ions G↓C4 . Using he (
a
50) cha ac e s @ he
aces o he I.R. associa ed wi h ~44!and ~45!# o-
ge he wi h Table I, we ind he ollowing esul s:
G↓C4 Dim
knn50; A1A11
~46!
knn50; B2B21
~k2m11,2n11;A1!A1%B12
~k2m,2n;A2!A2%B22
~k2m,2n11;B2!E2
~k2m11,2n;B1!E2
Pa icula a en ion should be paid o he hi d and ou h
ows in he abo e able, co esponding o he I.R. spanned by
he basis s a es ~10!and ~9!. These se s span wo-
dimensional I.R. in Gwhich educe o a di ec sum o ep-
esen a ions unde educ ion o C4 . The e o e, while he
degene acy associa ed wi h ~9!and ~10!is acciden al unde
he geome ic C4 g oup, unde G he degene acy u ns ou
o be na u al. Again, his si ua ion is analogous o he one
p esen in he non ela i is ic hyd ogenic a om, whe e SO(4)
and SO(3) play he oles o Gand C4 , espec i ely. The
g oup Ghas a s uc u e simila o ha o a space ~c ys allo-
g aphic!g oup,10,11 albei he ole o he ansla ions is
played by he in e nal ans o ma ions gene a ed by D(B1),
Eq. ~19!.
VI. CONCLUSIONS
We ha e analyzed he p oblem o a pa icle enclosed by an
impene able squa e box in wo dimensions, discussed i s
acciden al degene acy associa ed wi h he exis ence o a dy-
namical symme y, and cons uc ed he highe symme y
g oup G5D(1)∧C4 . The s uc u e o Gis simila o ha
o a space g oup, and we ha e exploi ed his ac in o de o
explici ly cons uc he I.R. spanned by he squa e-well wa e
unc ions. The doubly degene a e s a es o he sys em a e
now classi ied acco ding o di e en double- alued ep esen-
a ions o G, i.e., unde he new g oup he degene acy has
been ende ed no mal. Fu he mo e, we ha e shown ha
hese ep esen a ions co ec ly educe o he app op ia e I.R.
unde he C4 subg oup.
I should be ema ked ha ou analysis does no explain
addi ional degene acies @o he ‘‘Py hago ean’’ ype ~4!#
which, as ema ked upon in Sec. I, p obably equi e he es-
ablishmen o a connec ion be ween g oup heo y and num-
be heo y.7,12 Ou analysis can be gene alized o he h ee-
dimensional ee pa icle sys em enclosed by an
impene able cubic box, al hough he solu ion is mo e com-
plex. In his case he e is a six-dimensional degene acy no
ully explained by he Ohappa en symme y o he sys em,
and he e a e wo addi ional dynamical symme ies, which
should be combined wi h he ope a ions o he Ohg oup.
A simila analysis can be applied o o he sepa able sys-
ems o he kind
H51
2Px
21V~
u
x
u
!11
2Py
21V~
u
y
u
!,~47!
including he ee pa icle and he ha monic oscilla o sys-
em, bo h o which display highe symme ies han G.I is
pe haps ele an o no e ha he qua ic Hamil onian
H5n
ˆx
21n
ˆy
2,~48!
whe e n
ˆxand n
ˆya e he xand ynumbe ope a o s, displays
a symme y g oup isomo phic o G o he space spanned by
ha monic oscilla o wa e unc ions. In his space he s udy o
he addi ional degene acies ~4!may be simple o ca y ou .3
Finally, we no e ha he same kind o me hods can be ap-
plied o he analysis o he acciden al degene acies associa ed
wi h a ec angula box wi h commensu able sides, ha is,
whe e nL15mL2, whe e L1and L2a e he side leng hs and
nand ma e in ege s. In ha case he double degene acies
occu o he le els (n1n2) and (n1
8,n2
8) when nn15mn2
8,
mn25nn1
8, besides he appea ance o gene alized
‘‘Py hago ean’’ iden i ies. Since he pa en symme y in his
case is C2 , no double degene acies a e expec ed, so he
na u e o he p oblem is, in p inciple, qui e di e en . The
in eg al o he mo ion ~15!, howe e , s ill connec s he de-
gene a e wa e unc ions in he sys em. In his case he
double degene acy can be asc ibed o he p esence o a hid-
den disc e e symme y, which oge he wi h he in eg al o
he mo ion can explain he acciden al degene acy obse ed.13
ACKNOWLEDGMENTS
This wo k was suppo ed in pa by he Eu opean Com-
muni y unde Con ac No. CI1*-CT94-0072, DGAPA-
UNAM unde P ojec Nos. IN105194 and IN105595,
CONACyT unde P ojec No. 400340-5-3401E, and Spanish
DGICyT unde Con ac No. PB95-0533-A.
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2L. Hul hen, ‘‘U
¨be die Quan en Mechanische He lei ung de Balme -
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and he Coulomb P oblem ~Wiley In e science, New Yo k, 1972!;X.L.
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dimensional Hyd ogen A om,’’ Am. J. Phys. 59, 231–232 ~1991!.
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na ion o he symme y Lie algeb a o a Hamil onian wi h acciden al
degene acy,’’ Ann. Phys. 198, 103–131 ~1990!.
4See, e.g., V. I. Man’ko, ‘‘In a ian s and S a es Gene a ing Symme y o
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pp. 453–473.
5R. L. Libo , In oduc o y Quan um Mechanics ~Holden-Day!, pp. 297–
298, San F ancisco, Cali o nia, 1980.
6See, e.g., C. Cohen-Tannoudji, Quan um Mechanics,~Wiley In e science,
New Yo k, 1977!, Vol. I.
1093 1093Am. J. Phys., Vol. 65, No. 11, No embe 1997 Ley az e al.
7W.-K. Li, ‘‘Degene acy in he pa icle-in- he-squa e p oblem,’’ Am. J.
Phys. 50, 666 ~1982!.
8M. Hame mesh, G oup Theo y and i s Applica ion o Physical P oblems
~Do e , New Yo k, 1962!.
9R. W. Haymake and R. P. Rau, ‘‘Supe symme y in Quan um Mechan-
ics,’’ Am. J. Phys. 54, 928–936 ~1986!; F. Coope and B. F eedman,
‘‘Aspec s o Supe symme ic Quan um Mechanics,’’ Ann. Phys. 146, 262-
288 ~1983!; M. de C omb ugghe and V. Ri enbe g, ‘‘Supe symme ic
Quan um Mechanics,’’ ibid.151, 99–126 ~1983!.
10S. L. Al mann, Induced Rep esen a ions in C ys als and Molecules ~Aca-
demic, New Yo k, 1977!.
11W. Ludwig and C. Fal e , Symme ies in Physics, G oup Theo y Applied o
Physical P oblems, Sp inge Se ies in Solid-S a e Sciences Vol. 64
~Sp inge , New Yo k, 1988!.
12E. B. Bogomolny, B. Geo geo , M. J. Giannoni, and C. Schmi h, ‘‘Chao ic
Billia ds Gene a ed by A i hme ic G oups,’’ Phys. Re . Le . 69, 1477–
1480 ~1992!.
13A. F ank, R. Lemus, M. And es, and F. Ley az ~unpublished!.
1094 1094Am. J. Phys., Vol. 65, No. 11, No embe 1997 Ley az e al.