Accidental degeneracy in a simple quantum system: A new symmetry group for a particle in an impenetrable square-well potential

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. 1See, e.g., J. M. Jauch and E. L. Hill, ‘‘On he p oblem o degene acy in quan um mechanics,’’ Phys. Re . 57, 641–645 ~1940!; H. V. McIn osh, ‘‘Symme y and Degene acy,’’ in G oup Theo e ical Me hods and i s ap- plica ions, edi ed by E. M. Loebl ~Academic, New Yo k, 1990!, Vol. II, pp. 75–137; M. Moshinsky, ‘‘Acciden al Degene acies and Symme y G oups,’’ Found. Phys. 13, 73–79 ~1983!. 2L. Hul hen, ‘‘U ¨be die Quan en Mechanische He lei ung de Balme - e me,’’ Z. Phys. 86, 21–23 ~1933!; V. Fock, ‘‘Zu Theo ie des Wasse - s o a oms,’’ ibid.98, 145–154 ~1936!; M. J. Engle ield, G oup Theo y and he Coulomb P oblem ~Wiley In e science, New Yo k, 1972!;X.L. Yang, M. Liebe , and F. T. Chan, ‘‘The Runge Lenz Vec o o he Two- dimensional Hyd ogen A om,’’ Am. J. Phys. 59, 231–232 ~1991!. 3M. Moshinsky, C. Quesne, and G. Loyola, ‘‘A o Science: The de e mi- 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 Nons a iona y Sys ems,’’ in ‘‘Symme ies in Science, edi ed by B. G ube , L. C. Biedenha n, and H. D. Doebne ~Plenum, New Yo k, 1991!, Vol. V, 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.