Accidental degeneracy and hidden symmetry: Rectangular wells with commensurate sides

Acciden al degene acy and hidden symme y: Rec angula wells wi h commensu a e sides 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 A. F ank 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 and 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 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 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 ~Recei ed 28 July 1997; accep ed 26 No embe 1997! Two-dimensional quan um squa e wells and ec angula billia ds wi h commensu a e sides a e simple sys ems which exhibi acciden al degene acies. We show ha a ecen analysis o squa e wells can be simila ly applied o ec angula wells wi h commensu a e sides. © 1998 Ame ican Associa ion o Physics Teache s. I. INTRODUCTION In a ecen pape we ha e shown ha he impene able squa e-well po en ial exhibi s acciden al degene acy and ha i can be unde s ood in e ms o a hidden ‘‘dynamical symme y.’’ 1The ac ha pa o he double degene acies in his sys em is no explained by he pa en geome ical sym- me y o he squa e box is no widely app ecia ed,2since double degene acies a e expec ed o he C4 symme y g oup o he squa e box. In o he wo ds, he unexpec ed e- sul is ha some o he double s do no ans o m as he wo-dimensional ep esen a ion ‘‘E’’ o C4 , and hus he con inuous symme y o Re . 1 has o be in oked in o de o cons uc a la ge symme y g oup.1 The case o impene able ec angula wells is a mo e in- iguing example o acciden al degene acy, which occu s whene e he ec angle’s sides a e commensu a e, since in his si ua ion no geome ical ans o ma ion can be ound ha ans o ms he degene a e s a es in o each o he . F om a mo e echnical poin o iew, he pa en symme y g oup o a ec angula box has only one-dimensional ep esen a ions.3 How can we unde s and his peculia esul ? In his a icle we shall show ha ec angula boxes wi h commensu a e sides possess a disc e e hidden symme y ha leads o a e- ma kable analogy be ween i s s a es and hose o he squa e box, om he poin o iew o symme y. We shall hus be able o ela e he acciden al degene acy o he o me o ou p e ious wo k on he la e .1 II. COMMENSURATE RECTANGULAR-WELL POTENTIAL A ee pa icle enclosed by an impene able wo- dimensional ec angula well o sides L1and L2has he eigens a es c n1n2~x,y!52 A L1L2 sin n1 p x L1sin n2 p y L2,~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, when he o igin o he coo dina e sys em is chosen a he lowe le co ne , as shown in Fig. 1. The sys em’s Hamil onian H52 2 2m S ] 2 ] x21 ] 2 ] y2 D ~2! gi es ise o he eigen alues En1,n25 p 2 2 2 m F n1 2 L1 21n2 2 L2 2 G ,~3! which a e in gene al nondegene a e. Fo commensu a e side leng hs, howe e , i.e., nL15mL2[L0,n,min ege , ~4! we ind En1n25 p 2 2 2 m L0 2@n2n1 21m2n2 2#.~5! I is hen simple o see ha he s a es c n1,n2and c n1 8n2 8a e degene a e i he ollowing condi ions a e sa is ied: nn15mn2 8,mn25nn1 8.~6! Gi en nand m, Eqs. ~6!will be sa is ied only o a subse o all eigen alues ~3!. As in he squa e-box case, addi ional degene acies a e ound o ‘‘Py hago ean’’ iden i ies such as ~nn1!21~mn2!25~nn3!21~mn4!2,~7! which we shall no conside u he in ou discussion.1 The double degene acies En1,n25En1 8,n2 8, occu ing when (n1,n2) and (n1 8,n2 8) sa is y condi ions ~6!@and (n1 8,n2 8) Þ(n1,n2)#, a e he ones we would like o explain om he poin o iew o symme y. 629 629Am. J. Phys. 66 ~7!, July 1998 © 1998 Ame ican Associa ion o Physics Teache s The pa en symme y g oup o he ec angula box, wi h o wi hou commensu a e sides, is he g oup C2 , whose s uc- u e is depic ed in Fig. 2. I consis s o he wo e lec ions s ˆ a and s ˆ b, he o a ion h ough p abou he zaxis, and he iden i y E. The C2 cha ac e able is shown in Table I. Since all he ope a ions o C2 commu e, he g oup is Abe- lian, which in u n means ha all i s i educible ep esen a- ions ~i eps!a e one dimensional,3as men ioned in he In- oduc ion. The eigen unc ions ~1!span i educible ep esen a ions o he symme y g oup C2 in acco dance wi h he ollowing: I ep label Pa i y n1n2 A1odd odd ~8! A2e en e en B1e en odd B2odd e en as can be eadily deduced om he ac ion o he symme y ans o ma ions RPC2 on he eigens a es ~1!. F om he poin o iew o he pa en symme y o he ec angula box he e is no eason o he double degene acy encoun e ed. In he nex sec ion we ind a ‘‘hidden’’ symme y ope a ion ha leads o an explana ion o his ma e . III. HIDDEN SYMMETRY When we subs i u e condi ions ~6!in o he eigens a es ~1! and use he ela ion ~4!be ween he side leng hs, we ind c n1 8n2 8~x,y!52 A L1L2 sin n1 8 p x L1sin n2 8 p y L2 52 A L1L2 sin mn2 p x nL1sin nn1 p y mL2 52 A L1L2 sin n2 p x L2sin n1 p y L1 5 c n1,n2~y,x!. ~9! Equa ion ~9!leads o he ema kable esul ha when c n1 8n2 8(x,y) is degene a e o c n1n2(x,y), i ollows ha c n1 8n2 8~x,y!5 s ˆd a c ˆn1n2~x,y!,~10! whe e we ha e used he no a ion o Re . 1 and indica ed by s ˆd a he ope a ion ha in e changes he xand ycoo dina es in he wa e unc ions. As is clea om Fig. 1, he s ˆd aope a ion is no a pa en ~geome ic!symme y o he ec angula box, bu a he a hidden ~dynamic!symme y o he sys em. The p oblem is no ye ully sol ed, howe e , since, as explained in Re . 1, no all he cases o double degene acy can be explained by his symme y. To illus a e his s a e- men we shall conside he case n52, m51in~4!. The spec um o his ec angula box is shown in Fig. 3 up o he i s 12 le els. The s a es a e cha ac e ized by he quan um numbe s (n1n2) and he symme y label o C2 . The s a es ul illing he condi ions ~6!a e indica ed in bold ace (n1,n2). No e ha some s a es sa is y (n1n2)5(n1 8n2 8), so ha no degene acy appea s. We now u n ou a en ion o he doubly degene a e s a es. In o de o iden i y he cases whe e he ope a o s ˆd aexplains he degene acy, i is con enien o cons uc a ep esen a ion o he ope a o s. We i s conside he double ( c 14 B2, c 22 A2), which de ines a subspace spanning he ollowing ep esen a- ion o he ope a o s associa ed wi h C2 and he dynamical ope a o s ˆd a: Fig. 1. Selec ed coo dina e sys em o he ec angula -well po en ial. Fig. 2. Symme y elemen s o he g oup C2 . Table I. Cha ac e able o he g oup C2 . C2 EC2 s a s b A11 111z,x 2 ,y 2 A 2112121xy B1121121x,xz B2121211y,yz 630 630Am. J. Phys., Vol. 66, No. 7, July 1998 Lemus e al. D~C2!5 S 10 021 D ,D ~ s a ! 5 S 210 021 D , ~11! D~ s ˆd a!5 S 01 10 D , whe e we only need o speci y he gene a o s o C2 . F om he o m o he ep esen a ions ~11!one can show ha i is no possible o diagonalize simul aneously hese ma ices. This means ha he ep esen a ion ~11!is i educible and consequen ly ha he ope a o s ˆd acan be iden i ied as he hidden symme y ha explains he degene acy. I is neces- sa y, howe e , o de ine he new unc ions c 15 c 22 A21 c 14 B2, c 25 c 22 A22 c 14 B2,~12! since hey a e he ones ha ca y he labels co esponding o he ull symme y.1 On he o he hand, he ep esen a ion gene a ed by he unc ions ( c 16 B2 c 32 B2) is educible, as can be shown om i s explici o m D~C2!5 S 210 021 D , ~13! D~ s a!5 S 210 021 D ,D ~ s d ! 5 S 01 10 D . The change o basis 15 c 16 B21 c 32 B2, 25 c 16 B22 c 32 B2,~14! diagonalizes he ep esen a ion ~13!. This will happen when- e e he pa ne s ( c n1n2 G, c n1 8n2 8 G) ca y he same i educible ep esen a ion Go C2 . I ollows ha he ope a o s ˆd acan- no explain he degene acy o all cases. I u ns ou , how- e e , ha in comple e analogy o he squa e-box sys em,1 he addi ional degene acies can be explained by he in oduc ion o he in eg al o mo ion D ˆ~A1!5 ] 2 ] x22 ] 2 ] y2,~15! whe e (A1) indica es he C2 cha ac e o he ope a o . This C2 symme y implies ha he ope a o sa is ies he equi e- men ha i can connec he wa e unc ions ca ying he same C2 i educible ep esen a ion. In ac , he ac ion o D(A1)o e he space ~14!can be easily shown o be gi en by he ma ix i ^ i u D ˆA1 u j & i 5 S 01 10 D ,i,j51,2, ~16! which means ha he ope a o ~15!in e changes he degen- e a e s a es and hus ep esen s he addi ional hidden symme- y necessa y o explain he degene acy o he wa e unc- ions ( c n1n2 G, c n1 8n2 8 G). We ha e ound wo ope a o s, namely s ˆd aand D ˆ(A1), which ep esen hidden symme ies o he ec angula box wi h commensu a e sides. These ope a o s, oge he wi h C2 , gene a e a g oup which, howe e , is no he same as he one appea ing in he squa e-well sys em. This is a con- sequence o he ac ha in his case he symme y elemen s ˆd aand he C2 g oup elemen s do no in e sec , as can be seen in Fig. 2. The discussion o he g oup gene a ed by hese ope a o s is beyond he scope o his pape . We only wish o men ion ha such g oup co esponds o he plana space g oup4p4mm, which oge he wi h he in eg al o mo ion D ˆ(A1)gi es ise o he g oup D(1)∧p4mm, which is analogous o he g oup D(1)∧C4 appea ing in he squa e- box p oblem.1 IV. CONCLUSIONS In his pape we ha e ex ended ou p e ious analysis o acciden al degene acy in he squa e-box sys em o he case o ec angula wells wi h commensu a e side leng hs. Ou main esul s a e con ained in Eqs. ~9!and ~10!, which indi- ca e ha he subse o s a es in he la e sys em which sa is y ela ions ~6!a e closely linked o he s a es in he o me om he poin o iew o symme y. In his espec , he ec angula -well’s acciden al degene acy and i s explana ion in e ms o a connec ion o he squa e-box sys em pe haps ep esen a mo e sub le and in e es ing esul han ha ound o he squa e box i sel , whe e pa o he degene acy can be explained by he pa en C4 symme y.1 1F. Ley az, A. F ank, R. Lemus, and M. And e ´s ‘‘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,’’ Am. J. Phys. 65 ~11!, 1087–1094 ~1997!. 2Claude Cohen-Tannoudji, Be na d Dui, and F ank Laloe ¨,Quan um Me- chanics ~Wiley, New Yo k, 1977!. 3Mo on Hamme mesh, G oup Theo y and I s Applica ion o Physical P oblems ~Do e , New Yo k, 1989!. 4Ge ald Bu ns and A. M. Glaze , Space G oups o Solid S a e Scien is s ~Academic, New Yo k, 1990!. Fig. 3. Spec um associa ed wi h a ec angula box o n52 and m51in Eq. ~4!. 631 631Am. J. Phys., Vol. 66, No. 7, July 1998 Lemus e al.