Accidental degeneracy and hidden symmetry: Rectangular wells with commensurate sides

Accidental degeneracy and hidden symmetry: Rectangular wells with commensurate sides R. Lemus Instituto de Ciencias Nucleares, UNAM, Apdo. Postal 70-543, Circuito Exterior, C.U. 04510 Mexico, D.F., Mexico A. Frank Instituto de Ciencias Nucleares, UNAM, Apdo. Postal 70-543, Circuito Exterior, C.U. 04510 Mexico, D.F., Mexico and Instituto de Fı ´sica, Laboratorio de Cuernavaca, UNAM, Apdo. Postal 139-B, Cuernavaca, Morelos, Mexico M. V. Andre ´s Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear Facultad de Fı ´sica, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Spain F. Leyvraz Instituto de Fı ´sica, Laboratorio de Cuernavaca, UNAM, Apdo. Postal 139-B, Cuernavaca, Morelos, Mexico ~Received 28 July 1997; accepted 26 November 1997! Two-dimensional quantum square wells and rectangular billiards with commensurate sides are simple systems which exhibit accidental degeneracies. We show that a recent analysis for square wells can be similarly applied to rectangular wells with commensurate sides. © 1998 American Association of Physics Teachers. I. INTRODUCTION In a recent paper we have shown that the impenetrable square-well potential exhibits accidental degeneracy and that it can be understood in terms of a hidden ‘‘dynamical symmetry.’’ 1The fact that part of the double degeneracies in this system is not explained by the patent geometrical symmetry of the square box is not widely appreciated,2since double degeneracies are expected for the C4vsymmetry group of the square box. In other words, the unexpected result is that some of the doublets do not transform as the two-dimensional representation ‘‘E’’ of C4v, and thus the continuous symmetry of Ref. 1 has to be invoked in order to construct a larger symmetry group.1 The case of impenetrable rectangular wells is a more intriguing example of accidental degeneracy, which occurs whenever the rectangle’s sides are commensurate, since in this situation no geometrical transformation can be found that transforms the degenerate states into each other. From a more technical point of view, the patent symmetry group of a rectangular box has only one-dimensional representations.3 How can we understand this peculiar result? In this article we shall show that rectangular boxes with commensurate sides possess a discrete hidden symmetry that leads to a remarkable analogy between its states and those of the square box, from the point of view of symmetry. We shall thus be able to relate the accidental degeneracy of the former to our previous work on the latter.1 II. COMMENSURATE RECTANGULAR-WELL POTENTIAL A free particle enclosed by an impenetrable twodimensional rectangular well of sides L1and L2has the eigenstates c n1n2~x,y!52 A L1L2 sin n1 p x L1sin n2 p y L2,~1! where n1,n2are positive integers. These states satisfy the condition c n1n250 at the boundaries of the box, when the origin of the coordinate system is chosen at the lower left corner, as shown in Fig. 1. The system’s Hamiltonian H52 \2 2m S ] 2 ] x21 ] 2 ] y2 D ~2! gives rise to the eigenvalues En1,n25 p 2\2 2 m F n1 2 L1 21n2 2 L2 2 G ,~3! which are in general nondegenerate. For commensurate side lengths, however, i.e., nL15mL2[L0,n,minteger, ~4! we find En1n25 p 2\2 2 m L0 2@n2n1 21m2n2 2#.~5! It is then simple to see that the states c n1,n2and c n1 8n2 8are degenerate if the following conditions are satisfied: nn15mn2 8,mn25nn1 8.~6! Given nand m, Eqs. ~6!will be satisfied only for a subset of all eigenvalues ~3!. As in the square-box case, additional degeneracies are found for ‘‘Pythagorean’’ identities such as ~nn1!21~mn2!25~nn3!21~mn4!2,~7! which we shall not consider further in our discussion.1 The double degeneracies En1,n25En1 8,n2 8, occurring when (n1,n2) and (n1 8,n2 8) satisfy conditions ~6!@and (n1 8,n2 8) Þ(n1,n2)#, are the ones we would like to explain from the point of view of symmetry. 629 629Am. J. Phys. 66 ~7!, July 1998 © 1998 American Association of Physics Teachers The patent symmetry group of the rectangular box, with or without commensurate sides, is the group C2v, whose structure is depicted in Fig. 2. It consists of the two reflections s ˆv a and s ˆv b, the rotation through p about the zaxis, and the identity E. The C2vcharacter table is shown in Table I. Since all the operations of C2vcommute, the group is Abelian, which in turn means that all its irreducible representations ~irreps!are one dimensional,3as mentioned in the Introduction. The eigenfunctions ~1!span irreducible representations of the symmetry group C2vin accordance with the following: Irrep label Parity n1n2 A1odd odd ~8! A2even even B1even odd B2odd even as can be readily deduced from the action of the symmetry transformations RPC2von the eigenstates ~1!. From the point of view of the patent symmetry of the rectangular box there is no reason for the double degeneracy encountered. In the next section we find a ‘‘hidden’’ symmetry operation that leads to an explanation of this matter. III. HIDDEN SYMMETRY When we substitute conditions ~6!into the eigenstates ~1! and use the relation ~4!between the side lengths, we find 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! Equation ~9!leads to the remarkable result that when c n1 8n2 8(x,y) is degenerate to c n1n2(x,y), it follows that c n1 8n2 8~x,y!5 s ˆd a c ˆn1n2~x,y!,~10! where we have used the notation of Ref. 1 and indicated by s ˆd athe operation that interchanges the xand ycoordinates in the wave functions. As is clear from Fig. 1, the s ˆd aoperation is not a patent ~geometric!symmetry of the rectangular box, but rather a hidden ~dynamic!symmetry of the system. The problem is not yet fully solved, however, since, as explained in Ref. 1, not all the cases of double degeneracy can be explained by this symmetry. To illustrate this statement we shall consider the case n52, m51in~4!. The spectrum of this rectangular box is shown in Fig. 3 up to the first 12 levels. The states are characterized by the quantum numbers (n1n2) and the symmetry label of C2v. The states fulfilling the conditions ~6!are indicated in boldface (n1,n2). Note that some states satisfy (n1n2)5(n1 8n2 8), so that no degeneracy appears. We now turn our attention to the doubly degenerate states. In order to identify the cases where the operator s ˆd aexplains the degeneracy, it is convenient to construct a representation of the operators. We first consider the doublet ( c 14 B2, c 22 A2), which defines a subspace spanning the following representation for the operators associated with C2vand the dynamical operator s ˆd a: Fig. 1. Selected coordinate system for the rectangular-well potential. Fig. 2. Symmetry elements of the group C2v. Table I. Character table of the group C2v. C2vEC2 s v a s v b A11 111z,x 2 ,y 2 A 2112121xy B1121121x,xz B2121211y,yz 630 630Am. J. Phys., Vol. 66, No. 7, July 1998 Lemus et al. D~C2!5 S 10 021 D ,D ~ s v a ! 5 S 210 021 D , ~11! D~ s ˆd a!5 S 01 10 D , where we only need to specify the generators of C2v. From the form of the representations ~11!one can show that it is not possible to diagonalize simultaneously these matrices. This means that the representation ~11!is irreducible and consequently that the operator s ˆd acan be identified as the hidden symmetry that explains the degeneracy. It is necessary, however, to define the new functions c 15 c 22 A21 c 14 B2, c 25 c 22 A22 c 14 B2,~12! since they are the ones that carry the labels corresponding to the full symmetry.1 On the other hand, the representation generated by the functions ( c 16 B2 c 32 B2) is reducible, as can be shown from its explicit form D~C2!5 S 210 021 D , ~13! D~ s v a!5 S 210 021 D ,D ~ s d ! 5 S 01 10 D . The change of basis f 15 c 16 B21 c 32 B2, f 25 c 16 B22 c 32 B2,~14! diagonalizes the representation ~13!. This will happen whenever the partners ( c n1n2 G, c n1 8n2 8 G) carry the same irreducible representation Gof C2v. It follows that the operator s ˆd acannot explain the degeneracy for all cases. It turns out, however, that in complete analogy to the square-box system,1the additional degeneracies can be explained by the introduction of the integral of motion D ˆ~A1!5 ] 2 ] x22 ] 2 ] y2,~15! where (A1) indicates the C2vcharacter of the operator. This C2vsymmetry implies that the operator satisfies the requirement that it can connect the wave functions carrying the same C2virreducible representation. In fact, the action of D(A1)over the space ~14!can be easily shown to be given by the matrix i ^ f i u D ˆA1 u f j & i 5 S 01 10 D ,i,j51,2, ~16! which means that the operator ~15!interchanges the degenerate states and thus represents the additional hidden symmetry necessary to explain the degeneracy of the wave functions ( c n1n2 G, c n1 8n2 8 G). We have found two operators, namely s ˆd aand D ˆ(A1), which represent hidden symmetries of the rectangular box with commensurate sides. These operators, together with C2v, generate a group which, however, is not the same as the one appearing in the square-well system. This is a consequence of the fact that in this case the symmetry element s ˆd aand the C2vgroup elements do not intersect, as can be seen in Fig. 2. The discussion of the group generated by these operators is beyond the scope of this paper. We only wish to mention that such group corresponds to the planar space group4p4mm, which together with the integral of motion D ˆ(A1)gives rise to the group D(1)∧p4mm, which is analogous to the group D(1)∧C4vappearing in the squarebox problem.1 IV. CONCLUSIONS In this paper we have extended our previous analysis of accidental degeneracy in the square-box system to the case of rectangular wells with commensurate side lengths. Our main results are contained in Eqs. ~9!and ~10!, which indicate that the subset of states in the latter system which satisfy relations ~6!are closely linked to the states in the former from the point of view of symmetry. In this respect, the rectangular-well’s accidental degeneracy and its explanation in terms of a connection to the square-box system perhaps represent a more subtle and interesting result than that found for the square box itself, where part of the degeneracy can be explained by the patent C4vsymmetry.1 1F. Leyvraz, A. Frank, R. Lemus, and M. Andre ´s ‘‘Accidental degeneracy in a simple quantum system: A new symmetry group for a particle in an impenetrable square well potential,’’ Am. J. Phys. 65 ~11!, 1087–1094 ~1997!. 2Claude Cohen-Tannoudji, Bernard Dui, and Frank Laloe ¨,Quantum Mechanics ~Wiley, New York, 1977!. 3Morton Hammermesh, Group Theory and Its Application to Physical Problems ~Dover, New York, 1989!. 4Gerald Burns and A. M. Glazer, Space Groups for Solid State Scientists ~Academic, New York, 1990!. Fig. 3. Spectrum associated with a rectangular box for n52 and m51in Eq. ~4!. 631 631Am. J. Phys., Vol. 66, No. 7, July 1998 Lemus et al.