Accidental Degeneracies of the Particle in a Box C. A. Hollingswotth University of Pittsburgh, Pittsburgh. PA 15260 E(5rn,,5m2,n,)= E(7m,,m2,n,)
(3)
If an irreducihle representation of the symmetry group of a -Hamiltonian is k-dimensional, then those eigenfunctions (k in number) that form a basis for the representation all corresnond t o the same value of the energy. Therefore, that energ; state must be at least k-fold degenerate. If twoeigenfunctions belonginp . .. to the hases of two different irreducible representations correspond to the same energy, their degeneracv is called "accidental". some degeneracies appear to he accidental because the group used is really only a suhgroup of the group under which the Hamiltonian is invariant. For example, according to the three-dimensional rotation group, 0(3),ihe degeneracies of the hydrogen atom would he 21 1, where 1 is the azimuthal quantum numher. The actual degeneracies are n2, where n is the principal quantum numher. There appears to he some accidental degeneracy. However, it is known' that the Hamiltonian for the hydrogen atom is actually invariant under the four-dimensional rotation group, 0(4), and this larger group predicts the degeneracies n2. The three-dimensional isotropic harmonic oscillator and the spherical top are two other cases in which the group O(3) is just a suhgroup of the groups under which the Hamiltonians are in~ariant.'.~ Accidental degeneracies are not present when the larger groups are used. I t is tempting to speculate that all accidental degeneracies may he the result of the use of symmetry groups that do not include the whole symmetry of the Hamiltonian. However, there is reason to question the validity of that s ~ e c u l a t i o n . ~ It ib the purpose of this paper to describe the types of accidental degeneracies that occur for a single particle in a box with imprnetrahle walls. Theordinan three-dimensional geometric symmetry is used.
This degeneracy does not depend upon the parameter a and is not the result of oermutations allowed hv We . svmmetrv. . shall refer to i t as numerical accidental degeneracy. Actuallv. .. ea- 4 aives only a part of the degeneracy involved. The three permutations-in ~(5,5,n,)a n d t h e six permutations in E(1,7,nZ) give a total of nine equal energy states. This nine-fold degeneracy contains the numerical accidental degeneracy and some symmery dependent degeneracy. It is interesting to note that the numerical accidental degeneracy resulting from
The Rectangular Box The wave functions and energies are
is part of a 12-fold degeneracy.
+
This type of degeneracy depends upon the parameters a and b andnot upon the symmetry. We shall refer to it asparameter dependent accidental degeneracy, to distinguish i t from other types that we shall illustrate by considering the cubic box. The Cublc Box In this case a = b = c, so that all the permutations of n,, n,, and n, in +, given by eq 1, can give six distinct functions with the same energy. Then the degeneracy is six-fold. However, none of the irreducihle representations of the cuhic groups have dimension greater than three. I t follows that there is accidental degeneracy. Since this type of degeneracy depends upon the svmmetrv, hut not the value of the parameter q, weshall refer to i t as symmetry-dependent accidental degeneracy. The cuhic box gives another type of accidental degeneracy illustrated by E(5,5,n,) = E(1,7,n,)
(4)
The Spherlcal BoxS We mention here only that the spherical box shows no accidental degeneracy. The degeneracy is just the (21 1)fold degeneracy predicted by the group O(3).
+
wheren,=1,2,3 ,...;n,= 1,2,3,. . .;n,= 1,2,3,. . . andwhere a, b, and c are the lengths of the sides. Unless at least two of the values a, b, or c are integers, there is no degeneracy. This is in agreement with the prediction obtained hy the use of the point group Dzh, which has only one-dimensional irreducihle representations. However, if a t least two of the sides are of integral length, there is some degeneracy. For example, if a = ml and b = m2, where m, and mn are integers, then
'
Mclntosh, H. V. "Symmetry and Degeneracy" in Group Theory andits Applications. Loelb. Ed.; Academic: New York, 1971; Vol II. Moshinsky, M. Accidental Degeneracy and Symmetry Groups in Ouantum. Space and Time-The Quest Continues; Barut. A. O.,van der Merwe, A.; Vigier. J., Eds.; Cambridge University Press; Cambridge. 1984. FlOgge, S. Practical Quantum Mechanics; Springer-Verlag: New York. 1974: p 155.
Volume 67 Number 12 December 1990
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