Symmetry of Tops and Rotators

The whole subject of tops and rotators has had a long history and an extensive literature, especially in classical mechanics, and was highly developed long before the advent of wave mechanics. Perhaps the most distinguished treatment of the theory of tops is the monumental four-volume work of Klein and Sommerfeld (26), published in 1897, wherein the motion of tops, gyroscopes, and rotating systems in general is given an exhaustive treatment. These objects also figured prominently in the old quantum mechanics, since they form the natural model for the treatment of the rotation of molecules, and were very promptly investigated when the new quantum mechanics began to make its appearance around 1926. The proper formulation of Schrödinger's equation for the rigid rotator required some thought, simply because three-dimensional space is not its configuration space. Rather, the ``coordinates'' are Euler's angles or some similar rotational parameters, and it was necessary to understand how to write a proper wave mechanical analogue of Euler's equations for the motion of a rotating body. On the other hand, it was possible to treat the rotator at once in the matrix mechanics, due to the ease with which the principal moments of inertia could be used as coefficients to write the Hamiltonian as a sum of squares of the angular momentum operators, and its eigenvalues obtained by operational methods.

In the years around 1920 the old quantum mechanics was used for the study of tops and rotators, notably by Epstein (27), Reiche (28), Kramers (29), and Kramers and Pauli (30), some of it based on an analysis of the equations of motion made years earlier by Kolossoff (31). Not surprisingly, they were among the first examples taken, both in the matrix mechanics and in the wave mechanics, but it was not long before explicit treatments appeared. Dennison (32), Reiche and Radamacher (33), Witmer (34), Kronig and Rabi (35), and Lütgemeier (36) made some of the earliest contributions. From the point of view of symmetry, and especially of the spherical top, Hund's (37) treatment of the spherical top in quaternionic coordinates, and Klein's (38) 1929 determination of the commutation rules so that he could analyze the asymmetric rotor by operational methods are the most relevant.

The asymmetrical top received its most extensive treatment in a series of papers by Kramers and Ittman (39), also in 1929, with some further contributions by Wang (40), and Ray (41). Casimir's thesis of 1931 (42) discusses the quantization in terms of generalized coordinates, and the commutation rules of the angular momentum operators. There the matter rested for two decades, until the availability of microwave equipment in the late 1940s permitted a thoroughgoing experimental analysis of molecular rotational spectra, and with it further studies of the quantum mechanics of rotating systems.

The principal new techniques to have emanated from this recent period are operational methods, not only for the determination of energy levels and wave functions, but for dipole and quadrupole transition matrix elements and other such information. Additionally, the advent of electronic computers has permitted the application of all these techniques to an array of configurations which would previously have been completely impossible. Some of this material is to be found in the series of papers initiated by King el al. (43), in the Journal of Chemical Physics, while the diverse operational techniques were published by Burkhard (44), and Shaffer (45).

In a sense, the degeneracy and symmetry which one encounters in the family of tops is not accidental, any misapprehension to the contrary having most likely arisen from the hasty assumption that three-dimensional space is their configuration space, and that the Hamiltonian, rather than the Eulerian equations, have been applied. In reality, the four-dimensional unit sphere is the most convenient parameter space for the treatment of rigid motion and, if anything, one should expect the four-dimensional rotation group to be the one governing symmetries. For a better understanding of this situation one ought to take note of the different classes of ``rotors'' which the spectroscopist recognizes, as well as enumerating their characteristic degeneracies.

First of all, there is a difference between a ``rotator'' and a ``rotor'' or top. The former is merely a point rotating about some center, possessing an energy in virtue of its angular momentum. Its radial distance is constant; for example, if a massive point is connected to a fixed center by a light rigid rod. Its wave functions will be ordinary spherical harmonics, and its eigenvalues are those of the square of the angular momentum operator. It manifests a degeneracy in the $z$ component of its angular momentum on account of the spherical symmetry of such a configuration.

Such an arrangement, however, is not what one would customarily regard as a ``top'' because it describes the motion of a point, perhaps even a rod, but not the rotational motion of a solid body. At best, it can be regarded as characteristic of a body one of whose moments of inertia is zero. Turning to true tops, if all three moments of inertia are equal and nonzero, the body is called a spherical top. Quantized, it shows a very high degeneracy, characteristic of the four-dimensional and not the three-dimensional rotation group. Such degeneracy is $n^2$-fold, $n$ being a principal quantum number. Now, if only two moments of inertia are equal, but distinct from the third, the top is called a ``symmetrical'' top, and has a degeneracy $2(2\ell + 1)$, which is double that of a spherically symmetrical system, in spite of the fact that a body with such moments of inertia does not have spherical symmetry. Finally, when all three moments of inertia are different, the object is called an ``asymmetrical top'' and surprisingly enough it still shows a degeneracy in its energy levels $(2\ell + 1)$, characteristic of spherical symmetry.

What do we mean by the ``configuration space'' of a top? It is hardly three-dimensional space, which is suitable for describing the location of points comprising the top, but has to be formed from some sort of collective coordinates, such as their center of mass, or in the present case, parameters describing the orientation of the body in space. One very convenient set of parameters for this purpose are the Euler angles, which may be used to specify the orientation of a set of coordinates attached to the body, in terms of a fixed set of coordinates attached to the laboratory. Analogous parameters may describe rotations in a space of any number of dimensions, but in three dimensions, we first may rotate the $z$ axis to a new position, which requires two coordinates -- let us say the colatitude and azimuth of the new $z$-axis. This determines the orientation of the new $x$-$y$ plane, but not the location of the $x$ axis within the new plane, so that a third angle must be specified. The result is three coordinates, the three Euler angles, which must range over a three-dimensional manifold, and not a two-dimensional manifold which ordinarily specifies angular orientation.

There are many ways of parameterizing an element of the three-dimensional rotation group, and for many purposes one of the most convenient is the use of unit quaternions, whose coefficients in such a representation are the Cayley-Klein parameters of the rotation. Since the unit quaternions range over the surface of the hypersphere, a three-dimensional manifold, it is in this way that the hypersphere becomes the natural configuration space of the top problems. The relationship is a very curious accident of three dimensions, for it is only in this one exceptional instance that the parameter space for a rotation group is a sphere. Of course, one will obtain wave equations for tops of other dimensions, with their appropriate degeneracies, but there would not result quite the elegant picture which one obtains in three dimensions, that the motion of the spherical top is equivalent to force-free motion on the surface of a hypersphere. Since the motion of the spherical top can be so characterized, there results the rather nice uniform interpretation of the accidental degeneracy of three of the most important idealized systems in quantum mechanical theory, which was expounded by Saénz in the first half of his dissertation.

In explaining the accidental degeneracy of the hydrogen atom one has to introduce stereographic coordinates in momentum space, which allows the reduction of the motion to force-free motion on the surface of a hypersphere, for negative energies. For the harmonic oscillator, the action-angle variables for an uncommon, but not unreasonable, coordinate system again allow an interpretation of the motion as force-free motion on the surface of a hypersphere. For the spherical top, it may be demonstrated that the equations of motion are again those of force-free motion for a particle constrained to reside on the surface of the hypersphere. However, we have a rather more direct way in which to relate motion on the hypersphere to the actual movement of the top, which contrasts with the considerably more indirect interpretation which must be given to rotations of the hypersphere in the other two problems. Since rotation of a body is detected by noticing the difference in orientation between a system of coordinates fixed to the body and a set fixed in the laboratory, it seems that we may rotate either one or the other of these coordinate systems at will. Rotating the set attached to the laboratory results in a new motion derived from the old by a rotation, and hence is a symmetry of whatsoever top, spherical symmetrical or asymmetrical. Rotation of the set attached to the body is immaterial for the spherical top, is a symmetry if performed about ths symmetry axis of the symmetric rotor, and results in no symmetry at all for the asymmetric top. The four-dimensional rotation group is a direct product of two three-dimensional rotation groups, and in the case of the motion of a top, the two constituent factors can be identified with the external rotations (of the laboratory coordinates) and the internal rotations (of the body-fixed axes). Rotation of the laboratory coordinates will always be a symmetry operation, but the internal rotation group will be restricted according to the number of principal moments of inertia which are equal. Thus the spherical top will have the full four-dimensional rotation group as a symmetry group, the symmetric top will have the direct product of the three- dimensional group with the rotations about the symmetry axis, and the asymmetric rotor will only have a reflection group for its internal symmetries.