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Molecular Physics

The rotational dynamics of rigid bodies implemented with the Cayley Klein parametrization

W. N. Cottingham, D. D. DOYLE

Published Nov 20, 2001• 2 citations
Publisher

Abstract

Rigid body rotations are described using the Cayley Klein parametrization. On quantization, the wave functions are simply polynomials in the parameters. Also, half-integral as well as integral angular momenta are allowed. The Cayley Klein parametrization provides a simple representation of both the space fixed and body fixed angular momentum operators. The construction of the allowed energies and the associated wavefunctions basically involves only algebra; no reference needs be made to the Schrödinger partial differential equations of the Euler angle parametrization.

DOI
10.1080/00268970110083870

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Rigid body rotations are described using the Cayley Klein parametrization. On quantization, the wave functions are simply polynomials in the parameters. Also, half-integral as well as integral angular momenta are allowed. The Cayley Klein parametrization provides a simple representation of both the space fixed and body fixed angular momentum operators. The construction of the allowed energies and the associated wavefunctions basically involves only algebra; no reference needs be made to the Schrödinger partial differential equations of the Euler angle parametrization.
In the present paper we investigate an alternative two-axes decomposition method for rotations that has been proposed in our earlier research. It is shown to provide a convenient parametrization for many important physical systems. As an example, the kinematics of a rotating rigid body is considered and a specific class of solutions to the Euler dynamical equations are obtained in the case of symmetric inertial ellipsoid. They turn out to be related to the Rabi oscillator in spin systems well known in quantum computation. The corresponding quantum mechanical angular momentum and Laplace operator are derived as well with the aid of infinitesimal variations. Curiously, the coefficients in this new representation happen to depend only on one of the angles, which simplifies the corresponding system of ODE's emerging from separation of variables. Some applications of the hyperbolic and complex analogues of this construction in quantum mechanics and relativity are considered in a different paper cited below.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
We use the mathematical structure of group algebras and H+H^{+}H+-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2){\rm SU}(2)SU(2) and its quotient SO(3,R){\rm SO}(3,\mathbb{R})SO(3,R). The scheme developed is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. The other promising area of applications is Schrödinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrödinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of "large" quantum numbers, i.e., "quickly oscillating" functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of SU(2){\rm SU}(2)SU(2).
The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result. The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory. The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.