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Spin ½: the rigid rotator

Peter Holland

Published May 13, 1993• 0 citations

Abstract

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.

DOI: 10.1017/cbo9780511622687.011

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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 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.

In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schrödinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.

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.

Express the velocity of each point of a rigid body in terms of the time derivative of the rotation matrix applied to the initial position of the point and hence integrate its norm square over the entire rigid body to obtain the kinetic energy of the rigid body as a quadratic function of the time derivative of the rotation matrix with the quadratic function being determined in terms of the moment of inertia matrix of the body. Express the gravitational potential of the rigid body as a linear function of the rotation matrix and hence determine the total Lagrangian of the body in terms of the instantaneous rotation matrix and its time derivative. Perform the Legendre transformation on this Lagrangian to obtain the Hamiltonian of the rigid body in terms of the Lie algebra coordinates and their corresponding canonical momenta.

Nuclear rotation is revisited to seek for a new perspective.Review is made first for the basic concept of the nuclear rotation.Then, the quantization procedure of the rigid-body Hamiltonian is reviewed for a quantum mechanical understanding of the dynamics of rigid-body rotations.A linear rigid body is then investigated and its motion is studied from classical and quantum viewpoints.Finally, an application of the linear rigid-body rotation is attempted for a description of elongated nuclear systems, such as Beryllium 8, a two-alpha-cluster system.

The Hamiltonian describing a system of particles in a rotating coordinate system is derived and it is shown that the simple classical solution of rigid flow is forbidden in quantum mechanics, even at very low angular velocities. This effect is closely parallel to the Aharonov–Bohm effect, which likewise has its origin in the single-valuedness requirement of the wave function. An analytical approach to perturbation theory is used to include the effects of the Coriolis and centrifugal forces and to derive the current flows for some independent-particle systems, that is, for the Inglis cranking model. It is shown, by explicit construction, that the currents are not rigid even when the moment of inertia assumes the rigid-flow value, as it does for the harmonic oscillator single-particle potential under conditions of self-consistency. Furthermore, it is shown that, for a more general potential, even the moment of inertia is not rigid.