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Journal of Modern Physics

Statistical Wave Equation for Nonrelativistic Rigid Body Motions

G. H. Goedecke

Published Jan 1, 2017• 0 citations
Publisher

Abstract

In this work, the general nonrelativistic classical statistical theory presented in an earlier paper (J. Mod. Phys. 8, 786 (2017)) is applied in detail to the Euler angle and center-of-mass coordinates of an extended rigid body with arbitrary distributions of mass and electric charge. Results include the following: 1) The statistical theory spin angular momentum operators are independent of the body’s morphology; 2) These operators obey the usual quantum commutation rules in a non-rotating center-of-mass (CM) reference frame, but left-handed rules in a rotating body-fixed CM frame; 3) Physical boundary conditions on the Euler angle wavefunctions restrict all mixed spin wavefunctions to a superposition of half-odd-integer spin eigenstates only, or integer spin eigenstates only; 4) Spin s eigenfunctions are also Hamiltonian eigenfuctions only if at least two of the body’s principal moments of inertia are equal; 5) For a spin s body with nonzero charge density in a magnetic field, the theory automatically yields 2s+1 coupled wave equations, valid for any gyromagnetic ratio; and 6) For spin 1/2 the two coupled equations become a Pauli-Schrödinger equation, with the Pauli matrices appearing automatically in the interaction Hamiltonian.

DOI
10.4236/jmp.2017.812114

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In this work, the general nonrelativistic classical statistical theory presented in an earlier paper (J. Mod. Phys. 8, 786 (2017)) is applied in detail to the Euler angle and center-of-mass coordinates of an extended rigid body with arbitrary distributions of mass and electric charge. Results include the following: 1) The statistical theory spin angular momentum operators are independent of the body’s morphology; 2) These operators obey the usual quantum commutation rules in a non-rotating center-of-mass (CM) reference frame, but left-handed rules in a rotating body-fixed CM frame; 3) Physical boundary conditions on the Euler angle wavefunctions restrict all mixed spin wavefunctions to a superposition of half-odd-integer spin eigenstates only, or integer spin eigenstates only; 4) Spin s eigenfunctions are also Hamiltonian eigenfuctions only if at least two of the body’s principal moments of inertia are equal; 5) For a spin s body with nonzero charge density in a magnetic field, the theory automatically yields 2s+1 coupled wave equations, valid for any gyromagnetic ratio; and 6) For spin 1/2 the two coupled equations become a Pauli-Schrödinger equation, with the Pauli matrices appearing automatically in the interaction Hamiltonian.
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.
We prove that any nonrelativistic classical system must obey a statistical wave equation that is exactly the same as the Schrödinger equation for the system, including the usual “canonical quantization” and Hamiltonian operator, provided an unknown constant is set equal to . We show why the two equations must have exactly the same sets of solutions, whereby this classical statistical theory (CST) and nonrelativistic quantum mechanics may differ only in their interpretations of the same quantitative results. We identify some of the different interpretations. We show that the results also imply nonrelativistic Lagrangian classical mechanics and the associated Newtonian laws of motion. We prove that the CST applied to a nonrelativistic rigid rotator yields spin angular momentum operators that obey the quantum commutation rules and allow both integer and half-odd-integer spin. We also note that the CST applied to systems of identical massive particles is mathematically equivalent to nonrelativistic quantum field theory for those particles.
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schrödinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the NNN-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum NNN-vector replaced by −iℏ-i\hbar−iℏ times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. We also provide two illustrative examples and a discussion of how coordinate trajectories could be compatible with wave properties such as interference, diffraction, and tunneling.
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.
Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the angular momentum operators derived this way are the quantum version of Poincaré's vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the operators one obtains represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. The relevance of the proposed formalism for the progress in our understanding of the spin–statistics connection in nonrelativistic quantum mechanics is discussed. Introduction The connection between the spin of quantum particles and the statistics they obey is a remarkable example of a simply stated physical fact without the recognition of which many physical phenomena (ranging from the stability of matter and the electronic configuration of atoms to Bose–Einstein condensation and superconductivity) would not have an explanation. Nevertheless, the simplicity of the assertion “integer spin particles obey Bose statistics, and half-integer spin particles obey Fermi statistics” stands in bold contrast to its intricate physical origin. Indeed, Pauli's proof of the spin–statistics theorem [Pau40] (improving on earlier work by Fierz [Fie39]), showed that the spin–statistics connection was deeply rooted in relativistic quantum field theory. The path to a rigorous proof of this theorem (from the mathematical point of view) was a long one and involved the efforts of many people (see, e.g., the book by Duck and Sudarshan [DS98b]).
The global rotational degrees of freedom in the Schr\"odinger equation for an N-body system are completely separated from the internal ones. After removing the motion of the center of mass, we find a complete set of (2l+1)(2\mathcal{l}+1)(2l+1) independent base functions with angular momentum l.\mathcal{l}.l. These are homogeneous polynomials in the components of the coordinate vectors and the solutions of the Laplace equation, where the Euler angles do not appear explicitly. Any function with given angular momentum and given parity in the system can be expanded with respect to the base functions, where the coefficients are the functions of the internal variables. With the right choice of the base functions and the internal variables, we explicitly establish the equations for those functions. Only $(3N\ensuremath{-}6)$ internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.