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-Schrodinger equation, with the Pauli matrices appearing automatically in the interaction Hamiltonian.
Highlights
These treatments either postulate the usual commutation rules for the Cartesian components of the spin operator in the non-rotating coordinate system by analogy with the rules for orbital angular momentum, or they begin with an Euler angle description of rigid rotations, and postulate that the momenta conjugate to the angles become operators given by −i times derivatives with respect to the angles, in analogy with conjugate translational momenta
We showed that half-odd-integer spin is allowed. (Some further rationale for half-odd-integer spin is provided in the paragraphs following Equation (52) below.)
For spin 1/2, these coupled equations reduce to the Pauli-Schrdinger equation, with the usual Pauli matrix representation of the space-fixed system spin operators and with two-component spinors, but with arbitrary magnetogyric ratio and additional rigid rotator terms in the Hamiltonian
Summary
Many authors have considered classical spinning top models and their possible connections to quantum spin and magnetic moment; we refer to a few examples that seem important in regard to this work [2]-[12] These treatments either postulate the usual commutation rules for the Cartesian components of the spin operator in the non-rotating coordinate system by analogy with the rules for orbital angular momentum, or they begin with an Euler angle description of rigid rotations, and postulate that the momenta conjugate to the angles become operators given by −i times derivatives with respect to the angles, in analogy with conjugate translational momenta. We provide a new derivation of the fact that the Euler angle spin angular momentum operators are independent of the structure of the model rotator, whereby any object in a nonrelativistic rigid rotator spin eigenmode must have either odd-half-integer or integer spin.
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