Classical Spin Angular Momentum

In the past two decades, several researchers have identified spin angular momentum in classical physics wave phenomena. It has long been known that many types of waves have both intrinsic and wave momenta. Waves on a string have intrinsic momentum of the string moving back and forth, and also have wave momentum associated with energy propagation along the string. Waves in a three-dimensional solid clearly have both intrinsic (spin) and wave angular momenta as well. Fundamentally, incompressible intrinsic momentum density is equal to half the curl of spin density. Examples of simple momentum density profiles with azimuthal symmetry show that classical spin angular momentum is precisely what we think of as ordinary angular momentum. It is related to the coordinate-dependent “moment of momentum” through integration by parts. Furthermore, the second-order wave equation for spin density in an elastic solid can be factored to construct a first-order Dirac equation with the same angular momentum operators as in relativistic quantum mechanics (the Schrödinger equation is just a simplifying approximation of the Dirac equation). Therefore, teaching classical spin angular momentum will give students an understanding of the physical significance of quantum mechanical operators.