2 - Quantum and Classical Waves

Abstract

This chapter provides an overview of quantum and classical waves. The scalar wave equation differs from the electromagnetic and elastic wave equations in that the true classical waves have vector character. For the electromagnetic wave, there are two polarizations transverse to k, and for the elastic wave, there is in addition a longitudinal polarization parallel to k. In a uniform, isotropic medium, classical waves of different polarizations are each described by a scalar wave equation, with a velocity v that can be different for the elastic longitudinal wave and the elastic transverse (shear) waves because they rely on different material properties for their propagation. As different polarizations are decoupled from each other in a uniform, isotropic medium, the scalar wave equation is thus accurate for each polarization component of the classical waves. An important difference between the Schrödinger (quantum) wave equation and the scalar (classical) wave equation is the order of the time derivative. It is first order in the quantum case and it is second order in the classical case. The chapter concludes with the discussion of Green's function for waves in a uniform medium.