Hamilton–Jacobi formulation and quantum theory of thermal wave propagation in the solid state

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Abstract
A mathematical formalism has been developed for the description of the propagation of thermal waves in solids. The theory is based on the analogy between thermal waves and mechanics as manifested upon expressing the heat conduction equation in the Hamilton–Jacobi formalism. The transition of the classical formulation to quantum mechanics is accomplished by defining thermal wave operators for the generalized coordinates and the canonically conjugated momenta. The resulting theory shows that propagating, albeit heavily damped, thermal waves can be formally described by a quantum mechanical thermal harmonic oscillator (THO). The expectation values of observables, derived from Ehrenfest’s theorems, are found to be of practical importance for the description of the thermal wave field in solids with inhomogeneous thermal and thermodynamic properties.