Quantum Theory of Heat Transport in an Isotopically Substituted, One-Dimensional, Harmonic Crystal

We present a quantum mechanical treatment of thermal transport in a one-dimensional isotopically substituted harmonic lattice. This work is an extension of a classical mechanical treatment. We find that the difference between the quantum and classical expressions for the thermal conductivity of a random chain vanishes in the limit N → ∞, where N is the number of isotopes. Thus, as in the classical treatment, the thermal conductivity diverges as N1/2. For a periodic diatomic lattice, we derive explicit formulas for the heat current as a function of temperature. At very low temperatures, this quantum mechanical current exhibits Kapitsa behavior.