General Formula for the Efficiency of Quantum-Mechanical Analog of the Carnot Engine

An analog of the Carnot engine reversibly operating within the framework of pure-state quantum mechanics is discussed. A general formula is derived for the efficiency of such an engine with an arbitrary confining potential. Its expression is given in terms of an energy spectrum and shows how the efficiency depends on a potential as the analog of a working material in thermodynamics, in general. This non-universal nature results from the fact that there exists no analog of the second law of thermodynamics in pure-state quantum mechanics where the von Neumann entropy identically vanishes. A special class of spectra, which leads to a common form of the efficiency, is identified.