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We study two-dimensional quantum Carnot engines of spherical symmetry byconsidering the case of a particle on the surface of a sphere of changingradius. The Carnot cycle is built allowing the state of the system to changewith the specific constrains discussed in Bender's work for Carnot cycles.After studying the Carnot cycle, we maximize the output power and efficiency ofthe system to show that as it happens in one dimension systems:(i) theefficiency can be optimized; being its optimal value independent of theparameters describing the system and that the optimal output power at theoptimal efficiency is non-zero. (ii) The optimal efficiency of the sphericalsystem is much bigger than that of the one-dimensional quantum well consideredin Abe's work.

Maximum power of a two-dimensional quantum mechanical engine with spherical symmetry