  A precise definition for a quantum electron thermometer is given, as an
electron reservoir coupled locally (e.g., by tunneling) to a sample, and
brought into electrical and thermal equilibrium with it. A realistic model of a
scanning thermal microscope with atomic resolution is then developed, where the
resolution is limited in ultrahigh vacuum by thermal coupling to the
electromagnetic environment. We show that the temperatures of individual atomic
orbitals or bonds in a conjugated molecule with a temperature gradient across
it exhibit quantum oscillations, whose origin can be traced to a realization of
Maxwell's demon at the single-molecule level. These oscillations may be
understood in terms of the rules of covalence describing bonding in
\pi-electron systems. Fourier's law of heat conduction is recovered as the
resolution of the temperature probe is reduced, indicating that the macroscopic
law emerges as a consequence of coarse graining.
