  The proper definition of entropy is fundamental to the relationship between
statistical mechanics and thermodynamics. It also plays a major role in the
recent debate about the validity of the concept of negative temperature. In
this paper, I analyze and calculate the thermodynamic entropy for large, but
finite quantum mechanical systems. A special feature of this analysis is that
the thermodynamic energy of a quantum system is shown to be a continuous
variable, rather than being associated with discrete energy eigenvalues.
Calculations of the entropy as a function of energy can be carried out with a
Legendre transform of thermodynamic potentials obtained from a canonical
ensemble. The resultant expressions for the entropy are also able to describe
equilibrium between quantum systems having incommensurate energy-level
spacings. This definition of entropy preserves all required thermodynamic
properties, including satisfaction of all postulates and laws of
thermodynamics. It also demonstrates the consistency of the concept of negative
temperature with the principles of thermodynamics.
