  We formulate a thermodynamic theory applicable to both classical and quantum
systems. These systems are depicted as thermodynamic system-bath models capable
of handling isothermal, isentropic, thermostatic, and entropic processes. Our
approach is based on the use of a dimensionless thermodynamic potential
expressed as a function of the intensive and extensive thermodynamic variables.
Using the principles of dimensionless minimum work and dimensionless maximum
entropy derived from quasi-static changes of external perturbations and
temperature, we obtain the Massieu-Planck potentials as entropic potentials and
the Helmholtz-Gibbs potentials as free energy. These potentials can be
interconverted through time-dependent Legendre transformations. Our results are
verified numerically for an anharmonic Brownian system described in phase space
using the low-temperature quantum Fokker-Planck equations in the quantum case
and the Kramers equation in the classical case, both developed for the
thermodynamic system-bath model. Thus, we clarify the conditions for
thermodynamics to be valid even for small systems described by Hamiltonians and
establish a basis for extending thermodynamics to non-equilibrium conditions.
