  The positivity conditions of the relative entropy between two thermal
equilibrium states $\hat{\rho}_1$ and $\hat{\rho}_2$ are used to obtain upper
and lower bounds for the subtraction of their entropies, the Helmholtz
potential and the Gibbs potential of the two systems. These limits are
expressed in terms of the mean values of the Hamiltonians, number operator, and
temperature of the different systems. In particular, we discuss these limits
for molecules which can be represented in terms of the Franck--Condon
coefficients. We emphasize the case where the Hamiltonians belong to the same
system at two different times $t$ and $t'$. Finally, these bounds are obtained
for a general qubit system and for the harmonic oscillator with a time
dependent frequency at two different times.

Quantum Physics