  In this article, we derive a useful expectation identity using the language
of quantum statistical mechanics, where density matrices represent the state of
knowledge about the system. This identity allows to establish relations between
different quantum observables depending on a continuous parameter. Such a
parameter can be contained in the observables itself (e.g. perturbative
parameter) or may appear as a Lagrange multiplier (inverse temperature,
chemical potential, etc.) in the density matrix, excluding parameters that
modify the underlying Hilbert space. In this way, using both canonical and
grand canonical density matrices along with certain quantum observables
(Hamiltonian, number operator, etc.) we found new identities in the field,
showing not only its derivation but also its meaning. Additionally, we found
that some theorems of traditional quantum statistics and quantum chemistry,
such as the thermodynamical fluctuation-dissipation theorem, the Ehrenfest, and
the Hellmann-Feynman theorems, among others, are particular instances of our
aforementioned quantum expectation identity. At last, using a generalized
density matrix arising from the Maximum-Entropy principle, we derive
generalized quantum expectation identities: these generalized identities allow
us to group all the previous cases in a unitary scheme.
