  In this investigation, the displacement operator is revisited. We established
a connection between the Hermitian version of this operator with the well-known
Weyl ordering. Besides, we characterized the quantum properties of a simple
displaced harmonic oscillator, as well as of a displaced anisotropic
two-dimensional non-Hermitian harmonic oscillator. By constructing the
partition functions for both harmonic oscillators, we were able to derive
several thermodynamic quantities from their energy spectra. The features of
these quantities were depicted and analyzed in details.
