  The hydrodynamic interpretation of quantum mechanics treats a system of
particles in an effective manner. In this work, we investigate squeezed
coherent states within the hydrodynamic interpretation. The Hamiltonian
operator in question is time dependent, n-dimensional and in quadratic order.
We start by deriving a phase space Wigner probability distribution and an
associated equilibrium entropy for the squeezed coherent states. Then, we
decompose the joint phase space distribution into two portions: a marginal
position distribution and a momentum distribution that is conditioned on the
post-selection of positions. Our conditionally averaged momenta are shown to be
equal to the Bohm's momenta whose connection to the weak measurements is
already known. We also keep track of the corresponding classical system
evolution by identifying shear, magnification and rotation components of the
symplectic phase space dynamics. This allows us to pinpoint which portion of
the underlying classical motion appears in which quantum statistical concept.
We show that our probability distributions satisfy the Fokker-Planck equations
exactly. They can be used to decompose the equilibrium entropy into the missing
information in positions and in momenta as in the Sackur-Tetrode entropy of the
classical kinetic theory. Eventually, we define a quantum pressure, a quantum
temperature and a quantum internal energy which are related to each other in
the same fashion as in the classical kinetic theory. We show that the quantum
potential incorporates the kinetic part of the internal energy and the
fluctuations around it. This allows us to suggest a quantum conditional virial
relation. In the end, we show that the kinetic internal energy is linked to the
fractional Fourier transformer part of the underlying classical dynamics
similar to the case where the energy of a quantum oscillator is linked to its
Maslov index.
