  The Madelung equations are two equations that are equivalent to the one-body
time-dependent Schroedinger equation. In this paper, the Madelung equation,
whose gradient is an Euler equation, is refined by introducing interpretations
of functions that are shown to depend only on the real-part of the
complex-valued wavefunction. These interpretations are extensions of functions
from the recently derived generalized Bernoulli equation, applicable to
real-valued quantum-mechanical stationary states. In particular, the velocity
and pressure definitions are extended so that they depend on the real-part of a
time-dependent complex-valued wavefunction. The Bohn quantum potential is then
interpreted as the sum of two terms, one involving the kinetic energy and the
other involving the pressure. Substituting the interpreted quantum-potential
into the Madelung equation gives a refined equation containing two kinetic
energy terms, a pressure term, and the external potential. It is easily
demonstrated that the refined Madelung equation, applied to the hydrogen atom
states with a nonzero magnetic quantum number, gives a fluid velocity that
contains both a radial component and a free vortex. Hence, the fluid particles
have angular momentum and move on streamlines that terminate at infinity. It is
also demonstrated that the two velocities from the refined Madelung equation
are related: One is the real component and the other is the imaginary component
of a complex velocity. Furthermore, an Euler equation for quantum mechanical
systems is derived by taking the gradient of the refined Madelung equation.
