  Method of the quantum hydrodynamics has been applied in quantum plasmas
studies. As the first step in our consideration, derivation of classical
semi-relativistic (i. e. described by the Darwin Lagrangian on microscopic
level) hydrodynamical equations is given after a brief review of method
development. It provides better distinguishing between classic and quantum
semi-relativistic effects. Derivation of the classical equations is interesting
since it is made by a natural, but not very widespread method. This derivation
contains explicit averaging of the microscopic dynamics. Derivation of
corresponding quantum hydrodynamic equations is presented further. Equations
are obtained in the five-momentum approximation including the continuity
equation, Euler and energy balance equations. It is shown that relativistic
corrections lead to presence of new quantum terms in expressions for a force
field, a work field etc. The semi-relativistic generalization of the quantum
Bohm potential is obtained. Quantum part of the energy current, which is an
analog of the quantum Bohm potential for the energy evolution equation, is
derived. The Langmuir wave dispersion in semi-relativistic quantum plasmas,
corresponding to the Darwin Lagrangian, is also considered to demonstrate
contribution of semi-relativistic effects on basic plasma phenomenon.
