  If we study the quantum effects in plasmas in terms of traditional
hydrodynamics via the continuity and Euler equations we find the quantum Bohm
potential and the force of spin-spin interaction. However, if we extend the set
hydrodynamic equations beyond the 13-moments approximation, and include the
third rank tensor evolution equation along with the pressure evolution
equation, we obtain the quantum corrections to the Coulomb interaction. It is
found in contrast with the fact that hydrodynamic equations for the higher rank
tensors do not contain interaction in the classic plasmas studied in the
selfconsistent (meanfield) approximation. Therefore, we present the quantum
hydrodynamic model, where the quantum effects are studied beyond the quantum
Bohm potential. Developed model is considered in two regimes: all electrons in
plasmas are considered as the single fluid, and the separate spin evolution
regime, where electrons with different spin projections are considered as two
different fluids. To illustrate the fundamental meaning of found quantum
effects we demonstrate their contribution in the spectrum of the Langmuir waves
and the spin-electron-acoustic waves. It is worth to mention that the
application of the pressure evolution equation ensures that the contribution of
pressure in the Langmuir wave spectrum is proportional to $(3/5)v_{Fe}^{2}$,
unlike $(1/3)v_{Fe}^{2}$ appearing from hydrodynamics based on the continuity
and Euler equations, where $v_{Fe}$ is the Fermi velocity. Same correction
corresponds on other plasmas phenomena like the speed of sound for
spin-electron-acoustic waves. Moreover, it is found that novel quantum effects
provide the novel wave solutions.
