  Quantum turbulence that exhibits vortex creation, annihilation and
interactions is demonstrated as an exact solution of the time-dependent,
free-particle Schr\"odinger equation evolved from a smooth random-phased
initial condition. Relaxed quantum turbulence in 2D and 3D exhibits universal
scaling in the steady-state energy spectrum as k-1 in small scales. Due to the
lack of dissipation, no evidence of the Kolmogorov-type forward energy cascade
in 3D or the inverse energy cascade in 2D is found, but the rotational and
potential flow components do approach equi-partition in the scaling regime. In
addition, the 3D vortex line-line correlation exhibits universal behaviour,
scaled as \Deltar^-2, where \Deltar is the separation between any two vortex
line elements, in fully developed turbulence. We also show that the quantum
vortex is not frozen to the matter, nor is the vortex motion induced by other
vortices via Biot-Savart's law. Thus, the quantum vortex is actually a
nonlinear wave, propagating at a speed very different from a classical vortex.
