  Currently, simulations of many-body quantum systems are known to be
computationally too demanding to be solved on classical computers. The main
problem is that the computation time and memory necessary for performing the
calculations usually grow exponentially with the number of particles $N$. An
efficient approach to simulate many-body quantum systems is the use of
classical approximation. However, it is known that at least at low
temperatures, the allowed spin fluctuations in this approach are overestimated
what results in enhanced thermal fluctuations. It is therefore timely and
important to assess the validity of the classical approximation. To this end,
in this work, we compare the results of numerical calculations of small
Mn$^{3+}$ magnetic clusters in GaN, where the Mn spins are treated classically
with those where they are treated quantum-mechanically (crystal field model).
In the first case, we solve the Landau-Lifshitz-Gilbert (LLG) equation that
describes the precessional dynamics of spins represented by classical vectors.
On the other hand, in the crystal field model, the state of Mn$^{3+}$ ion
($d^4$ configuration with $S=2$, $L=2$) is characterized by the set of orbital
and spin quantum numbers $|m_s,m_L>$. Particular attention is paid to use
numerical parameters that ensure the same single ion magnetic anisotropy in
both classical and quantum approximation. Finally, a detailed comparative study
of magnetization $\mathbf{M}(\mathbf{H}, T)$ as a function of the magnetic
field $\mathbf{H}$, temperature $T$, number of ions in a given cluster $N$ and
the strength of super-exchange interaction $J$, obtained from both approaches
will be presented.
