  The one-dimensional system of particles with a $1/x^2$ repulsive potential is
known as the Calogero-Moser system. Its classical version can be generalised by
substituting the coupling constants with additional degrees of freedom, which
span the $\mathfrak{so}(N)$ or $\mathfrak{su}(N)$ algebra with respect to
Poisson brackets. We present the quantum version of this generalized model. As
the classical generalization is obtained by a symplectic reduction of a free
system, we present a method of obtaining a quantum system along similar lines.
The reduction of a free quantum system results in a Hamiltonian, which
preserves the differences in dynamics of the classical system depending on the
underlying, orthogonal or unitary, symmetry group. The orthogonal system is
known to be less repulsive than the unitary one, and the reduced free quantum
Hamiltonian manifests this trait through an additional attractive term
$\sum_{i<j}\frac{-\hbar^2}{(x_i-x_j)^2}$, which is absent when one performs the
straightforward Dirac quantization of the considered system. We present a
detailed and rigorous derivation of the generalized quantum Calogero-Moser
Hamiltonian, we find the spectra and wavefunctions for the number of particles
$N=2,3$, and we diagonalize the Hamiltonian partially for a general value of
$N$.
