  A simple system of two particles in a bidimensional configurational space $S$
is studied. The possibility of breaking in $S$ the time independent
Schr\"{o}dinger equation of the system into two separated one-dimensional
one-body Schr\"{o}dinger equations is assumed. In this paper, we focus on how
the latter property is countered by imposing such boundary conditions as
confinement in a limited region of $S$ and/or restrictions on the joint
coordinate probability density stemming from the sign-invariance condition of
the relative coordinate (an impenetrability condition). Our investigation
demonstrates the reducibility of the problem under scrutiny into that of a
single particle living in a limited domain of its bidimensional configurational
space. These general ideas are illustrated introducing the coordinates $X_c$
and $x$ of the center of mass of two particles and of the associated relative
motion, respectively. The effects of the confinement and the impenetrability
are then analyzed by studying with the help of an appropriate Green's function
and the time evolution of the covariance of $X_c$ and $x$. Moreover, to
calculate the state of the single particle constrained within a square, a
rhombus, a triangle and a rectangle the Green's function expression in terms of
Jacobi $\theta_3$-function is applied. All the results are illustrated by
examples.
