  In this paper an exact transparent boundary condition (TBC) for the
multidimensional Schr\"odinger equation in a hyperrectangular computational
domain is proposed. It is derived as a generalization of exact transparent
boundary conditions for 2D and 3D equations reported before. A new exact fully
discrete (i.e. derived directly from the finite-difference scheme used) 1D
transparent boundary condition is also proposed. Several numerical experiments
using an improved unconditionally stable numerical implementation in the 3D
space demonstrate propagation of Gaussian wave packets in free space and
penetration of a particle through a 3D spherically asymmetrical barrier. The
application of the multidimensional transparent boundary condition to the
dynamics of the 2D system of two non-interacting particles is considered. The
proposed boundary condition is simple, robust and can be useful in the field of
computational quantum mechanics, when an exact solution of the multidimensional
Schr\"odinger equation (including multi-particle problems) is required.
