  This paper addresses the numerical implementation of the transparent boundary
condition (TBC) and its various approximations for the free Schr\"odinger
equation on a rectangular computational domain. In particular, we consider the
exact TBC and its spatially local approximation under high frequency assumption
along with an appropriate corner condition. For the spatial discretization, we
use a Legendre-Galerkin spectral method where Lobatto polynomials serve as the
basis. Within variational formalism, we first arrive at the time-continuous
dynamical system using spatially discrete form of the initial boundary-value
problem incorporating the boundary conditions. This dynamical system is then
discretized using various time-stepping methods, namely, the
backward-differentiation formula of order 1 and 2 (i.e., BDF1 and BDF2,
respectively) and the trapezoidal rule (TR) to obtain a fully discrete system.
Next, we extend this approach to the novel Pad\'e based implementation of the
TBC presented by Yadav and Vaibhav [arXiv:2403.07787(2024)]. Finally, several
numerical tests are presented to demonstrate the effectiveness of the boundary
maps (incorporating the corner conditions) where we study the stability and
convergence behavior empirically.
