  Stein's method is used to study discrete representations of multidimensional
distributions that arise as approximations of states of quantum harmonic
oscillators. These representations model how quantum effects result from the
interaction of finitely many classical "worlds," with the role of sample size
played by the number of worlds. Each approximation arises as the ground state
of a Hamiltonian involving a particular interworld potential function. Such
approximations have previously been studied for one-dimensional quantum
harmonic oscillators, but the multidimensional case has remained unresolved.
Our approach, framed in terms of spherical coordinates, provides the rate of
convergence of the discrete approximation to the ground state in terms of
Wasserstein distance. The fastest rate of convergence to the ground state is
found to occur in three dimensions. This result is obtained using a discrete
density approach to Stein's method applied to the radial component of the
ground state solution.
