Calculations are presented for microscopic quantum stress tensors for selected closed-shell atomic systems. The stress tensor in spherically symmetric atoms is expressed as a diagonal dyadic in terms of spherical-polar unit vectors. In equilibrium the balancing of momentum and electric flux implies a relationship between the radial and tangential components of the stress tensor. This relationship is used to express the microscopic pressure as a total differential of a pressure virial and to analyze the effects of finite numerical precision and algorithm errors. Effects of non-self-consistency are illustrated by a comparison of local-density-approximation and Hartree-Fock models for the stress tensor.