We consider in this paper spectral and pseudospectral approximations
using Hermite functions for PDEs on the whole line.  We first develop
some basic approximation results associated with the projections and
interpolations in the spaces spanned by Hermite functions.  These
results play important roles in the analysis of the related spectral
and pseudospectral methods. We then consider, as an example of
applications, spectral and pseudospectral approximations of the Dirac
equation using Hermite functions.  In particular, these schemes
preserve the essential conservation property of the Dirac equation. We
also present some numerical results which illustrate the effectiveness
of these methods.