  We investigate the asymptotic behavior of the solutions to the Klein-Gordon
and Dirac equations using the local spatial averaging approach to Bohr's
correspondence principle in the large principal quantum number regime. The
procedure is applied in two basic problems in $1+1$-dimensions, the
relativistic quantum oscillator and the relativistic particle in a box. In the
harmonic oscillator cases, we find that the corresponding probability densities
reduce to their respective classical single-particle distributions plus a
series of terms suppressed by powers of the $\hbar$ constant, while particle in
a box cases show a different structure for the quantum corrections.
