We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr-Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. In this talk, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set.