  The path integral formulation can reproduce the right energy spectrum of the
harmonic oscillator potential, but it cannot resolve the Coulomb potential
problem. This is because the path integral cannot properly take into account
the boundary condition, which is due to the presence of the scattering states
in the Coulomb potential system. On the other hand, the Sommerfeld quantization
can reproduce the right energy spectrum of both harmonic oscillator and Coulomb
potential cases since the boundary condition is effectively taken into account
in this semiclassical treatment. The basic difference between the two schemes
should be that no constraint is imposed on the wave function in the path
integral while the Sommerfeld quantization rule is derived by requiring that
the state vector should be a single-valued function. The limitation of the
semiclassical method is also clarified in terms of the square well and
$\delta(x)$ function potential models.
