The two-component approach to the one-dimensional Dirac equation is applied to the Woods-Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E = −m) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero-momentum when the corresponding well supports a half-bound state at E = −m is demonstrated.

It is shown that a Dirac particle of mass m and arbitrarily small momentum will tunnel without reflection through a potential barrier V = U(c)(x) of finite range provided that the potential well V = -U(c)(x) supports a bound state of energy E = -m. This is called a supercritical potential well.

It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is −1 and hence the transmission coefficient T = 0 in general. If however the potential supports a half-bound state at momentum k = 0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T = 1. Therefore in some circumstances a Dirac particle of arbitrarily small momentum can tunnel without reflection through a potential barrier.

We review the analytic results for the phase shifts δ l (k) in non-relativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the non-relativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E = m. Resonances resulting from scattering from a barrier can be explained in terms of the 'crossing' theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extract phase shifts for general short range potentials is introduced and illustrated by considering relativistic scattering from a Gaussian potential well and barrier.