a relativistic single particle. However, as we will see later, this equation is quite meaningful as a relativistic field equation.
1.3.1 Klein paradox. Let us consider a charged scalar particle described by the Klein-Gordon equation (1.40) in an external electromagnetic field. We recall that the coupling of a charged particle to an electromagnetic field is given by the minimal coupling
where we have used the coordinate representation for the momentum as in (1.33) and Aµ denotes the vector potential associated with the electromagnetic field. In this case, therefore, the scalar particle will satisfy the minimally coupled Klein-Gordon equation (e > 0, namely, the particles are chosen to carry positive charge)
As a result, the probability current density in (1.46) can be determined to have the form
where we have defined
With this general description, let us consider the scattering of a charged scalar (Klein-Gordon) particle with positive energy from a constant electrostatic potential. In this case, therefore, we have
For simplicity, let us assume the constant electrostatic potential to be of the form
and we assume that the particle is incident on the potential along the z-axis as shown in Fig. 1.2.
Figure 1.2: Klein-Gordon particle scattering from a constant electrostatic potential.
The dynamical equations will now be different in the two regions, z < 0 (region I) and z > 0 (region II), and have the forms (see (1.54))
In region I, there will be an incident as well as a reflected (plane) wave so that we can write (remember that the incident particle has positive energy)
while in region II, we only expect a transmitted wave (traveling to the right) of the form
where A, B are related respectively to reflection and transmission coefficients. We note here that the continuity of the wave function at the boundary z = 0 requires that the energy be the same in the two regions.
For the wave functions in (1.60) and (1.61) to satisfy the respective equations in (1.59), we must have
Here we have used the fact that the energy of the incident particle is positive and, therefore, the square root in the first equation in (1.62) is with a positive sign. However, the sign of the square root in the second relation remains to be fixed.
Let us note from the second relation in (1.62) that p′ is real for both E − eΦ0 > m (weak potential) and for E − eΦ0 < −m (strong potential). However, for a potential of intermediate strength satisfying −m < E − eΦ0 < m, we note that p′ is purely imaginary. Thus, the behavior of the transmitted wave depends on the strength of the potential. As a result, in this second case, we must have
in order that the wave function is damped in region II. To determine the sign of the square root in the cases when p′ is real, let us note from the second relation in (1.62) that the group velocity of the transmitted wave is given by
Since we expect the transmitted wave to be travelling to the right, we determine from (1.64) that
This, therefore, fixes the sign of the square root in the second relation in (1.62) for various cases.
Matching the wave functions in (1.60) and (1.61) and their first derivatives at the boundary z = 0, we determine
so that we determine
Let us next determine the probability current densities associated with the different waves. From (1.55) as well as the form of the potential in (1.58) we obtain
where we have used (1.67) as well as the fact that, while p is real and positive, p′ can be positive or negative or even imaginary depending on the strength of the potential
