April 15, 1998

PHY 442 - Problem Set # 18

Consider eigenstates of the one-dimensional Dirac equation:

$$[ \alpha_z p_z + \beta mc^2 + V(z) ] \Psi(z) = E \Psi(z),$$

where

$$V(z) = \left\{ \begin{array} {ll} -V_0 & \;\;\;\; {\rm{for}}... .../2 \\ 0 & \;\;\;\;\;\;{\rm{otherwise.}} \end{array} \right.$$

1.

For $0 \leq V_0 \ll mc^2$, set derive the transcendental equations for the bound states of the system and show how the results are related to the corresponding non-relativistic system.

2.

Consider what happens to the solutions when V₀ > 2mc².