Stephan Zakowicz
Square-Integrable Wave Packets from the Volkov Solutions
(361K, Postscript)
ABSTRACT. Rigorous mathematical proofs of some properties of the Volkov solutions are presented, which describe the motion of a relativistic charged Dirac particle in a classical, plane electromagnetic wave. The Volkov solutions are first rewritten in a convenient form, which clearly reveals some of the symmetries of the underlying Dirac equation. Assuming continuity and boundedness of the electromagnetic vector potential, it is shown how one may construct square-integrable wave packets from momentum distributions in the space $\mathcal{C}^{\infty}_0(\mathbb{R}^3)^4$. If, in addition, the vector potential is $\mathcal{C}^1$ and the derivative is bounded, these wave packets decay in space faster than any polynomial and fulfill the Dirac equation. The mapping which takes momentum distributions into wave packets is shown to be isometric with respect to the $L^2(\mathbb{R}^3)^4$ norm and may therefore be continuously extended to a mapping from $L^2(\mathbb{R}^3)^4$. For a momentum function in $L^1(\mathbb{R}^3)^4 \cap L^2(\mathbb{R}^3)^4$, an integral representation of this extension is presented.