High-energy tail of the linear momentum distribution inthe ground state of hydrogen atoms or hydrogen-like ions

Abstract

A long-standing dispute concerning the high-energy tail of thelinear momentum distribution (HTMD) in the ground state ofhydrogen atoms/hydrogen-like ions (GSHA) has beenunresolved up to now. A possible resolution of the abovedispute might be connected to the problem of the role ofsingular solutions of quantal equations, which is afundamental problem in its own right. The paradigm isthat, even allowing for finite nuclear sizes, singular solutions of the Dirac equation for the Coulomb problem should be rejected for nuclear charges Z < 1/α≈137. In this paper we break this paradigm. First, we derive a generalcondition for matching a regular interior solution with asingular exterior solution of the Dirac equation forarbitrary interior and exterior potentials. Then wefind explicit forms of several classes of potentials that allowsuch a match. Finally, we show that, as an outcome, the HTMD forthe GSHA acquires terms falling off much slower than the1/p6-law prescribed by the previously adoptedquantal result. Our results open up a unique way to testintimate details of the nuclear structure by performing atomic(rather than nuclear) experiments and calculations.