Relativistic kinematics momentum

The result about the velocity obtained above is not what we expect from classical relativistic kinematics. In classical mechanics, the connection between the kinietic energy E, the momentum p, and the velocity v can be described by the formula... 

Obviously, the relativistic momentum pi is also a Lorentz 4-vector, where we have introduced the relativistic (kinematic) 3-momentum... 

It is useful to clarify the various ways in which relativity enters the nucleon-nucleus scattering problem. With respect to kinematics, the nucleon-nucleus center-of-momentum (COM) system wave number and reduced mass, which appear in the Schrodinger equation, are replaced by corresponding relativistic quantities. Also the transformation of the NN scattering amplitude in the NN COM system (where it is known) to the nucleon-nucleus COM frame (where it is needed for the calculation) is done using a proper Lorentz boost [Me 83a]. Both of these procedures for accounting for relativistic kinematics are included in the nonrelativistic scattering calculations done here and shown in this work. 

In fig. 10 the separate effects of Wigner rotation and the NN COM energy shift are shown. These results, from ref. [Me 83a], are from nonrelativistic impulse approximation on-shell tp model calculations. Three calculations are considered. The first uses only the M0ller factor the second additionally includes the Wigner rotation matrix the third uses the full, relativistic kinematic transformation by additionally including the shift in the NN COM system energy as a function of momentum... 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

The relativistic correction to the potential is no more singular than the potential itself in this limit and therefore will support bound states. In the small momentum limit, when the electron is far from the nucleus, the potential goes as 1 /r and is therefore a short-range potential. It can be seen that the kinematic factors provide a cutoff to the potential that is absent in the Pauli approximation and that permits variational calculations with the free-particle Foldy-Wouthuysen transformed Hamiltonian. 

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