The Schrödinger Potential Function

Abstract

The notion of odd and even operators introduced by Schrodinger is generalized and developed in this paper. It is shown that, accepting the hypothesis of Schrodinger that "only even operators exist" the undesirable negative energy solutions of the Dirac equation are eliminated since they do not combine with the observed states, while at the same time the relativistically invariant form of the Dirac equation can be retained. The fine structure formula for the energy levels remains exact and not merely approximate. However, it is proposed to relinquish these apparent advantages in order to substitute for Schrodinger's hypothesis one easier to accept but practically equivalent. This is that the fundamental potential between charges is an even operator. Possible ways of forming even operators out of classical mixed ones are discussed. The decomposition of the fundamental operators, ai the "spin" operator, and %i the coordinate operator, indicates a remarkably simple form for the odd and even parts. The even parts prove to be immediately connected with observable properties of the electron. This leads one to believe that the division into odd and even parts is a useful performance aside from its application to the question of negative energy solutions. While the positive to negative transitions are made to vanish identically, the direct transitions between positive states remain quite unaffected. The "Klein difficulty" is examined and it proves to disappear with the negative energy states. Possible experimental tests of the new theory are shown to exist. The values of hyperfine structure separations and also the x-ray levels of heavy atoms should depend markedly on whether the Coulomb potential or just its even part acts on the electron. Some interesting possibilities suggested by the theory are mentioned at the end of the paper.T HE most serious difficulty which till recently beset the theory of the electron has now been removed by Schrodinger. 1 This difficulty lay in the fact that whereas the Dirac theory accounted in exact detail for all the known properties of the electron, it also predicted certain others so strange that no one even thought of testing them out. The unbelievable properties referred to were all connected with the fact that there were solutions (hereafter called negative solutions) of the equations corresponding to total energies which in spite of the large positive term mc 2 were numerically less than zero.The behavior of a negative electron in a state described by one of these solutions was similar to that expected of a positive charge. A possible interpretation for some of these features has been suggested by Dirac 2 but enough discordance with observation remained to make the situation quite unsatisfactory. It should also be recalled that Dirac's suggestion left the Klein 3 diffi-1 E. Schrodinger, Sitz. Ber. d.