Sommerfeld’s Fine Structure Constant and Born’s Reciprocity

Summary

The problem of the electronic charge distribution together with that of the nagnitude of Sommerfeld’s fine structure constant α = 2πe ² /hc = I/I37 is reduced to a linear homogeneous integral equation for the density amplitude ψ. The kernel depends on the parameter α, and every value of α determines an eigenfunction. One eigen value α and one eigenfunction is selected by an additional condition which is, the energy balance. The results are based on a hyperbolic sine relation between the momentum measurements of an inside and an outside observer, and on a similar relation between the space measurements of the two observers. Whereas the relations in momentum space are derived from the familiar Einstein energy-momentum equation, the space geometry of the particle is obtained by an application of the principle of reciprocity of M. Born.¹