A new proof of existence of a bound state in the quantum Coulomb field II

The wave function of the bound state of the quantum Coulomb field is explicitly constructed. We define a sequence of obviously legal states in the Hilbert space of the quantum theory of the Coulomb field and show that for 0 < e2 / πħc < 1 this sequence does have the Cauchy property while for e2 / πħc > 1 it does not have this property. The average value of the first Casimir operator C1 = −(½)MμνMμν is shown to converge to the previously calculated eigenvalue z(2−z), 0 < z = e2 / πħc < 1.