Summary: 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 \(0 < e^2/\pi\hbar c < 1\) this sequence does have the Cauchy property while for \(e^2/\pi\hbar c > 1\) it does not have this property. The average value of the first Casimir operator \(C_1 = -(\frac12) M_{\mu\nu}M^{\mu\nu}\) is shown to converge to the previously calculated eigenvalue \(z(2-z)\), \(0 < z = e^2/\pi\hbar c < 1\).
Parts I and III, Acta Phys. Pol. B 35, 2249–2260 (2004), and ibid. 40, No. 12, 3771– 3775 (2009).