Unitary Principle and Real Solution of Dirac Equation

The exact solution of the Dirac equation with the exact boundary condition is further investigated in the present paper. We introduce the unitary principle that can be used to disclose all logic paradoxes hidden in natural science and mathematics. By the unitary principle, we show that the existence of solution of differential equation requires an important modification to the constant of angular momentum in the original Dirac equation. Applying it to the hydrogen-like atoms, we derive the real solution involving the zero energy and the quantum proton radius as well as the neutron ionization energy. However, the expression for the discrete energy levels in the real solution only depends on the radial quantum number. How to treat the difference between the real expression for the discrete energy levels of the Dirac-Coulomb equation and the experimental observation of the atomic spectrums of hydrogen is not resolved yet.