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The Dirac equation with the Coulomb potential is studied. It is shown thatthere exists a new invariant in addition to the known Dirac and Johnson-Lippmanones. The solution of the Dirac equation, using the generalized invariant, andexplicit expressions for the bispinors corresponding to the three sets of theinvariants, their eigenvalues and quantum numbers are obtained. The generalsolution of the Dirac equation with the Coulomb potential is shown to containfree parameters, whose variation transforms one particular solution into anyother and controls spatial electron probability amplitude and spinpolarization. The electron probability densities and spin polarizations areobtained in the general form and calculated explicitly for some electron statesin the hydrogen-like energy spectrum. The spatial distributions of thesecharacteristics are shown to depend essentially on the invariant set,demonstrating physical difference of the states corresponding to differentinvariants.

General solution of the Dirac equation with the Coulomb potential