We have shown in the section on conserved quantities that the operator
The operator may be written in several ways.
Assume that the eigenvalues of are given by
The eigenvalues of are
We can use commutation and anticommutation relations to write in terms of separate angular and radial operators.
Note that the operators and act only on the angular momentum parts of the state. There are no radial derivatives so they commute with . Lets pick a shorthand notation for the angular momentum eigenstates we must use. These have quantum numbers , , and . will have and must have the other possible value of which we label . Following the notation of Sakurai, we will call the state . (Note that our previous functions made use of particularly in the calculation of and .)
The effect of the two operators related to angular momentum can be deduced. First, is related to . For positive , has . For negative , has . For either, has the opposite relation for , indicating why the full spinor is not an eigenstate of .
We now have everything we need to get to the radial equations.
This is now a set of two coupled radial equations. We can simplify them a bit by making the substitutions and . The extra term from the derivative cancels the 1's that are with s.
These equations are true for any spherically symmetric potential. Now it is time to specialize to the hydrogen atom for which . We define and and the dimensionless . The equations then become.
With the guidance of the non-relativistic solutions, we will postulate a solution of the form
For the lowest order term , we need to have a solution without lower powers. This means that we look at the recursion relations with and solve the equations.
As usual, the series must terminate at some for the state to normalizable. This can be seen approximately by assuming either the 's or the 's are small and noting that the series is that of a positive exponential.
Assume the series for and terminate at the same . We can then take the equations in the coefficients and set to get relationships between and .
The final step is to use this result in the recursion relations for to find a condition on which must be satisfied for the series to terminate. Note that this choice of connects and to the rest of the series giving nontrivial conditions on . We already have the information from the next step in the recursion which gives .
Using the quantum numbers from four mutually commuting operators, we have solved the radial equation in a similar way as for the non-relativistic case yielding the exact energy relation for relativistic Quantum Mechanics.
We can identify the standard principle quantum number in this case as . This result gives the same answer as our non-relativistic calculation to order but is also correct to higher order. It is an exact solution to the quantum mechanics problem posed but does not include the effects of field theory, such as the Lamb shift and the anomalous magnetic moment of the electron.
Relativistic corrections become quite important for high atoms in which the typical velocity of electrons in the most inner shells is of order .
Jim Branson 2013-04-22