In response to Phil's latest post, I think I'd like to clarify why the values are so agreeable between semiclassical Bohr theory, and relativistic Bohr theory. Let's start by deriving the allowed Bohr orbital radii in the relativistic correction. Assuming circular orbits, we balance the centripetal force with the (Lorentz invariant) Coulomb force:

\[\frac{\gamma m_e v^2}{r}=\frac{e^2}{r^2}\]

where here I'm using Gaussian units for notational ease, \(m_e\) is the mass of an electron, and \(\gamma=\frac{1}{\sqrt{1-v^2/c^2}}\). We also make the Bohr assumption that angular momentum is quantized in units of Dirac's constant:

\[L=\gamma m_e v r=n\hbar\quad n=1,2,3,\dots\]

Eliminating the radius from the system of equations gives us \(v/c=\alpha/n\), where \(\alpha=\frac{e^2}{\hbar c}\) is the fine structure constant. Solving for the radius and using the definition of gamma, we find:

\[r=\frac{n^2 a_0}{\gamma}=n a_0\sqrt{n^2-\alpha^2}\]

Where \(a_0=\frac{\hbar^2}{m_e e^2}=52.9\) pm is the Bohr radius. In particular, the ground state is \(r_1=a_0\sqrt{1-\alpha^2}\), about 99.997% of the Bohr radius. So we see that the relativistic correction makes the ground state radius slightly smaller, a factor easily lost in rounding of values. This result also gives an interesting insight into atomic structure: consider a hydrogenic (one electron) atom, of atomic number Z. Then everywhere in our calculations, we have \(e^2\to Ze^2\). This is of particular interest in our expression for the velocity of the electron in the ground state:

\[v_1=Z\alpha c\]

Note that if Z is large enough (greater than 137), the velocity of the electron would exceed the speed of light. This puts an intrinsic limit on the size of atoms, namely that atomic numbers above 137 are essentially singular. Looking at the periodic table, we see that we have identified elements up to 118, so the results are in agreement.