(6162) The fine structure of the Hydrogen spectrum
Consider the Hydrogen atom with
\( V_0(r)=-\alpha c /r \)
.
By expanding the exact solution of the Dirac equation
with respect to
\( \alpha \)
one obtains
\( \frac{E_{n,\ell,j,m}}{Mc^2} = 1 - \frac{\alpha^2}{2n^2} + \frac{1}{2} \left( \frac{\alpha^2}{2n^2} \right)^2 \left[ 3 - \frac{4n}{j+(1/2)} \right] + ... \)
Recover this expression by calculating the following corrections to the non-relativist result:
The kinetic correction
\( W_{K} = - \frac{1}{8m^3c^2}p^4 \)
,
the Darwin correction
\( W_{D} = \frac{1}{8m^2c^2}\nabla^2 V_0 \)
,
and the spin-orbit correction
\( W_{S} \)
.
Find also the
\( \alpha^5 \)
corrections due to Lamb shift.
These arise because of two reasons:
an extra term
\( W_{L}=-\frac{\alpha}{15\pi m^2c^2}\nabla^2 V_0 \)
,
and an anomalous spin-orbit coupling
\( g\approx 2+(\alpha/\pi) \)
.
All the results should be expressed in terms of
\( \alpha \)
and the quantum numbers
\( (n,\ell,j) \)
.