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\begin{document}

\( H_{E}=-eE^{z}z \). No coupling to \( \mu ^{i} \) so can ignore spin. In the ground state \( L^{i}S^{i}=0 \) so \( \triangle E^{(1)}=-eE^{z}<nlm_{l}|z|nlm_{l}> \).
For the ground state, this is 0 to first order.

In second order, you get: \( \triangle E^{(2)}=e^{2}E^{z2}\sum _{nlm}\frac{|<nlm|z|100>|^{2}}{E_{n}-E_{0}} \). numerator not 0 for l=1 and n\(>\)1.
Approximation gives: \( \triangle E^{(2)}\leq e^{2}E^{z2}\sum _{nlm}\frac{|<nlm|z|100>|^{2}}{E_{2}-E_{0}} \)

\( =e^{2}E^{z2}\frac{1}{\frac{3}{4}Ry}\sum _{nlm}<100|z|nlm><nlm|z|100> \)

\( =e^{2}E^{z2}\frac{1}{\frac{3}{4}Ry}<100|z^{2}|100> \)\( =e^{2}E^{z2}\frac{1}{\frac{3}{4}Ry}\frac{1}{3}<100|r^{2}|100> \)

\( =e^{2}E^{z2}\frac{a_{0}^{2}}{\frac{3}{4}Ry} \)\( =\frac{8}{3}E^{z2}a_{0}^{3} \)= quadratic stark effect. Polarization= \( \frac{\partial \triangle E}{\partial E} \). Polarizability= \( \frac{\partial }{\partial E} \)polarization.
This is \( \frac{16}{3}a_{0}^{3} \). Actual calculation gives \( \frac{9}{2}a_{0}^{3} \) which agrees well with experimentation.

For excited states:

For n=2, there is 8 fold degeneracy

With spin orbit coupling, you get a 6 fold degeneracy on \( 2P_{\frac{3}{2}} \).

Weak fileds -\(>\) non-degenerate P.T. =\(>\) 2nd order quadratic.

Strong fields, then 2s and 2p are degenerate.

\( <210|-eE^{z}z|200>=-3eE_{z}a_{0} \). Constructing a 4x4 matrix, you get:

\( \begin{array}{ccccc}
0 & -3eE_{z}a_{0} & 0 & 0 & |200>\\
-3eE_{z}a_{0} & 0 & 0 & 0 & |210>\\
0 & 0 & 0 & 0 & |211>\\
0 & 0 & 0 & 0 & |21-1>
\end{array}
 \). Diagonilizing, you get: the roots, 0,0,\( \pm 3eE^{z}a_{0} \). The eigen states are
\( \frac{1}{\sqrt{2}}(|200>+|210>) \), \( \frac{1}{\sqrt{2}}(|200>-|210>) \). This is the linear stark effect. Only shows up for strong fields,
when lamb shifts and spin orbit coupling are neglectable. This means
\( E^{z}\sim 20kV/cm \).

\end{document}
