How to directly calculate the polarizability

[sandf]

Dear all,
I want to directly calculate the polarizability using the following perturbation expression,
when the ground state wave function |0> is known.
for example, for He atom, r₁ and r₂ two electrons.

$\alpha = \frac{2}{3}\left\langle {0|({r_1} + {r_2})\frac{1}{{H - {E_0}}}({r_1} + {r_2})|0} \right\rangle$

How to deal with the term of $\frac{1}{{H - {E_0}}}$ ?

Any help or references will be appreciated.

Best regards.
Youzhao Lan

 

[vanhees71]

That's a bit a longer story. First, have a look on time-independent perturbation theory in a good textbook on quantum theory. The treatment in Sakurai is very good. Then look for the application on the Stark effect (2nd order perturbation theory).

 

[sandf]

Thank you very much, I get it.

Best regards.
Lan

 

[DrDu]

Usually, you insert a resolution of the identity in terms of eigenfunctions of H, i.e. $(H-E_0)^{-1}=\sum_i |i\rangle (E_i-E_0)^{-1}\langle i |$.

 

[sandf]

Dear DrDu,
Thank you very much! The infinite summation seems to be another difficult task for obtaining exact results.
Best regards.
Lan