A Electrostatic interaction inside and outside the source

AI Thread Summary
The discussion focuses on calculating the electrostatic interaction between an electron and a nucleus, considering the electron's potential presence within the nucleus. The user presents a double integral involving the electron and nuclear coordinates and seeks to simplify the expression for the interaction term, specifically the term ##\frac{1}{|r_e-r_n|}##. A suggested approach involves expanding this term using Legendre polynomials, which allows for a clearer representation of the interaction. The expansion can facilitate further simplification and truncation as needed for practical calculations. The conversation emphasizes the mathematical techniques necessary for handling complex integrals in quantum mechanics.
kelly0303
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Hello! I want to get the electrostatic interaction (between and electron and a nucleus), while accounting for the fact that the electron can also be inside the nucleus (e.g. in an S##_{1/2}## state). I ended up with this double integral:

$$\int_{r_e=0}^{r_e=\infty}\int_{r_n=0}^{r_n=R}\frac{\rho(r_n)}{|r_e-r_n|}d^3r_ed^3r_n$$

where ##r_e## and ##r_n## are the electron and nuclear coordinates and ##R## is the nuclear radius. Please note that we are not necessarily assuming that the nucleus is a perfect sphere (although it is usually very close to it). How can I expand the ##\frac{1}{|r_e-r_n|}## and get this into a simpler form that I can also truncate as needed? Thank you!
 
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That function is expanded in Legendre polynomials as
$$\frac{1}{|{\bf r}-{\bf r'}|}=\sum_l\frac{r'^l}{r^{l+1}}P_l(\cos\theta)$$,
with the smaller r in the numerator.
 
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