How to Simplify the Hamiltonian for a Homogeneous System in Scaled Coordinates?

[greisen]

I have to show that the hamiltonian for a homogeneous system can be simplified in scaled coordinates.
The first two terms I can convert to scaled coordinates <T>+<V> whereas I have some trouble for the last term

-½*\int d³r d³r&#039; \frac{n²}{|r-r&#039;|}

where n is the density. The scaled coordinates can be expressed as \tilde{r}=\frac{a_0}{r_s} - r_s is a average distance between electrons and the expression can be written as

-\frac{3}{4\pi} \int d³\tilde{r} \frac{1}{\tilde{r}}

I have some troubles getting the last part - how can the two d³r d³r' be reduced to d³\tilde{r} - any hints or advise appreciated
thanks in advance

 

[greisen]

I can see that some of the exponents have vanished so the integral which gives me problems is

- \int dr^{3} dr'^{3} \frac{n^{2}}{|r-r'|}

which in scaled coordinates can be written as

- \frac{3}{4*pi}\int d\tilde{r}^{3} \frac{1}{\tilde{r}}