- #1

greisen

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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*[tex] \int d³r d³r' \frac{n²}{|r-r'|} [/tex]

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

[tex] -\frac{3}{4\pi} \int d³\tilde{r} \frac{1}{\tilde{r}} [/tex]

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

The first two terms I can convert to scaled coordinates <T>+<V> whereas I have some trouble for the last term

-½*[tex] \int d³r d³r' \frac{n²}{|r-r'|} [/tex]

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

[tex] -\frac{3}{4\pi} \int d³\tilde{r} \frac{1}{\tilde{r}} [/tex]

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

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