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Homework Help: Hamiltonian for homogeneous system

  1. Feb 12, 2008 #1
    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
    Last edited by a moderator: Feb 13, 2008
  2. jcsd
  3. Feb 15, 2008 #2
    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}}
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