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Homework Help: QM, central potential, system collapse

  1. Feb 12, 2008 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations
    (this is ~Fetter & Walecka Quantum theory of many-particle systems problem 1.2b)
    Homogeneous system of spin 1/2 particles, potential V.
    Expectation value of Hamiltonian in the non interacting ground state is
    [tex] E^{(0)} + E^{(1)} = 2 \sum_k^{k_F} \frac{\hbar^2 k^2}{2m} + 1/2 \sum_{k \lambda}^{k_F} \sum_{k' \lambda'}^{k_F} \big[ <k\lambda k'\lambda'|V|k\lambda k'\lambda'> - <k\lambda k'\lambda'|V|k'\lambda' k\lambda>\big] [/tex]

    Assume V is central and spin independant
    V(|x_1-x_2|) < 0 for all |x_1-x_2|
    The intergral of |V(x)| is finite

    Prove that the system will collapse.
    Hint: start from [tex] ( E^{(0)} + E^{(1)} ) / N [/tex] as a function of density

    3. The attempt at a solution

    If the energy per particle, [tex] ( E^{(0)} + E^{(1)} ) / N [/tex], is reduced when the density increases (which I guess means that N increases) the system will collapse.
    But how do I show that, I don't see what I can do.
  2. jcsd
  3. Feb 13, 2008 #2
    If the particle number increases, the sums might have a few more terms, how does that reduce the number of particles?
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