# QM, central potential, system collapse

1. Feb 12, 2008

### WarnK

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
$$E^{(0)} + E^{(1)} = 2 \sum_k^{k_F} \frac{\hbar^2 k^2}{2m} + 1/2 \sum_{k \lambda}^{k_F} \sum_{k' \lambda'}^{k_F} \big[ <k\lambda k'\lambda'|V|k\lambda k'\lambda'> - <k\lambda k'\lambda'|V|k'\lambda' k\lambda>\big]$$

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 $$( E^{(0)} + E^{(1)} ) / N$$ as a function of density

3. The attempt at a solution

If the energy per particle, $$( E^{(0)} + E^{(1)} ) / N$$, 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. Feb 13, 2008

### WarnK

If the particle number increases, the sums might have a few more terms, how does that reduce the number of particles?