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WarnK

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## Homework Statement

## Homework 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

## 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.