Statistical Mechanics of Coupled Particles in the Schrödinger Representation

A perturbation treatment is presented for the operator $u\left(t\right)\mathrm{}$ giving the evolution of the state vector of a system of $N$ particles coupled by 2-body forces. The total Hamiltonian is divided into a diagonal part $H_0$ and off-diagonal part $H_1$ in the representation with $N$-particle plane-wave states as basis. An effective pair-interaction operator ${\overline{H}}_1$ is defined which includes "vertex" corrections of all orders, and the off-diagonal part of $u$ is expressed exactly by an expansion which involves only ${\overline{H}}_1$ and the exact diagonal part of $u$. A closed set of equations determining ${\overline{H}}_1$ and the diagonal part of $u$ are obtained by retaining only the leading terms in the expansion. These equations include all the corrections included in the Brueckner approximation and, in addition, they contain 3-body effects, iterated to all orders, which are omitted in that approximation. No one-to-one correspondence between the eigenstates of $H_0$ and those of the total Hamiltonian is appealed to, and the ground state plays no special role. The connection with statistical mechanics follows the fact that the partition function is $\mathrm{Tr}\left[u\right(-\frac{i}{\mathrm{kT}}\left)\right]$. The theory simplifies greatly for very large $N$.