Many-body problems in quantum mechanics

An important idea used in our approach to the many-body problem is to exploit the non-individuality of identical particles to relate the N-body problem to certain specially constructed 2-body problems. We study systems of N identical particles bound by pair potentials. We let H be the Hamiltonian for the system, with the centre-of-mass KE removed. Since the symmetrization postulate (for bosons or fermions) is in the individual particle indices, the expression of the Bose or Fermi symmetry in the translation-invariant many-body wave function may be quite complicated. In suitable units, and for either species of particle, we have <H> = (N-1)<K + (N/2)V>, where K and V are (reduced) two-body operators. Thus there is a close relation between H/(N-1) and the corresponding (reduced) two-body Hamiltonian H = K + (N/2)V. This relation allows one to derive general lower-bound energy formulas for the N-body problem: for strongly bound systems, these lower bounds are often surprisingly good; for harmonic oscillators, they may be exact. In the case of fermions, the attempt to express the anti-symmetry of the many-body wave function in expansions over two-body states leads to the use of non-orthogonal relative coordinates. These methods can also be applied to the excited N-body states.

Specific topics and the locations of their treatment in this work:

Boson systems  [2][5][25][39][41][42][43][53][60]

Fermion systems  [3][20][60]

Exact N-body solutions  [12][13][14][23][60]

Uniqeness of harmonic oscillator: lower bound is exact  [8][9]

Non-local interactions  [19]

Translation-invariant atoms  [15]

Nuclei  [2][6]

Quark systems  [18][24][27][32][40][41]

Relation to delta lower bound  [44]

Relation between E_N and E_K, K < N.   [30]

Excited N-body states  [7][20]

Very general energy-bound formulas by use of spectral geometry  [60]