## 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]

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