The Pauli principle in the center-of-mass system

Abstract

The internal states of motion of an arbitrary closed system are studied in terms of a set of 3 N − 3 pairs of internal canonical variables. The complexities involved in constructing states which satisfy the Pauli principle are reduced by expressing these states in terms of ν sets of internal variables which are associated with the ν ways in which one of ν identicals particle may be eliminated. A formalism is set up wherby a simplified expression for the scalar product (and hence matrix elements) is obtained. For purposes of illustration the method is applied to systems of N spinless bosons and fermions interacting through oscillator forces and constrained to move in one dimension. For N identical particles it is shown that the internal Hamiltonian assumes an independent “particle” form only for a system of oscillators. In a section concerned with internal operators, several useful properties of the internal motion of an arbitrary closed system are discussed.