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Starting from an arbitrary orthonormal basis {|uk〉} of the state space for one particle, we constructed in § C-3-d of Chapter XIV a basis of the state space for N identical particles. Its vectors are characterized by the occupation numbers ni, with:

(A-4)

where n₁ is the occupation number of the first basis vector |u₁〉 (i.e. the number of particles in |u₁〉), n₂ that of |u₂〉, ..,nk that of |uk〉. In this series of numbers, some (even many) may be zero: a given state has no particular reason to always be occupied. It is therefore often easier to specify only the non-zero occupation numbers, which will be noted ni, nj, .., ni,.. . This series indicates that the first basis state that has at least one particle is |ui〉 and it contains ni particles; the second occupied state is |uj〉 with a population nj, etc. As in (A-4), these occupation numbers add up to N.

Comment:

In this chapter we constantly use subscripts of different types, which should not be confused. The subscripts i, j, k, l, ..denote different basis vectors {|ui〉} of the state space ₁ of a single particle; they span values given by the dimension of this state space, which often goes to infinity. They should not be confused with the subscripts used to number the particles, which can take N different values, and are labeled q, q′, etc. Finally the subscript α distinguishes the different permutations of the N particles, and can therefore take N! different values.