Editing Indistinguishable particles (section)

=== ''N'' particles ===

The above discussion generalizes readily to the case of ''N'' particles. Suppose there are ''N'' particles with quantum numbers ''n''<sub>1</sub>, ''n''<sub>2</sub>, ..., ''n''<sub>''N''</sub>. If the particles are bosons, they occupy a '''totally symmetric state''', which is symmetric under the exchange of ''any two'' particle labels:
: <math>|n_1 n_2 \cdots n_N; S\rang = \sqrt{\frac{\prod_n m_n!}{N!}} \sum_p \left|n_{p(1)}\right\rang \left|n_{p(2)}\right\rang \cdots \left|n_{p(N)}\right\rang </math>

Here, the sum is taken over all different states under [[permutation]]s ''p'' acting on ''N'' elements. The square root left to the sum is a [[normalizing constant]]. The quantity ''m<sub>n</sub>'' stands for the number of times each of the single-particle states ''n'' appears in the ''N''-particle state. Note that {{nowrap|1=Σ<sub>''n''</sub> ''m''<sub>''n''</sub> = ''N''}}.

In the same vein, fermions occupy '''totally antisymmetric states''':
: <math>|n_1 n_2 \cdots n_N; A\rang = \frac{1}{\sqrt{N!}} \sum_p \operatorname{sgn}(p) \left|n_{p(1)}\right\rang \left|n_{p(2)}\right\rang \cdots \left|n_{p(N)}\right\rang\ </math>

Here, {{math|sgn(''p'')}} is the [[parity of a permutation|sign]] of each permutation (i.e. <math>+1</math> if <math>p</math> is composed of an even number of transpositions, and <math>-1</math> if odd). Note that there is no <math>\Pi_n m_n</math> term, because each single-particle state can appear only once in a fermionic state. Otherwise the sum would again be zero due to the antisymmetry, thus representing a physically impossible state. This is the [[Pauli exclusion principle]] for many particles.

These states have been normalized so that
: <math> \lang n_1 n_2 \cdots n_N; S | n_1 n_2 \cdots n_N; S\rang = 1, \qquad \lang n_1 n_2 \cdots n_N; A | n_1 n_2 \cdots n_N; A\rang = 1. </math>