12 Quantum statistics

Properly symmetrized states of more than two bosons are obtained by adding all distinct permutations of the individual states and dividing by the root of the number of such permutations. If two of the three states of three bosons of the same type are identical, we thus obtain something like

(|aab> + |aba> + |baa>) /√3,

and if all three states are distinct, we obtain something like

(|abc> + |acb> + |bac> + |bca> + |cab> + |cba>) /√6.

Suppose now that n bosons have been found in possession of the same complete set of properties X, and that one boson has been found in possession of a different complete set of properties Y. What is the probability of finding all n+1 bosons in possession of X?

The initial state is

(|X…XY> + ··· + |X…XYX…X> + ··· + |YX…X>) /√(n+1).

n+1 terms are being summed, with Y in a different position in each term. The final state is |X…X>, which has X in all n+1 positions. The transition amplitude thus is

(<X…X|X…XY> + ··· + <X…X|YX…X>) /√(n+1).

Since the n+1 terms in brackets are all equal to <X|X>n <X|Y>, the transition amplitude boils down to

√(n+1) <X…X|X…XY>,

so that the wanted probability turns out to be

(n+1) |<X…X|X…XY>|2.

If instead of dealing with indistinguishable bosons — bosons of the same type, fully described by X or Y — we were dealing with distinguishable bosons — bosons with distinguishing properties (“identity tags”) represented by their position in the state of the composite system (first, second, third, …) — then the initial state would be, say, |X…XY>, and the corresponding transition probability would be |<X…X|X…XY>|2.

The upshot: if n bosons have been found in possession of the same set of properties X and one boson has been found in possession of a different set of properties Y, then the probability of subsequently finding all n+1 bosons in possession of X is n+1 times as large if the sets of properties X and Y are complete, than it is if the positions of the individual boson states in the state of the composite system represent differences that exist in the actual world.

While identical (indistinguishable) bosons are gregarious, identical (indistinguishable) fermions are solitary. Since fermion amplitudes change sign each time states are swapped between particles, fermion states must be antisymmetrized. Antisymmetric states are obtained by adding all even permutations of the individual states and subtracting all odd permutations, before dividing by the root of the number of permutations. (An even permutation of elements x1, x2, x3, … is obtained by swapping an even number of elements, whereas an odd permutation is obtained by swapping an odd number of elements.)

Since no antisymmetric state can be formed of two identical fermion states, the probability of finding two fermions in possession of the same complete set of properties is zero. This is the content of the exclusion principle, which was first formulated by Wolfgang Pauli.

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