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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