= Solution
<identical particles> have the same intrinsic properties. Exchanging their bookkeeping labels cannot distinguish a physical preparation or change a measurable prediction. This is stronger than merely failing to track classical trajectories. For ordinary <bosons> and <fermions>, the many-particle <wavefunction> has respectively <bosonic exchange symmetry> or <fermionic exchange symmetry> under every <particle exchange operator>, acting on spatial and <spin> coordinates together. The <Spin-statistics theorem> connects integer <spin> to <bosons> and half-integer <spin> to <fermions>; this connection uses relativistic quantum-field assumptions, rather than following from the nonrelativistic <Schrödinger equation> alone.
For <one-particle states> $u,v$ satisfying <orthonormality>, the two alternatives form a <quantum superposition>:
$$
\Psi_\pm(1,2)=\frac{u(1)v(2)\pm v(1)u(2)}{\sqrt2}.
$$
Its <probability density> contains an exchange interference term, absent from an incoherent mixture of labelled alternatives. The antisymmetric expression vanishes when both <one-particle states> are identical: \b[one complete <one-particle state> can contain at most one <fermion>]. This <Pauli exclusion principle> explains <electron shell> filling, the electronic <Fermi sea> and <degeneracy pressure>. Opposite <spins> are different <one-particle states>, so two <electrons> may share an <atomic orbital>. Spatially separated <wave packets> with negligible overlap can often be treated as distinguishable for local measurements; the underlying exchange symmetry remains.
In <thermal equilibrium>, occupation of a level of <energy> $\epsilon$ obeys
$$
\bar n_{\rm B}=\frac1{e^{(\epsilon-\mu)/(k_BT)}-1},\qquad
\bar n_{\rm F}=\frac1{e^{(\epsilon-\mu)/(k_BT)}+1}.
$$
<bosons> can accumulate macroscopically in one state, giving <Bose-Einstein condensation>; <fermions> instead fill successive states up to the <Fermi energy> at zero <temperature>. When occupation is dilute, both expressions reduce to the <Maxwell-Boltzmann distribution>. The identical-particle classical <partition function> contains $1/N!$, avoiding spurious <entropy> from permutations and the <Gibbs paradox>. Exchange effects become conspicuous when the <thermal de Broglie wavelength> is comparable to the interparticle spacing.
An operational test is the <Hong-Ou-Mandel effect>. Down-converted <photon> pairs enter separate ports of a balanced <beam splitter>; indistinguishable alternatives for one <photon> per output cancel. Scanning the relative arrival delay restores coincidences away from overlap. Matching <polarization>, spectral filtering and spatial alignment controls distinguishability; measuring delayed accidental coincidences controls background. The observed dip tests two-photon interference, and its finite depth must not automatically be interpreted as a failure of bosonic symmetry. The original apparatus varied beam-splitter position and subtracted a separately measured accidental background.<Measurement of Subpicosecond Time Intervals between Two Photons by Interference by C. K. Hong and others>[pp. 2044–2046, Fig. 1, Eq. (2), coincidence-background procedure and Fig. 2]{ref}
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