= Solution
<Indistinguishable particles> have the same intrinsic properties and cannot be assigned physically observable permanent labels. Exchanging their mathematical coordinates leaves all physical predictions unchanged. In ordinary three-dimensional quantum mechanics the total state, including <spin>, obeys
$$
\boxed{P_{12}\Psi=+\Psi\quad\text{for bosons},\qquad P_{12}\Psi=-\Psi\quad\text{for fermions}.}
$$
The <Spin-statistics theorem> associates integer <spin> with bosons and half-integer <spin> with fermions. Exchange does not mean physically transporting two particles along some prescribed trajectories; it is the symmetry of the many-particle state under permutation of its arguments.
For <orthonormal> one-particle states $u,v$, normalized two-particle spatial combinations are $[u(1)v(2)\pm v(1)u(2)]/\sqrt2$. They illustrate coherent exchange amplitudes. If both fermions have the same complete one-particle state, the antisymmetric combination vanishes. This is the <Pauli exclusion principle>: a fermionic mode has occupation zero or one. Bosonic modes can have arbitrarily large occupation. A many-fermion product basis is antisymmetrized by a <Slater determinant>.
It is the full spin-spatial state that must have the appropriate symmetry. For two electrons, an antisymmetric <spin> <singlet state> can accompany a symmetric spatial orbital, so opposite-spin electrons can occupy the same spatial orbital. A symmetric <spin> triplet instead needs an antisymmetric spatial state. Thus exclusion concerns identical complete states, not an unconditional prohibition on sharing a <position> or spatial orbital.
Counting allowed occupations gives <Bose-Einstein statistics> and <Fermi-Dirac statistics>. For an ideal gas in <thermal equilibrium>,
$$
\overline n_i=\frac1{e^{(\epsilon_i-\mu)/(k_BT)}-1}\quad\text{for bosons},\qquad \overline n_i=\frac1{e^{(\epsilon_i-\mu)/(k_BT)}+1}\quad\text{for fermions}.
$$
Bosonic accumulation permits <Bose-Einstein condensation>; exclusion gives a filled <Fermi sea> and degeneracy pressure even at zero <temperature>. In the dilute classical limit the occupations are small and both expressions approach the <Maxwell-Boltzmann distribution>. Exchange effects become important when wavepackets overlap appreciably, often characterized by number density times the cube of the <thermal wavelength>.
Composite particles follow their total exchange statistics while remaining intact. The <composite statistics of neutral atoms> depend on electrons as well as nucleons, as in the two lithium isotopes above. Symmetrization can produce bunching or antibunching in joint detection probabilities even in the absence of an interaction potential; it is not an additional classical force.
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