= Solution
\b[<identical particles> have the same intrinsic properties], including mass, charge and <spin>. Their labels are bookkeeping devices rather than measurable identities. When <wave functions> overlap, <quantum mechanics> supplies no observable trajectory that permanently identifies “particle 1” independently of “particle 2”. A many-particle description must therefore treat exchange consistently. The <Hamiltonian> and physical <observables> are invariant under relabelling <identical particles>.
For two particles, the <exchange operator> obeys $\hat P_{12}^2=\hat I$. The usual three-dimensional <symmetrization postulate> selects states symmetric under exchange for <bosons> and antisymmetric for <fermions>. In relativistic quantum theory, the <Spin-statistics theorem> associates integer <spin> with <bosons> and half-integer <spin> with <fermions> under its locality and positivity assumptions; the connection is not derived merely from $\hat P_{12}^2=\hat I$ in elementary nonrelativistic mechanics. For orthonormal one-particle states $u$ and $v$, the two-particle spatial combinations are
$$
\Psi_\pm(1,2)=\frac{u(1)v(2)\pm v(1)u(2)}{\sqrt2}.
$$
The sign concerns the \b[complete state], including <spin> and other internal variables, not necessarily the spatial factor alone.
The antisymmetric state vanishes if both <fermions> occupy the same one-particle state: this is the \b[<Pauli exclusion principle>]. For $N$ <fermions> in orthonormal <spin orbitals> $\chi_j$, the normalized <Slater determinant> $\Psi=(N!)^{-1/2}\det[\chi_j(i)]$ supplies the required exchange sign. Bosonic occupation has no corresponding upper limit, and symmetrized products describe many <bosons>. Fermionic occupation is zero or one per complete one-particle state; two electrons can still occupy the same spatial <atomic orbital> with opposite <spin>.
<Spin> and spatial symmetry are linked. A two-electron <spin singlet state> is antisymmetric, so its spatial factor must be symmetric. A symmetric <triplet state> instead requires an antisymmetric spatial factor. Exchange then changes pair correlations and the energies available when interactions depend on particle separation, even without introducing a new classical “exchange <force>”. It contributes to atomic shell structure, bonding and magnetic behaviour.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2016/paper-1-exchange-correlations.png]
{title=Symmetric and antisymmetric two-particle spatial states have different joint-position correlations}
{source=paper-1-exchange-correlations.py}
In an occupation-number description, bosonic <creation and annihilation operators> satisfy $[\hat a_i,\hat a_j^\dagger]=\delta_{ij}$, whereas fermionic operators satisfy $\{\hat c_i,\hat c_j^\dagger\}=\delta_{ij}$ and anticommute among themselves. The latter imply $(\hat c_i^\dagger)^2=0$. This formalism implements the same exchange constraints without assigning permanent particle labels.
For noninteracting particles in <thermal equilibrium>, the mean occupation of a state of energy $\epsilon$ is
$$
\bar n_B=\frac1{e^{(\epsilon-\mu)/(k_BT)}-1},\qquad\bar n_F=\frac1{e^{(\epsilon-\mu)/(k_BT)}+1}.
$$
<Fermi-Dirac statistics> produces a filled Fermi sea at zero <temperature> and <degeneracy pressure>; <Bose-Einstein statistics> permits macroscopic <Ground state> occupation when the density of available states and thermodynamic conditions allow condensation. The different statistics therefore have measurable collective consequences, not just different notation.<Statistical Physics by David Tong>[Chapter 3, §§3.5–3.6: Bose and Fermi distributions, condensation and degenerate Fermi gases]{ref}
When phase-space occupation is dilute, both distributions reduce to the <Maxwell–Boltzmann statistics> limit. Likewise, well-separated localized <wave packets> have negligible overlap and small exchange corrections, so effectively labelled classical trajectories can become a useful approximation. In two spatial dimensions more general exchange statistics are possible; the standard boson/fermion account here concerns ordinary three-dimensional particles and their total states.
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