Identical Particles and the Symmetry of a Many-Body State
For ordinary bosons and fermions in three dimensions, the full state is symmetric or antisymmetric under exchange. Antisymmetry forbids two fermions occupying the same complete one-particle state. Spin and spatial symmetry affect two-electron Coulomb energies.
What a learner can do afterwards
- Builds symmetric and antisymmetric two-particle states from single-particle ones
- Derives the exclusion principle from antisymmetry instead of stating it as a rule
- Explains how exchange symmetry shifts the energy of a two-electron system
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Exchanging identical particles leaves physical predictions unchanged. For ordinary particles in three dimensions, bosons have symmetric full states and fermions have antisymmetric full states. These statistics are additional physical input, not a consequence of indistinguishability alone. For a pure state, |psi|² is probability density in the chosen coordinates; integrate it over a region to obtain a probability.
Let a and b be distinct orthonormal complete one-particle states. Labels 1 and 2 identify the arguments, not tracking tags. Write ab=a(1)b(2) and ba=b(1)a(2). Adding the exchanged products gives a symmetric state. Their orthogonality makes its normalized form (ab+ba)/sqrt(2). Exchange reverses the terms but preserves the sum.
An antisymmetric state changes sign when its particle arguments are exchanged: psi(2,1)=-psi(1,2). For distinct orthonormal complete states a and b, form [a(1)b(2)-b(1)a(2)]/sqrt(2). After exchange, the numerator is b(1)a(2)-a(1)b(2), which is minus the original. The full state changes sign, not magnitude, so its physical predictions stay unchanged.
Bosonic symmetry permits repeated occupation of one complete state or mode. A product with both bosons in state a is a(1)a(2); exchanging its arguments leaves it unchanged, and it need not vanish. Photons are bosons, so multiple photons may occupy one mode. Exchange symmetry imposes no two-photon occupancy cap. This permission alone does not explain how a laser produces light.
If both fermions occupy complete state a, the antisymmetric numerator is a(1)a(2)-a(1)a(2)=0. A physical state must have norm one, and no normalization can turn zero into such a state. This gives exclusion. It is cancellation under antisymmetrization, not addition of identical products or a rule that changes the sign of just one occupied particle.
For electrons, a complete one-particle state includes spin. Two electrons with the same spatial orbital and opposite spin states occupy different complete states, so this does not violate exclusion. The same orbital and the same spin state would repeat the complete state and is excluded. Exclusion limits orbital occupancy; lowest-energy filling then builds shells.
Use fixed normalized orthogonal spatial orbitals and neglect spin-dependent terms. Electron singlet spin is antisymmetric, with symmetric space; triplet space is antisymmetric. Total expectations are E_s=E0+D+K and E_t=E0+D-K. E0 sums the one-electron energy expectations; D is direct Coulomb repulsion and K the exchange integral. Only D±K is the Coulomb contribution. Subtract the totals to compare them. This describes an exchange contribution, not every helium splitting.
Exchange preserves predictions. Fermionic antisymmetry excludes identical complete states. Electron spin symmetry fixes spatial symmetry, which changes the exchange energy.
Original lesson, practice questions and flashcards authored for this dataset.
Original reading notes and flashcards authored for this dataset.
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