Questions: Identical Particles and Exchange Symmetry

Question 1

Two electrons are described by single-particle states φ_a and φ_b (a ≠ b). A student writes the two-particle state as ψ = φ_a(r₁)φ_b(r₂). What is the fundamental problem with this description?

• A. The wavefunction is not normalized
• B. The states must be orthogonal before combining
• C. This treats the electrons as distinguishable; the correct state must be antisymmetrized
• D. Electron wavefunctions must be symmetric, not antisymmetric

Answer: (C)

Electrons are identical fermions — there is no physical meaning to 'electron 1 is in state φ_a and electron 2 is in state φ_b.' The correct state is the Slater determinant: ψ_F = [φ_a(r₁)φ_b(r₂) − φ_a(r₂)φ_b(r₁)]/√2. Option D is backwards: electrons (spin-1/2) are fermions and require antisymmetric wavefunctions, not symmetric ones.

Question 2

According to the spin-statistics theorem, which pairing correctly connects spin to exchange symmetry?

• A. Integer spin → antisymmetric wavefunction (fermions); half-integer spin → symmetric wavefunction (bosons)
• B. Integer spin → symmetric wavefunction (bosons); half-integer spin → antisymmetric wavefunction (fermions)
• C. All particles are bosons at low energy and fermions at high energy
• D. Symmetry is determined by the particle's charge, not its spin

Answer: (B)

The spin-statistics theorem connects spin to exchange symmetry: integer-spin particles (0, 1, 2, …) are bosons with symmetric wavefunctions; half-integer spin particles (1/2, 3/2, …) are fermions with antisymmetric wavefunctions. Electrons (spin-1/2) are fermions; photons (spin-1) are bosons. Charge and energy have no bearing on this classification.

Question 3

The indistinguishability of identical quantum particles is a practical limitation — with sufficiently precise instruments, one could in principle track and label individual electrons.

• T. True
• F. False

Answer: False

Quantum indistinguishability is not an experimental limitation but a fundamental feature of the theory. There is no measurement, even in principle, that can assign a label to 'electron 1' versus 'electron 2.' Quantum particles of the same type share all intrinsic properties (mass, charge, spin), and wavefunctions describe probability amplitudes for configurations, not trajectories of labeled particles. Classical billiard balls are distinguishable in principle even if identical in appearance; electrons are not.

Question 4

The antisymmetric two-particle wavefunction ψ_F = [φ_a(r₁)φ_b(r₂) − φ_a(r₂)φ_b(r₁)]/√2 cannot be written as a product φ(r₁)φ(r₂) even when the particles do not interact.

• T. True
• F. False

Answer: True

The antisymmetrized wavefunction is entangled — it cannot be factored into a product of two independent single-particle wavefunctions. This is not caused by any physical interaction between the particles; it is purely a consequence of the exchange symmetry requirement. Even two non-interacting identical fermions are quantum-mechanically correlated, a purely quantum effect with no classical analogue.

Question 5

Explain why the Pauli exclusion principle is not an independent postulate but rather a mathematical consequence of the antisymmetry requirement for fermion wavefunctions.

Model answer: If two fermions occupy the same single-particle state (a = b), the antisymmetric wavefunction becomes ψ_F = [φ_a(r₁)φ_a(r₂) − φ_a(r₂)φ_a(r₁)]/√2 = 0. The wavefunction vanishes identically — there is no quantum state for this configuration. The exclusion principle ('no two fermions can share all quantum numbers') is thus forced by the mathematics of antisymmetry, not added as a separate rule.

The chain runs from indistinguishability → antisymmetry requirement → Pauli exclusion. Understanding this derivation reveals why the exclusion principle is so foundational: it is not assumed but derived, and it explains phenomena ranging from the structure of atoms to the Fermi pressure that supports neutron stars.