Two-Particle Symmetry
Joint probability density P(x₁, x₂) = |ψ(x₁,x₂)|² — 1D box, states φ₁ and φ₂
Mode:
Click any heatmap cell to inspect
Bosons (Symmetric)
ψ_sym = (1/√2)[φ₁(x₁)φ₂(x₂) + φ₂(x₁)φ₁(x₂)]
Fermions (Antisymmetric)
ψ_anti = (1/√2)[φ₁(x₁)φ₂(x₂) − φ₂(x₁)φ₁(x₂)]
Distinguishable
ψ_dist = φ₁(x₁)φ₂(x₂)
Inspector — click any heatmap cell to read off P values at that (x₁, x₂) point.
Bunching Ratio: P_sym / P_dist
Values > 1 → bosonic enhancement (red); < 1 → suppression
Anti-correlation Ratio: P_anti / P_dist
Zero on diagonal = Pauli exclusion node
Marginal Density ρ(x₂) = ∫₀¹ P(x₁,x₂) dx₁
All three cases produce the identical marginal: ½φ₁²(x₂) + ½φ₂²(x₂) — bosons and fermions are indistinguishable by single-particle measurement