Photon Number States (Fock States)

|n⟩ — eigenstates of the photon number operator N̂ = â†â
Wigner Function W(x, p)
|0⟩ Fock state
W(x,p) = (1/π) ∫ ⟨x+y|ρ|x−y⟩ e²ⁱᵖʸ dy
W_n(r²) = (−1)ⁿ/π · L_n(2r²) · e^(−r²)
Negativity of W ↔ non-classical state
Position Probability |⟨x|n⟩|²
ψ_n(x) = (1/√(2ⁿn!)) · (mω/πħ)^(1/4) · H_n(x) · e^(-x²/2)
n nodes in wavefunction, n+1 lobes
Energy: E_n = ħω(n + ½)
Fock states are the "particle-number eigenstates" of light — definite photon count but completely undefined phase. They exhibit sub-Poissonian photon statistics (Fano factor = 0 for |n⟩). The Wigner function has negative regions for n≥1 — a signature of quantum non-classicality impossible to explain classically.