Airy equation: d²ψ/dx² = x·ψ — arises for a particle in a linear potential V(x) = F·x. The turning point is at x = E/F where V = E.
Ai(x) decays exponentially for x>>0 (classically forbidden), oscillates for x<<0 (classically allowed).
Bi(x) diverges for x>>0.
WKB: ψ ≈ (k(x))^(-1/2) · sin(∫k dx + π/4) for x < x_turn; exponential decay for x > x_turn.
Top panel: Airy functions Ai(z) and Bi(z). Bottom panel: Quantum wavefunction ψ(x) for V(x)=Fx with energy E, showing turning point and tunneling tail.