1D Quantum Tunneling

📋 Teacher notes

• •Quantum tunneling is the phenomenon where a particle passes through a potential energy barrier that it classically cannot surmount. It arises from the wave nature of matter encoded in the Schrödinger equation.
• •The wavefunction ψ(x,t) is not zero inside the barrier even when E < V₀. It decays exponentially as ψ ∼ exp(−κx), where κ = √(2m(V₀−E))/ℏ. This evanescent wave carries no probability current, yet it connects to a non-zero transmitted wave on the far side.
• •Transmission probability for a thick rectangular barrier: T ≈ 16(E/V₀)(1−E/V₀) exp(−2κw). The exponential dependence on width w and height V₀ makes tunneling extremely sensitive to atomic-scale geometry.
• •Classically, a particle with E < V₀ is always reflected. The classical turning point is where E = V(x). Quantum mechanics permits non-zero |ψ|² beyond this point because the particle does not have a definite position — only a probability amplitude.
• •When E > V₀, transmission is not 100%. The wave partially reflects from the barrier boundaries due to impedance mismatch (change in wavelength), creating interference oscillations in T(E). This is purely a wave phenomenon.
• •Resonant tunneling occurs in double-barrier structures. Quasi-bound states in the quantum well between barriers allow T ≈ 1 even when each barrier individually would give T ≪ 1. This is the basis of resonant tunneling diodes (RTDs) and flash memory.
• •The scanning tunneling microscope (STM) relies on vacuum tunneling. The tunneling current I ∝ exp(−2κd), where d is tip-sample distance. A 1 Å change in d changes current by an order of magnitude, giving atomic resolution.
• •Alpha decay (Gamow 1928) is tunneling of an α-particle through the nuclear Coulomb barrier. The Geiger-Nuttall law relates half-life to decay energy: log τ ∝ 1/√E. Higher-energy α particles tunnel faster — dramatically so.
• •In fusion (e.g., the Sun), protons tunnel through the Coulomb repulsion barrier at temperatures where classical thermodynamics predicts essentially zero fusion. Without tunneling, stars would not shine.
• •The WKB approximation generalizes tunneling to arbitrary barriers: T ≈ exp(−2 ∫√(2m(V(x)−E)) dx / ℏ). It works when V(x) varies slowly compared to the de Broglie wavelength.
• •Heisenberg uncertainty explains tunneling qualitatively: borrowing energy ΔE for a time Δt ∼ ℏ/ΔE allows the particle to temporarily surmount the barrier. This is a hand-waving picture but useful for intuition.
• •Heavy particles tunnel less. The exponent 2κw scales as √m, so a proton tunnels far more readily than a macroscopic object. This explains why we do not observe people tunneling through walls.
• •In the classical limit (m → ∞ or ℏ → 0), κ → ∞ and T → 0. Tunneling is a purely quantum effect that vanishes in the correspondence limit, consistent with the Ehrenfest theorem.
• •Superconducting Josephson junctions and quantum computing qubits exploit controlled tunneling between states. The ability to tune barrier transparency with voltage or magnetic flux is central to quantum technology.