Quantum Tunneling Through Rectangular Barriers

Core Idea

When a quantum particle encounters a potential barrier higher than its energy, the wavefunction does not abruptly vanish—it decays exponentially inside the barrier. If the barrier has finite width, the wavefunction is non-zero on the far side, giving a non-zero probability of finding the particle there. The tunneling probability is exponentially sensitive to barrier width and height.

How It's Best Learned

Solve the time-independent Schrödinger equation in three regions (before, inside, and after the barrier) and match boundary conditions to find transmission coefficients. Calculate tunneling probability for specific barriers and particle energies.

Common Misconceptions

Tunneling requires the particle energy to be below the barrier top (it happens when E < V, not E > V). The particle does not gain energy inside the barrier; it is temporary and violates energy conservation only by ΔE ~ ℏ/Δt consistent with uncertainty principle.

Explainer

From your study of quantum tunneling, you know that a particle can pass through a classically forbidden region. The rectangular barrier makes this quantitative. Imagine a particle moving in the +x direction toward a wall of height V₀ and width d, where the particle's total energy E is less than V₀. In classical mechanics, the particle simply bounces back — it can never enter the region where kinetic energy would be negative. In quantum mechanics, the particle is described by a wavefunction, and wavefunctions don't hard-stop at boundaries.

The tool for calculating what happens is the time-independent Schrödinger equation, which you solved for infinite and finite wells. For the rectangular barrier, divide space into three regions: Region I (before the barrier, x < 0), Region II (inside the barrier, 0 < x < d), and Region III (after the barrier, x > d). In Regions I and III, E > 0 and the kinetic energy is positive, so the Schrödinger equation gives oscillatory solutions — plane waves of the form Ae^ikx + Be^-ikx, where ℏk = √(2mE). These represent incoming, reflected, and transmitted waves.

Inside the barrier, E < V₀, so the kinetic energy is negative. The Schrödinger equation becomes d²ψ/dx² = κ²ψ where κ = √(2m(V₀−E))/ℏ. The solutions are real exponentials: Ce^κx + De^-κx. This exponential decay is the key to tunneling — the wavefunction doesn't vanish abruptly but fades. If the barrier is thin enough, a non-zero amplitude survives to Region III, meaning there is a non-zero probability of finding the particle on the far side.

The transmission coefficient T is found by matching ψ and dψ/dx at both boundaries x=0 and x=d. Matching at both interfaces gives four equations relating the amplitudes. After solving, the result for E < V₀ is approximately T ≈ e^−2κd, where κ = √(2m(V₀−E))/ℏ. This single formula captures why tunneling is exponentially sensitive to barrier width d and height (V₀−E): doubling the width roughly squares the transmission probability, and a taller barrier increases κ, suppressing T even faster.

This exponential sensitivity has profound physical applications. In alpha decay, a nucleus emits an alpha particle that tunnels through the Coulomb barrier — tiny changes in barrier height explain why some isotopes have half-lives of microseconds while others have half-lives of billions of years. The scanning tunneling microscope (STM) exploits the same sensitivity: a metallic tip brought nanometers from a surface passes a tunneling current that falls by a factor of ~10 for each additional ångström of distance, allowing atomic-resolution imaging. Tunnel diodes use controlled barrier engineering for fast electronic switching. In all these cases, the physics is the same rectangular-barrier calculation you've now learned to solve.