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Home/Chemistry/Rectangular Barrier Tunneling

Rectangular Barrier Tunneling

Analytic transmission T(E) vs energy for a 1D barrier; tunneling below V₀ and resonances above (ℏ = m = 1).

Rectangular barrier

4
1.2
2.5

Analytic T(E) for a 1D rectangular barrier (continuity of ψ and ψ′). Below V₀, tunneling is possible; above V₀, T(E) oscillates and hits T = 1 when sin(qa) = 0 (Fabry–Pérot-like resonances).

Measured values

Transmission T(E)0.0571
RegimeE < V₀ (tunneling)

About this model

A plane wave with energy E encounters a one-dimensional rectangular barrier of height V₀ and width a. The simulator plots the standard analytic transmission coefficient T(E) from continuity of ψ and dψ/dx at the two interfaces, using ħ = m = 1. Below V₀, classically forbidden transmission (tunneling) is still nonzero; above V₀, T(E) oscillates because of interference, reaching T = 1 when sin(qa) = 0 (resonant transmission) and dropping where reflected amplitudes add destructively.

Who it's for: Undergraduate quantum mechanics students learning barrier penetration and resonant transmission.

Key terms

  • Tunneling
  • Transmission Coefficient
  • Rectangular Barrier
  • Continuity Conditions
  • Resonance

How it works

Transmission coefficient T(E) for a plane wave incident on a rectangular barrier in one dimension (ħ = m = 1). Compare under-barrier tunneling with above-barrier resonances.

Frequently asked questions

Why can T(E) be less than 1 even when E > V₀?
Above the barrier top, partial reflection still occurs at each interface. Depending on the phase accumulated across the barrier (related to sin(qa) with q = √(2m(E−V₀))/ħ), reflected waves interfere so T(E) oscillates between minima and T = 1 when sin(qa) = 0 (resonant transmission), not when the sine is largest.
What do ħ = m = 1 mean on this page?
They are model units: you should treat E, V₀, and a as dimensionless parameters illustrating the functional form of T(E). Reinstating physical ħ and m rescales energy and length consistently with your problem setup.