Rectangular Barrier Tunneling
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.
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