Quantum Tunneling
Exact rectangular-barrier transmission, stationary density and comparison with the WKB exponential.
About this tool
V₀, a and E update the stationary model directly; sliders complement the exact input fields. All values use model units ħ=m=1. Zero barrier width or height removes the barrier. Reset restores the default inputs. WKB changes only the comparison, with its values and limitations in Options. The spatial density and potential share the same horizontal scale; the spectrum shows the selected energy. Only raw input text and the WKB choice are saved, including blank, invalid or comma drafts. Legacy numeric inputs migrate. Calculated profiles and results are rebuilt, never saved.
For E<V₀: T=[1+V₀²sinh²(κa)/(4E(V₀−E))]⁻¹, κ=√[2(V₀−E)]. For E>V₀ replace sinh by sin and V₀−E by E−V₀. At E=V₀ the continuous limit is T=[1+V₀a²/2]⁻¹. Above-barrier resonances at E=V₀+n²π²/(2a²) have T=1. R=1−T because the exterior wave numbers agree. Very small T is shown in scientific notation, not rounded to zero.
The stationary wave matches ψ and its derivative at both boundaries, with unit incoming amplitude and no incoming wave from the right. Incident and reflected waves interfere on the left; the transmitted density is constant T on the right. Density can exceed one and is not a normalized finite packet probability. Density and potential use separate vertical scales. The state has time-independent density; no moving-packet claim is made.
WKB here means only exp(−2κa), below the barrier. It omits interface prefactors and is unreliable near the top or for abrupt barriers; it does not predict above-barrier reflection. The packet link opens the real time-dependent Schrödinger simulator with its own settings, finite Gaussian momentum spread, numerical grid and boundary limits. Those finite-packet fractions need not equal a single-energy T.