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Potential
Packet and barrier
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Model check
The science
What this page shows
A single quantum particle (an electron by default) is a Gaussian wave packet moving towards a potential barrier. The page integrates the one-dimensional time-dependent Schrödinger equation iℏ ∂ψ/∂t = -(ℏ²/2m) ∂²ψ/∂x² + V(x) ψ numerically on a 1024-point grid (60 nm box, split-step Fourier method, time step 0.025 fs), so the probability density |ψ|² you see is the real solution, not an animation.
Tunnelling
For a particle with energy E below a barrier height V0 classical mechanics forbids passing. The wave function instead decays inside the barrier as ψ ~ e-κx with κ = √(2m(V0-E))/ℏ, so a thin barrier leaves a small but non-zero amplitude on the far side. For one rectangular barrier of width a the exact plane-wave transmission is T = 1 / (1 + V0² sinh²(κa) / (4E(V0-E))), which for κa >> 1 reduces to T ≈ 16 (E/V0)(1-E/V0) e-2κa. That exponential makes T extremely sensitive to the width and (through κ) to the particle mass.
Why a packet, not a plane wave
A packet of width σx contains a spread of momenta, |φ(k)|² ~ exp(-2σx²(k-k0)²). Because T rises steeply with energy, the faster components tunnel preferentially, so the packet-averaged transmission ⟨T⟩ = ∫|φ(k)|² T(ℏ²k²/2m) dk is larger than T at the mean energy. The page computes both and compares them with the simulated outcome.
Other shapes
Over the barrier (E > V0) the transmission still oscillates, T = 1/(1 + V0² sin²(k'a)/(4E(E-V0))), with reflections from the two edges interfering (quantum "partial reflection"). A double barrier forms a resonant cavity: at the quasi-bound level of the well between the barriers T rises towards 1 (resonant tunnelling, the principle of the resonant-tunnelling diode). The smooth Gaussian barrier and the potential step are computed with exact piecewise-constant transfer matrices (160 slices for the Gaussian).
Honest simplifications
One spatial dimension, spinless particle, no decoherence, no interaction. Absorbing layers in the outer 4 nm of the box stop the packet from wrapping round the periodic grid. The default barrier is much thinner than a real tunnel junction would be to keep the effect visible; the atomic scale is about 0.1 nm. Transmission is read after the packet has left the barrier region; for very wide packets or energies close to V0 the packet and barrier overlap for longer, so the measured T lags the asymptotic value by a few per cent.
Reading the 3D scene
The coloured curtain standing on the bench is the probability density |ψ|²; its colour is the quantum phase of ψ (one full colour cycle per de Broglie wavelength, so the colour banding of a moving packet shows its momentum). The helix above it traces ψ in the complex plane: height is Re ψ, depth is Im ψ. The violet slab is V(x) at 0.6 scene units per eV and the pink plane is the particle's mean energy E: a packet whose energy plane sits below the top of the slab is classically forbidden to pass. The two glass gauges show the probability left and right of the barrier; the detector screen glows with the transmitted probability. The bench is 48 nm long and one scene unit along the bench is exactly 1 nm.
Same numbers as the 2D page. Both pages import the identical physics.js (split-step Fourier solver and exact transfer matrices), so the same parameters give the same transmission. Three checks run in the page: the transfer-matrix code against the closed-form sinh / sin barrier formulas, a free packet against the Gaussian spreading law, and a full simulation of your current setting against the packet-averaged exact transmission.