Approach
A Gaussian packet — a spread of possibility, not a point — arrives at momentum ħk. Classically, if its energy is less than the barrier, this is as far as it goes.
A study in quantum tunneling
Some barriers
only look absolute.
Every particle that reaches this wall carries a wavefunction, and a wavefunction never fully stops at an edge. Some fraction of it always continues — faint, real, and measurable on the other side.
Live scattering solution · ψ(x)
Re ψ
Im ψ
|ψ|²
0.45Energy E
0.39Transmission T
0.61Reflection R
1.05Decay κ
What actually happens at the wall
Three regions, one wavefunction
The particle never chooses. It exists a little in all three places at once, and the barrier just changes how much.
Beneath the barrier
Inside, the wavefunction stops oscillating and starts to fade — a smooth exponential decay, the way sound thins through a thick wall. Nothing collides. It simply becomes less likely, continuously, with depth.
Beyond
On the far side, a real wave packet reappears. Smaller, but whole — same energy, same shape, just less probable. That's not an approximation. That's the exact solution.
Six controls, no shortcuts
The instrument
Everything below maps directly to a live control in the simulation — nothing is simplified for the sake of the page.
EEnergy
0.051.50 ħω
Kinetic energy of the incoming wave packet.
V₀Barrier height
0.003.00 ħω
Potential energy of the barrier region.
dBarrier width
0.003.00 a₀
Thickness of the potential barrier.
mMass
0.105.00 mₑ
Effective mass of the particle.
σPacket width
0.202.00 a₀
Spatial spread of the initial Gaussian packet.
—Speed
0.1×5.0×
Playback rate of the simulation clock.
Exact, not approximated
Solved in closed form
The instrument evaluates the Schrödinger equation for a rectangular barrier analytically — a genuine scattering solution, not a numerically integrated one, so there's no accumulated error to tune away.
iℏ ∂ψ/∂t = [ −ℏ²/2m · ∂²ψ/∂x² + V(x)ψ ]
The transmission coefficient T is solved directly from E, V₀, d and m, then the incident, evanescent, and transmitted regions are matched at both boundaries so amplitude and phase stay continuous. The moving packet is reconstructed from those exact scattering amplitudes — it evolves continuously, with no artificial reset when it reaches the barrier.
- ψ(x,t)
- The wavefunction — a complex amplitude whose squared magnitude gives the probability of finding the particle at x.
- V(x)
- The potential — zero everywhere except the barrier region, where it takes the value V₀.
- ħ
- The reduced Planck constant, set to 1 in the simulation's natural units.
- m
- The effective mass of the particle — heavier particles tunnel less readily at the same energy.