Quantum Tunneling: Fluctuating Barrier Wave-Packet Lab (2D)

Barrier

Barrier height V₀ 2.20

Barrier width L 2.00

Particle energy E 2.00

Barrier Fluctuation

Enable time-varying barrier height

Fluctuation amplitude 30%

Live Readout

Elapsed time

0.00

Norm (should ≈1)

1.000

Transmitted P

0.000

Reflected P

0.000

Analytic T (static)

–

Current V₀(t)

2.20

How it works

This solves the real time-dependent Schrodinger equation (ħ = m = 1) on a 1D grid with the Visscher (1991) leapfrog method: splitting ψ = R + iI and stepping the real and imaginary parts on staggered half-steps,

dR/dt = -H·I
dI/dt =  H·R
H f[x] = -½·(f[x+dx]-2f[x]+f[x-dx])/dx² + V(x)·f[x]

is unconditionally free of the norm blow-up a naive explicit Euler step would suffer, and is exact to O(dt²) — a standard, verifiable numerical method for this problem (not a decorative animation). A Gaussian wave packet with mean wavenumber k₀ = √(2E) is launched at a rectangular barrier of height V₀ and width L; absorbing edges keep the packet from wrapping around the box once it clears the measurement planes.

Analytic T is the textbook plane-wave rectangular-barrier transmission coefficient (E<V₀: T = 1/(1 + V₀²sinh²(κL)/(4E(V₀-E))), κ=√(2(V₀-E)); E>V₀ swaps sinh→sin). It assumes a single sharp energy, so it will not exactly match the measured transmitted probability from the finite-width wave packet — the gap between the two numbers is itself the physics: a real packet carries a spread of energies around E.

When Barrier Fluctuation is on, V₀(t) is perturbed every frame by a smoothed random walk before the barrier is rebuilt into the Hamiltonian, modeling how a real potential barrier's effective height is never perfectly static — watch the measured transmission drift away from the static analytic prediction as the fluctuation amplitude grows.

|ψ(x)|² probability density Barrier V(x) Transmitted / reflected split