Lab 08 — Quantum Mechanics

Quantum Tunneling

A gaussian wave packet — quantum mechanics' portrait of a particle — with energy E slams into a barrier of height V₀. Classically, V₀ > E means 100% reflection. But in quantum mechanics, part of the packet seeps through the wall and emerges on the far side. Watch the transmission probability T accumulate in real time during the collision. Under the hood, the time-dependent Schrödinger equation is solved with the split-step Fourier method — 1,024 grid points, every step keeping total probability at exactly 100% (exactly unitary).

i\hbar\,\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\,\frac{\partial^2 \psi}{\partial x^2} + V(x)\,\psi \qquad T \approx e^{-2\kappa L}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}

Raise the wall above E — T still refuses to hit zero. That is tunneling

Barrier heightV₀ = 1.20 E

Barrier widthL = 3

Transmitted T0.00%Reflected R0.00%

Tunneling is everywhere: solar fusion (protons crossing the Coulomb barrier — the wall of their electric repulsion), flash-memory writes, scanning tunneling microscopes, alpha decay. Transmission falls exponentially with barrier thickness — $T \approx e^{-2\kappa L}$ , with κ growing like $\sqrt{V_0 - E}$ — so a slightly thicker wall becomes dramatically harder to cross.