Quantum Tunneling — Schrödinger Equation

Barrier height V₀ — 1.0

Barrier width w — 20

Packet energy E — 0.80

Packet width σ — 40

Barrier shape

Transmit

—%

Reflect

—%

t = 0

Tunneling probability T ≈ e^{−2κw} where κ=√(2m(V₀−E))/ℏ for E<V₀. Even classically forbidden particles have nonzero transmission — the hallmark of quantum mechanics. Used in: STM, alpha decay, tunnel diodes, flash memory.