Wave 27: Quantum Tunnelling, Fluid Vorticity & Game Theory

⛛️ Quantum Tunnelling — Schrödinger Equation & T(E)

A quantum particle incident on a classically forbidden potential barrier has a nonzero probability of being found on the other side. This simulation solves the 1D time-independent Schrödinger equation exactly using the transfer matrix method (TMM), computing the transmission coefficient T(E) and the full wavefunction |ψ|² profile across the barrier in real time as sliders are moved.

Three modes

• Square Barrier — the canonical case with exponential decay inside
• Double Barrier — resonant tunnelling: T = 1 at discrete energies (basis of the tunnel diode)
• Step Potential — quantum reflection even above the step

Transfer Matrix Method

For each piecewise-constant region of potential V_i and width d_i, the wavefunction is ψ_i(x) = A_i e^ik_ix + B_i e^−ik_ix where k_i = √(2m(E−V_i)/ℏ²) (real if E > V_i, imaginary for tunnelling). Boundary conditions (continuity of ψ and ψ′) at each interface are encoded as a 2×2 matrix. The total transfer matrix M = M_N … M₁ gives t = 1/M₀₀ and r = M₁₀/M₀₀.

The T(E) curve panel scans 400 energy points from 0 to 2V₀ on every parameter change. For the double-barrier case, resonance peaks (T = 1) are visible at the discrete eigenvalues of the quantum well formed between the two barriers — the same physics that makes resonant tunnelling diodes (RTDs) work.

🌊 Fluid Vorticity & Kelvin–Helmholtz Instability

Place N point vortices in a line along a shear interface and watch them roll up into characteristic spirals — the Kelvin–Helmholtz (KH) instability. This is one of the most universal phenomena in fluid dynamics: the same curling mechanism creates cloud billows, ocean mixing layers, solar corona jets, and the boundary turbulence in accretion discs around black holes.

Physics: Biot–Savart for Point Vortices

A continuous vortex sheet of surface vorticity ω(x) is approximated by N discrete point vortices of circulation Γ. The induced velocity at vortex i from all other vortices j follows the 2D Biot–Savart law (regularised with a finite core ε to prevent numerical blow-up at close encounters):

The integration uses a forward Euler scheme (tuneable step dt). Each animation frame advances 3 substeps for smooth visual rollup. Vortex trails (45-frame ring buffer) trace the Lagrangian paths, making the spiral rollup visually striking. A sparse velocity-field overlay (redrawn every 3 frames) shows the induced flow pattern.

Three initial conditions

• Single layer — N positive vortices along y = 0.5 with sinusoidal + random perturbation → classic KH rollup
• Double layer — counter-rotating row above → Kármán vortex street
• Random vortices — chaotic mix of ±Γ vortices → turbulent-like interaction

🤝 Evolutionary Game Theory — Prisoner’s Dilemma

A spatial evolutionary simulation of the iterated Prisoner’s Dilemma on a 2D grid. Each cell adopts one of five strategies — ALL-D (always defect), ALL-C (always cooperate), Tit-for-Tat (TFT), Grim Trigger (GRIM), and RAND — and competes with its eight neighbours. After each generation, every cell copies the strategy of its most successful neighbour. Mutation introduces random strategy switches.

Payoff matrix (standard Prisoner’s Dilemma)

The spatial structure is crucial: in a well-mixed population ALL-D always wins. But on a lattice, TFT and GRIM clusters can resist invasion by forming cooperative blocks where defectors are punished by neighbours. This is Nowak & May’s 1992 finding (Nature 359:826) that spatial structure alone can sustain cooperation without kin selection or repeated interaction at the population level.