🎱 Quantum Billiard — Particle in a Box
A quantum billiard is a particle confined inside a hard-walled cavity. For a rectangle the eigenstates are separable products of sine functions (ψnx,ny ∝ sin(nπx/L)·sin(mπy/L)); for the Bunimovich stadium the classical orbits are chaotic and the quantum spectrum shows level repulsion — a signature of quantum chaos. Choose a geometry, pick quantum numbers, and watch the probability density |ψ|² rendered as a heat map. The time-dependent superposition shows wavepacket interference.
Geometry
Quantum numbers
Visualization
Info
(2/L)·sin(nπx/L)·sin(mπy/L)
En,m = E₁(n²+m²)
E₁ = ℏ²π²/(2mL²)
About Quantum Billiards
The rectangular billiard is integrable: constants of motion pₓ and pᵧ quantize independently so scarring, nodal lines, and regular lattice spectra result. The Bunimovich stadium has completely chaotic classical orbits, and quantum mechanically its eigenvalue spacings obey GUE/GOE statistics (random-matrix universality class) — a key result of quantum chaos theory. Scarring (Heller 1984) shows that some eigenstates concentrate probability along unstable classical periodic orbits, violating naive ergodic expectations.