Quantum Chaos

1. The paradox: chaos in a linear theory

Classical chaos, as seen in the Lorenz attractor or the double pendulum, requires a nonlinear equation of motion and exponential sensitivity to initial conditions (a positive Lyapunov exponent). The Schrödinger equation, however, is strictly linear in the wavefunction ψ:

Linearity means two solutions can be added to form a new solution, and the overlap between two initially close quantum states grows at most linearly in time — never exponentially. Strictly speaking, there is no such thing as "quantum chaos" in the same sense as classical chaos: wavefunctions don't have trajectories to diverge.

The field of quantum chaos (more precisely called quantum chaology by Michael Berry) instead studies: the quantum-mechanical behaviour of systems whose classical limit is chaotic. It asks how the fingerprints of classical chaos show up in energy spectra, wavefunctions and quantum dynamics.

2. The correspondence principle

Bohr's correspondence principle says quantum mechanics should reproduce classical mechanics in the limit ℏ → 0 (or for large quantum numbers). For a classically chaotic system — say, a particle bouncing inside a stadium billiard (a rectangle capped with two semicircles) — the classical trajectories are exponentially sensitive to the starting angle.

The corresponding quantum problem is finding the eigenvalues Eₙ and eigenfunctions ψₙ of the time-independent Schrödinger equation inside that same billiard shape, with ψ = 0 on the walls. There is nothing "random" about this — it's an entirely deterministic boundary-value problem. Yet the resulting spectrum {Eₙ} behaves statistically like the eigenvalues of a random matrix, while a classically integrable billiard (a rectangle, or a circle) gives a spectrum that behaves like independent random numbers (Poisson statistics).

3. Level spacing statistics

The key diagnostic tool of quantum chaos is the distribution of spacings s = Eₙ₊₁ − Eₙ between neighbouring energy levels (after "unfolding" the spectrum to a uniform mean density).

"GOE" and "GUE" stand for the Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble — two of the classical random-matrix ensembles introduced by Eugene Wigner in the 1950s to model nuclear energy levels, decades before anyone connected them to classical chaos.

4. The Bohigas-Giannoni-Schmit conjecture

In 1984, Oriol Bohigas, Marie-Joya Giannoni and Charles Schmit proposed what is now the central conjecture of quantum chaos:

This is a remarkable claim: a chaotic hydrogen atom in a strong magnetic field, a chaotic microwave cavity and a chaotic atomic nucleus should all show the same statistical level-repulsion law, even though their Hamiltonians look nothing alike. The BGS conjecture has been confirmed numerically and experimentally hundreds of times (microwave billiards, quantum dots, Rydberg atoms) but remains, formally, an unproven conjecture rather than a theorem.

5. Quantum scarring

Even inside a fully chaotic billiard, some eigenfunctions |ψₙ|² are not spread out uniformly. Instead they show anomalous enhancement along the path of a short, unstable periodic orbit of the classical system — a phenomenon discovered by Eric Heller in 1984 and named quantum scarring.

Scars are surprising because a classically chaotic trajectory should be "forgotten" by any nearby quantum state — periodic orbits are unstable and measure zero in phase space. Yet quantum wave interference constructively reinforces the wavefunction near these orbits, leaving a visible "scar" in the probability density. Scarring has direct practical relevance: it affects conductance fluctuations in chaotic quantum dots and has recently resurfaced in the study of many-body quantum scars, exceptional states in interacting quantum systems that evade full thermalization.

6. Semiclassical methods: Gutzwiller trace formula

Martin Gutzwiller's 1971 trace formula is the bridge between the classical periodic orbits of a chaotic system and its quantum energy spectrum. Schematically, the density of states can be split into a smooth average part plus an oscillating sum over classical periodic orbits:

Here S_p is the classical action of orbit p, and A_p depends on its stability (Lyapunov exponent). Each unstable periodic orbit contributes a decaying oscillation to the spectrum — summing over the (infinitely many, exponentially proliferating) periodic orbits of a chaotic system in principle reconstructs the exact quantum spectrum from purely classical data. In practice the sum only converges for short orbits, but it already explains why individual short periodic orbits leave scars.

7. Pseudocode: nearest-neighbour spacing

// Given a sorted array of eigenvalues E[0..N-1]
// 1. Unfold the spectrum to unit mean spacing
function unfold(E):
  // fit a smooth polynomial N(E) to the cumulative counting function
  Ncum = cumulativeCount(E)
  smooth = polyFit(E, Ncum, degree=5)
  return E.map(e => smooth(e))  // unfolded levels, mean spacing = 1

// 2. Compute nearest-neighbour spacings
function spacingHistogram(Eu):
  spacings = []
  for i = 0 to Eu.length - 2:
    spacings.push(Eu[i+1] - Eu[i])
  return histogram(spacings, bins=40)

// 3. Compare to Poisson (integrable) vs Wigner-Dyson (chaotic)
poisson = s => Math.exp(-s)
wignerDyson = s => (Math.PI*s/2) * Math.exp(-Math.PI*s*s/4)