Scientist: denario Date: 2026-06-02
Research project on the anharmonic oscillator.
Scope: Study the classical and quantum anharmonic oscillator, i.e. a particle in a potential that deviates from the pure harmonic form by an anharmonic (typically quartic) term:
V(x) = (1/2) m ω^2 x^2 + (1/4) λ x^4
(and optionally a cubic term, V(x) = (1/2) m ω^2 x^2 + (1/3) g x^3 + (1/4) λ x^4).
Goals / questions of interest:
- Classical dynamics: dependence of oscillation period and trajectory shape on amplitude and anharmonicity λ; emergence of higher harmonics; phase-space portraits.
- Quantum spectrum: energy eigenvalues E_n(λ) of the quartic oscillator, anharmonic shifts relative to the harmonic ladder, and the strong-coupling regime.
- Perturbation theory vs. exact/numerical results: divergence of the perturbative series in λ and comparison against numerical diagonalization (matrix-element / shooting / variational methods).
- Optional: WKB approximation, semiclassical quantization, and the double-well case (negative quadratic term) with tunneling splitting.
Data: No external observational dataset. Data will be generated numerically — classical trajectories from ODE integration of the equation of motion, and quantum energy levels from numerical diagonalization of the Hamiltonian in a truncated harmonic-oscillator basis (or finite-difference / shooting). Units are natural/dimensionless (set m = ω = ħ = 1) with λ as the primary control parameter.
Deliverable: A reproducible study comparing analytic, perturbative, and numerical treatments of the anharmonic oscillator, with figures (spectra vs. λ, period vs. amplitude, phase portraits) and a written paper.