A phase space investigation of the quartic oscillator ground state

The symmetry and dynamics of the full solution of anharmonic oscillator with λ q⁴ type anharmonicity and unit oscillator frequency is studied numerically for intermediate values of λ using the Wigner function formalism. The calculations show that for any λ > 0 the ground state of the system quickly develops non-perturbative quantum fluctuations and beyond λ ∼ 0.5 any effective mean field assumption using correlated Gaussians is expected to fail. We briefly discuss the validity of the mean field solution below λ = 0.5 and compare with the numerical results. We further examine the marginal phase probability distribution corresponding to that of the exact phase operator. The marginal phase probability distrubution proves to be a valuable tool to extract the properties of the phase operator and it has potential use in building the necessary intuititive ground for the quantum action-angle formalism of the non-integrable quantum systems.