Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Journal Article · Thu Jun 15 04:00:00 EDT 2006 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2209769· OSTI ID:20860485

Liverts, E Z; Mandelzweig, V B; Tabakin, F ^[1]

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1. Racah Institute of Physics, Hebrew University, Jerusalem 91904 (Israel)

Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schroedinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground state wave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the quartic and pure quartic oscillators.

• OSTI ID: 20860485
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 6 Vol. 47; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
COUPLING CONSTANTS
GROUND STATES
ITERATIVE METHODS
NONLINEAR PROBLEMS
OSCILLATORS
RICCATI EQUATION
SCHROEDINGER EQUATION
SINGULARITY
WAVE FUNCTIONS