Unfolding the Quartic Oscillator (original) (raw)
The``exact WKB method'' is applied to the general quartic oscillator, yielding rigorous results on the ramification properties of the energy levels when the coefficients of the fourth degree polynomial are varied in the complex domain. Simple though exact``model forms'' are given for the avoided crossing phenomenon, easily interpreted in terms of complex branch points in the``asymmetry parameter.'' In the almost symmetrical situation, this gives a generalization of the Zinn Justin quantization condition. The analogous``model quantization condition'' near unstable equilibrium is thoroughly analysed in the symmetrical case, yielding complete confirmation of the branch point structure discovered by Bender and Wu. The numerical results of this analysis are in excellent agreement with those computed by Shanley, overtaking the most optimistic expectations of the realm of validity of semiclassical models. 1997 Academic Press The aim of this article is to present exact results on how the energy levels of the one-dimensional quartic oscillator, _ & 2 d 2 dq 2 +V(q) & 9=E9, (0) where V is the general fourth degree polynomial 1 V(q)=q 4 +:q 2 +;q, (V) depend on the coefficients of the polynomial V. By the``quasihomogeneity'' property of Eq. (0), which is invariant under the substitution q [ *q, [ * 3 , : [ * 2 :, ; [ * 3 ;, E [ * 4 E, (V) no generality would be lost by setting =1. But will play a prominent role in our techniques of proof, which will be based on the complex WKB method, or more precisely its``exact asymptotic'' version initiated by Voros [Vo1] and further Article No. PH975737 180 0003-4916Â97 25.00
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On the example of the semi-classical expansion for the levels of the quartic oscillator–(d 2/dq 2)+ q 4, we show how the complex WKB method provides information about the singularities of the Borel transform of the semi-classical series. In this problem there occurs a tunneling effect between complex turning points, by which those singularities generate exponentially small, yet detectable, corrections to the energy levels.
Quasilinearization method and summation of the WKB series
Physics Letters A, 2005
Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. Expansion of the pth QLM iterate in powers ofh reproduces the structure of the WKB series generating an infinite number of the WKB terms with the first 2 p terms reproduced exactly. The QLM quantization condition leads to exact energies for the Pöschl-Teller, Hulthén, Hylleraas, Morse, Eckart potentials, etc. For other, more complicated potentials the first QLM iterate, given by the closed analytic expression, is extremely accurate. The iterates converge very fast. The sixth iterate of the energy for the anharmonic oscillator and for the two-body Coulomb-Dirac equation has an accuracy of 20 significant figures.
Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators
Journal of Mathematical Physics, 2006
Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method ͑QLM͒. 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.
On The Exact and JWKB Solution of 1D Quantum Harmonic Oscillator by Mathematica
Although being the fundamental semiclassical approximation method mainly used in quantum mechanics and optical waveguides, JWKB method along with the application of the associated JWKB asymptotic matching rules is known to give exact solutions for the Quantum Harmonic Oscillator (QHO). Asymptotically matched JWKB solutions are typically accurate or exact in the entire domain except for a narrow domain around the classical turning points where potential energy equals the total energy of the related quantum mechanical system. So, one has to cope with this diverging behavior at the classical turning points since it prohibits us from using continuity relations at the related boundaries to determine the required JWKB coefficients. Here, a computational diagram and related mathematica codes to surmount the problem by applying parity matching for even and odd JWKB solutions rather than boundary continuities are being presented. In effect, JWKB coefficients as well as the conversion factor for the dimensionless form of the Schrodingers equation, which is common to both exact and JWKB solutions, is being successfully obtained. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Complex WKB analysis of energy-level degeneracies of non-Hermitian Hamiltonians
Journal of Physics A: Mathematical and General, 2001
The Hamiltonian H = p 2 +x 4 +iAx, where A is a real parameter, is investigated. The spectrum of H is discrete and entirely real and positive for |A| < 3.169. As |A| increases past this point, adjacent pairs of energy levels coalesce and then become complex, starting with the lowest-lying energy levels. For large energies, the values of A at which this merging occurs scale as the three-quarters power of the energy. That is, as |A| → ∞ and E → ∞, at the points of coalescence the ratio a = |A|E −3/4 approaches a constant whose numerical value is a crit = 1.1838363072914 · · ·. Conventional WKB theory determines the high-lying energy levels but cannot be used to calculate a crit . This critical value is predicted exactly by complex WKB theory.
Quasilinearization method and WKB
Computer Physics Communications, 2006
Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. While the WKB method generates an expansion in powers ofh, the quasilinearization method (QLM) approaches the solution of the nonlinear equation obtained by casting the Schrödinger equation into the Riccati form by approximating nonlinear terms by a sequence of linear ones. It does not rely on the existence of any kind of smallness parameter. It also, unlike the WKB, displays no unphysical turning point singularities. It is shown that both energies and wave functions obtained in the first QLM iteration are accurate to a few parts of the percent. Since the first QLM iterate is represented by the closed expression it allows to estimate analytically and precisely the role of different parameters, and influence of their variation on the properties of the quantum systems. The next iterates display very fast quadratic convergence so that accuracy of energies and wave functions obtained after a few iterations is extremely high, reaching 20 significant figures for the energy of the sixth iterate. It is therefore demonstrated that the QLM method could be preferable over the usual WKB method.
Complex Trajectories in the Quartic Oscillator and Its Semiclassical Coherent-State Propagator
Annals of Physics, 1996
The semiclassical approximation of the coherent-state propagator requires the computation of complex trajectories satisfying special boundary conditions. In this paper we present a method for the determination of such trajectories for one-dimensional polynomial potentials. We also compute the semiclassical propagator for the case V(q)=*q 2 Â2+;q 4 and compare the results with an``exact'' calculation.
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