Numerical investigation of quasilinearization method in quantum mechanics (original) (raw)
Related papers
Quasilinearization approach to quantum mechanics
Computer Physics Communications, 2003
The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precision of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.
Quasilinearization approach to computations with singular potentials
Computer Physics Communications, 2008
We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrödinger equation with singular potentials. The spiked harmonic oscillator r 2 + λr −α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrödinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated 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. The choice of zero iteration is based on general features of solutions near the boundaries. We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.
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.
Quasilinearization approach to the resonance calculations: the quartic oscillator
Physica Scripta, 2008
We pioneered the application of the quasilinearization method (QLM) to resonance calculations. The quartic anharmonic oscillator (kx 2 /2) + λx 4 with a negative coupling constant λ was chosen as the simplest example of the resonant potential. The QLM has been suggested recently for solving the bound state Schrödinger equation after conversion into Riccati form. In the quasilinearization approach the nonlinear differential equation is treated 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. The choice of zero iteration is based on general features of solutions near the boundaries. Comparison of our approximate analytic expressions for the resonance energies and wavefunctions obtained in the first QLM iteration with the exact numerical solutions demonstrate their high accuracy in the wide range of the negative coupling constant. The results enable accurate analytic estimates of the effects of the coupling constant variation on the positions and widths of the resonances.
Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs
Computer Physics Communications, 2001
The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some kind of a small parameter.
Quasilinear Approximation and WKB
Few-Body-Systems, 2004
Quasilinear solutions of the radial Schrödinger equation for different potentials are compared with corresponding WKB solutions. For this study, the Schrödinger equation is first cast into a nonlinear Riccati form. While the WKB method generates an expansion in powers of , the quasilinearization method (QLM) approaches the solution of the Riccati equation by approximating its nonlinear terms by a sequence of linear iterates. Although iterative, the QLM is not perturbative and does not rely on the existence of any kind of smallness parameter. If the initial QLM guess is properly chosen, the usual QLM solution, unlike the WKB, displays no unphysical turning point singularities. The first QLM iteration is given by an analytic expression. This allows one to estimate analytically the role of different parameters, and the influence of their variation on the boundedness or unboundedness of a critically stable quantum system, with much more precision than provided by the WKB approximation, which often fails miserably for systems on the border of stability. It is therefore demonstrated that the QLM method is preferable over the usual WKB method.
Annals of Physics, 2007
High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrö dinger equation into the 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 treated 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. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10 À4 and 10 À8 for the energies and, correspondingly, 10 À2 and 10 À4 for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.
International Journal of Modern Physics A, 2005
A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction HI(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H0 generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues En are known and (ii) equality of quantum averages (EQA): 〈n|HI(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and En are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H0 as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(HI(ϕ) - V(ϕ)), as the perturbation wher...
Abstract: A self-consistent, non-perturbative approximation scheme is proposed which i s potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction H I (φ) by a suitable 'potential' V(φ) which satisfies the following two basic requirements, (i) exact solvability (ES): the 'effective' Hamiltonian, H 0 generated by V(φ) is exactly solvable i.e., the spectrum of states | n > and the eigen-values E n are known and (ii) equality of quantum averages (EQA): < n | H I (φ)| n > = < n | V(φ)| n > for arbitrary 'n'. The leading order (LO) results for | n > and E n are thus readily obtained and are found to be accurate to within a few percent of the 'exact' results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H 0 as the unperturbed Hamiltonian and the modified interaction, λH ′ (φ) ≡ λ (H I (φ)-V(φ)), as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that < n | λ H ′ (φ) | n > = 0 for arbitrary 'λ' and 'n'. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naïve perturbation theory where the original interaction λH I (φ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic-and double well potentials, e.g. quartic-, sextic-and octic-anharmonic oscillators and quartic-, sextic-double well oscillators. Uniformly accurate results for the energy levels over the full allowed range of 'λ' and 'n' are obtained. The results compare well with the exact results predicted by super symmetry for the case of the sextic anharmonic-and the double well partner potentials. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.