On the power series solution of the Schrödinger equation

A numerical method for solving the Schrödinger equation for a one-dimensional potential expressed as a function which increases in both directions away from its minimum is proposed. The basic assumption relies on the asymptotic properties of the solution. We exemplify the method calculating energies and expectation values for the quartic anharmonic oscillator.     

Remark on the solution of the Schrödinger equation for anharmonic oscillators via the Feynman path integral

We give a simple proof of Feynman's formula for the Green's function of the n-dimensional harmonic oscillator valid for every time t with Im t ≤ 0. As a consequence the Schrödinger equation for the anharmonic oscillator is integrated and expressed by the Feynman path integral on Hilbert space.     

Analytic solutions of the Schrödinger equation for the modified quartic oscillator

No analytic solutions of the Schrödinger equation are known for the quartic anharmonic oscillator. We show in this paper that there are closely related modified quartic oscillators with the potential depending on |x| for which analytic solutions for some states exist. These results can be extended to the higher order oscillators     

Light-Front Holography and the Light-Front Coupled-Cluster Method

We summarize the light-front coupled-cluster (LFCC) method for the solution of field-theoretic bound-state eigenvalue problems and indicate the connection with light-front holographic QCD. This includes a sample application of the LFCC method and leads to a relativistic quark model for mesons that adds longitudinal dynamics to the usual transverse light-front holographic Schrödinger equation.     

Perturbation theory with band-matrix propagators

A new version of the Rayleigh-Schrödinger type of perturbation theory is presented. It is based on a rearrangement of hamiltonians H=T+λV containing a non-diagonal (band-matrix, strong-coupling) zero-order component T. Its efficiency is illustrated on the anharmonic oscillator.     

Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge-Kutta methods

In this Letter we present an explicit symplectic method for the numerical solution of the Schrödinger equation. We also develop a modified symplectic integrator with the trigonometrically fitted property based on this method. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential.     

Quartic oscillator potential in the γ-rigid regime of the collective geometrical model

A prolate γ-rigid version of the Bohr-Mottelson Hamiltonian with a quartic anharmonic oscillator potential in β collective shape variable is used to describe the spectra for a variety of vibrational-like nuclei. Speculating the exact separation between the two Euler angles and the β variable, one arrives at a differential Schrödinger equation with a quartic anharmonic oscillator potential and a centrifugal-like barrier. The corresponding eigenvalue is approximated by an analytical formula depending only on a single parameter up to an overall scaling factor. The applicability of the model is discussed in connection to the existence interval of the free parameter, which is limited by the accuracy of the approximation, and by comparison with the predictions of the related X(3) and X(3)-β ² models. The model is applied to qualitatively describe the spectra for nine nuclei which exhibit near-vibrational features.     

Calculation of the energy levels of excited vibrational states of the HD16O molecule by summing divergent series of the Rayleigh-Schrödinger perturbation theory. The shift of zero-order levels

The Rayleigh-Schrödinger perturbation theory is applied to calculation of the energy levels of excited vibrational states of the HD¹⁶O molecule. The model of coupled anharmonic oscillators is considered, with the anharmonic part of potential energy being taken into account as the perturbation. The calculations are carried out for the vibrational states that correspond to three-to seven-fold vibrational excitations. Since the perturbation series diverge in the case of strong resonance interactions and their approximations by the Padé and Padé-Hermite methods do not yield sufficiently correct results, a calculation technique is applied that allows the zero-order approximation to be modified. The zero-order Hamiltonian is modified by shifting the vibrational frequencies, which decreases the mixing of states. The new Rayleigh-Schrödinger series can be summed using the quadratic Padé-Hermite approximation method.     

非谐振子势的精确解和双波函数描述

求解了非谐振子势V(x)=x²/2+g/2x²的本征方程，给出了精确的能谱方程和归一化波函数.应用双波函数理论，得到了在非谐振子势场中单粒子运动状态的力学量的时间演化方程. 关键词：     

Moment method and the Schrödinger equation in the large N limit

A new technique is presented for solving the Schrödinger equation in the framework of the 1/N expansion. Based on recursion relations satisfied by moments of the coordinate operator, this method which allows to compute energy levels and wavefunctions is applied to four examples: the harmonic oscillator, the rotating harmonic oscillator, a linear plus Coulomb potential and a logarithmic one.     

Noether symmetries and the quantization of a Liénard-type nonlinear oscillator

The classical quantization of a Liénard-type nonlinear oscillator is achieved by a quantization scheme (M. C. Nucci. Theor. Math. Phys., 168:994–1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation in the momentum space as given in (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light on the apparently remarkable connection with the linear harmonic oscillator.     

On inferring quantum energies and expectation values

A maximum entropy-like inference approach is proposed that yields ground and excited states energies and expectation values without solving Schrödinger's equation. The method is tested with reference to the quartic anharmonic oscillator and other model potentials and yields good results without undue numerical effort.     

Soliton solutions of the non linear Schrödinger equation with an external time-dependent field

A functional transformation between solutions of the one-dimensional nonlinear Schrödinger equation with time- and coordinate-dependent coefficients and solutions of the conventional nonlinear Schrödinger equation (NSE) is constructed. Exact solutions of the NSE with a homogeneous time-dependent external electric field and the NSE with oscillator potential are obtained.     

The Schrödinger and Pauli-Dirac Oscillators in Noncommutative Phase Space

We investigate the non-relativistic Schrödinger and Pauli-Dirac oscillators in noncommutative phase space using the five-dimensional Galilean covariant framework. The Schrödinger oscillator presented the correct energy spectrum whose non isotropy is caused by the noncommutativity with an expected similarity between this system and the particle in a magnetic field. A general Hamiltonian for the 3-dimensional Galilean covariant Pauli-Dirac oscillator was obtained and it presents the usual terms that appears in commutative space, like Zeeman effect and spin-orbit terms. We find that the Hamiltonian also possesses terms involving the noncommutative parameters that are related to a type of magnetic moment and an electric dipole moment.     

Electron resonance dynamics in a double quantum well

We consider the two-level electron dynamics in a double quantum well in a periodic, anharmonic external electric field. We propose a method for solving the Schrödinger equation, which is based on the generalization of conventional resonance approximation for a system with an arbitrary number of resonances. The method is used for the case of both weak and strong fields. We obtain expressions for the quasi-energy wave functions and the electron dipole moment. It is shown that the dependence of the dipole moment on the constant component of external field is quasi-periodic, and the dipole moment changes sign at different half-periods.     

Model of a relativistic oscillator in a generalized Schrödinger picture

In a generalized Schrödinger picture, we consider the motion of a relativistic particle in an external field (like in the case of a harmonic oscillator). In this picture the analogs of the Schrödinger operators of the particle are independent of both the time and the space coordinates. These operators induce operators which are related to Killing vectors of the Anti de Sitter (AdS) space. We also consider the nonrelativistic limit.     

A test of perturbation theory for determining anharmonic vibrational corrections to properties of diatomic molecules

The convergence behaviour of the Rayleigh-Schrödinger perturbation series for obtaining anharmonic vibrational corrections to properties of diatomic molecules is studied. Various procedures for ordering these corrections are examined. It is verified numerically that an asymptotically divergent energy series results when the complete anharmonic potential is taken as the first-order hamiltonian. By contrast, when each term in a Taylor series expansion of the energy about the equilibrium distance is identified with successively higher orders of perturbation, convergence problems are essentially obviated. Properties other than the energy are also investigated with regard to convergence.     

Calculation of vibrational energy levels of water molecule by summing divergent perturbation theory series

The Rayleigh-Schrödinger perturbation theory is applied to a calculation of vibrational energy levels of the H₂O molecule for isolated states and the states involved in the anharmonic Fermi and Darling-Dennison resonances. It is shown that in spite of the rapid divergence of the perturbation theory series caused by the resonances, the use of the summation methods of Padé, Padé-Borel, and Padé-Hermite and the moments method allows one to obtain quite satisfactory results.     

On local symmetries of the Schrödinger equation

The multi-symplectic approach to the Schrödinger equation with a potential V = V(t,x^k) is given. The condition for a vector field X in the multi-symplectic space to be a symmetry field is found. For a spherically symmetrical potential all such symmetry fields are effectively found.The one-to-one correspondence between solutions of the free Schrödinger equation and solutions of the oscillator problem is given. This enables us to give a new geometric interpretation of the non-typical, given by A.O. Barut, symmetry of the Schrödinger equation.