Anharmonic solution of Schrödinger time-independent equation

We present here a mathematical explanation of how the Schr?dinger equation for a class of harmonic oscillators possesses exact solutions. Some of the extended potentials used here are not present in the literature.     

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

In this paper we have derived the fractional-order Schrödinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrödinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrödinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero – rather they are non-zero with a minimum spread. This type of non-zero spread arises in the microscopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials dx (and dt) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (Δx) ^α (and (Δt) ^α ) with 0 < α < 1; called as ‘fractional differentials’. For arbitrarily small Δx and Δt (tending towards zero), these ‘fractional’ differentials are greater than Δx (and Δt), i.e. (Δx) ^α > Δx and (Δt) ^α > Δt. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.     

The fractional discrete nonlinear Schrödinger equation

We examine a fractional version of the discrete nonlinear Schrödinger (dnls) equation, where the usual discrete laplacian is replaced by a fractional discrete laplacian. This leads to the replacement of the usual nearest-neighbor interaction to a long-range intersite coupling that decreases asymptotically as a power-law. For the linear case, we compute both, the spectrum of plane waves and the mean square displacement of an initially localized excitation in closed form, in terms of regularized hypergeometric functions, as a function of the fractional exponent. In the nonlinear case, we compute numerically the low-lying nonlinear modes of the system and their stability, as a function of the fractional exponent of the discrete laplacian. The selftrapping transition threshold of an initially localized excitation shifts to lower values as the exponent is decreased and, for a fixed exponent and zero nonlinearity, the trapped fraction remains greater than zero.     

Analytic presentation of a solution of the Schrödinger equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into the nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.     

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.     

A numerical method for fractional Schrödinger equation of Lennard-Jones potential

Depending on fractional analysis, we find a numerical algorithm to solve the time-independent fractional Schrödinger equation in case of Lennard-Jones potential in one dimension. We apply the algorithm for multiple values of the fractional parameter of the space-dependent fractional Schrödinger equation and multiple values of the system's energy to find the wave function and the probability in these cases.     

Lagrangian form of Schrödinger equation

Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein–Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.     

A solution spectrum of the nonlinear Schrödinger equation. II

It was shown in a previous communication that the nonlinear Schrödinger equation exhibits a spectrum of eigenfunctions of the form = _k,A _k (coshkx) ^–k and = _k B _k (coshkx) ^–k–1sinhkx, and the corresponding eigenvalues of the energy are related to a band structure with a characteristic energy gap as a significant feature. In the present paper, it is shown that a further spectrum exists exhibiting the general structure = _k=0 A _k(cosh kx)^–k–1/2and = _k=0 B_k(cosh kx)^–k–3/2sinhkx and yielding also a band structure. An extension of the solution spectrum to a nonlinear Klein-Gordon equation and a nonlinear Dirac equation does not imply essential difficulties, and the corresponding characteristic band structure has to be related to a mass spectrum.     

Exact solution of the discrete Schrödinger equation for ferromagnetic chains

It is shown that Bethe's exact solution for the finite Heisenberg ferromagnetic chain can be obtained via a direct Fourier transform. Unlike the Bethe ansatz approach, the latter is not confined to one space dimension and opens the possibility of obtaining exact solutions in higher dimensions.     

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

The continuous states of the l-wave Schrödinger equation for the diatomic molecule represented by the hyperbolical function potential are carried out by a proper approximation scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the hyperbolical function potential are presented and the corresponding calculation formula of phase shifts is derived. Also, we interestingly obtain the corresponding bound state energy levels by analyzing analytical properties of scattering amplitude.     

Time-space fractional Schrödinger like equation with a nonlocal term

In this paper a time-space fractional Schr?dinger equation containing a nonlocal term has been studied. The time dependent solutions have been obtained in terms of the H-function. New general results include the results of integer Schr?dinger equation with a nonlocal term and the well-known quantum formulae for a free particle kernel.     

Lagrangian formulation of the Schrödinger equation

We recast the Schrödinger equation in a new Lagrangian formulation. The equation is —i?dψ (x,t)/dt = Lψ (x,t), whereL is the Lagrangian operator. Expressions forL and ford/dt — ⊥ are derived in terms of coordinate and momentum operators.     

Direct solution of Schrödinger equation by genetic algorithm: test cases

A stable and generalizable recipe is proposed for directly solving Schrödinger equation by genetic algorithm. The recipe is tested for obtaining (i) the radial wave function of the H-atom in the ground and excited states, (ii) ground and excited states in a symmetric double well potential, (iii) ground and excited states of two coupled harmonic oscillators. The possibility of generalising and extending the recipe to more complex problems is analysed.     

Exact solution of D-dimensional Schrödinger equation generated from certain central power-law potentials

By applying Extended Transformation method we have generated exact solution of D-dimensional radial Schrödinger equation for a set of power-law multi-term potentials taking singular potentials $V(r) = ar^{ - \tfrac{1} {2}} + br^{ - \tfrac{3} {2}}$ , $V(r) = ar^{\tfrac{2} {3}} + br^{ - \tfrac{2} {3}} + cr^{ - \tfrac{4} {3}}$ , V(r) = ar + br ^?1 + cr ² and V(r) = ar ²+br ^?2+cr ^?4+dr ^?6 as input reference. The restriction on the parameters of the given potentials and angular momentum quantum number ? are obtained. The multiplet structure of the generated exactly solvable potentials are also shown.     

Exact solution of N-dimensional radial Schrödinger equation for the fourth-order inverse-power potential

Radial Schrödinger equation in N-dimensional Hilbert space with the potential V(r)=ar^-1+br^-2+cr^-3+dr^-4 is solved exactly by power series method via a suitable ansatz to the wave function with parameters those also exist in the potential function possibly for the first time. Exact analytical expressions for the energy spectra and potential parameters are obtained in terms of linear combinations of known parameters of radial quantum number n, angular momentum quantum number l, and the spatial dimensions N. Expansion coefficients of the wave function ansatz are generated through the two-term recursion relation for odd/even solutions.     

Dark-in-the-Bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms

We present new type of Dark-in-the-Bright solution also called dipole soliton for the higher order nonlinear Schrödinger (HNLS) equation with non-Kerr nonlinearity under some parametric conditions and subject to constraint relation among the parameters in optical context. This equation could be a model equation of pulse propagation beyond ultrashort range in optical communication systems. The solitary wave solution is composed of the product of bright and dark solitary waves. This type of pulse shape to be formed both the group velocity dispersion and third-order dispersion must be compensated. We also investigated the stability of the solitary wave solution under some initial perturbation on the parametric conditions. We have shown that the shape of pulse remains unchanged up to 20 normalized lengths even under some very small violation in parametric conditions.     

Nonperturbative analytical solution of the time fractional nonlinear Burger’s equation

A non-perturbative analytical solution is derived for the time fractional nonlinear Burger’s equation by using Adomian Decomposition Method (ADM). The present method performs extremely well in terms of accuracy, efficiency and simplicity.     

A canonical dilation of the Schrödinger equation

In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.