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Journal of Physics A Mathematical and General

An iterative solution of Schrodinger's equation

M. Cohen

Published Jun 1, 1984• 2 citations
Publisher

Abstract

The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H 0 + lambda H 1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.

DOI
10.1088/0305-4470/17/8/020

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The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H 0 + lambda H 1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.
Abstract The Hamiltonian and wavefunctions of two-dimensional two-electron quantum dots (2D2eQD) in parabolic confinement are determined. The ground and excited state energies are calculated solving the Schrödinger equation analytically and numerically. To determine the energy eigen-value of the system variational method is employed due to the large coupling constant λ ≈ 1.1 \lambda \approx 1.1 . The trial wavefunctions are developed for both ground and excited states. The ground state wave function is a para state and the excited state wavefunctions belong to both para and ortho states based on the symmetry and antisymmetry of spatial wavefunctions. Using the obtained energy eigen-values at the two states, the first- and third-order nonlinear absorption coefficient and refractive index are analytically obtained with the help of density matrix formalism and iterative procedure.
Quantum integrability has been applied to a large variety of low dimensional Hamiltonians in Quantum Field Theory, Condensed Matter Physics, and Statistical Mechanics to obtain exact expressions for the spectrum and thermodynamics of these systems. In most of these studies the coupling constants are constant in time. Here we present an exact solution of the nonstationary Schrödinger equation for the Kondo Hamiltonian with a time-dependent spin-exchange coupling J(t) J ( t ) of the form \lambda t + p(t) ± \sqrt{(\lambda t + p(t))^2 + 16/3} λ t + p ( t ) ± ( λ t + p ( t ) ) 2 + 16 / 3 , where p(t) p ( t ) is an arbitrary periodic function, under periodic boundary concditions. Unlike previously studied time-dependent integrable models, which are rooted in the classical Yang–Baxter structure and associated Knizhnik–Zamolodchikov equations, our approach is based on the quantum Knizhnik–Zamolodchikov framework and the quantum Yang–Baxter algebra. Our results broaden the domain of time-dependent integrability to a genuinely quantum class of models and provide new tools for exploring coherent nonequilibrium dynamics in strongly correlated systems.