The FFT-symplectic integrator (SI) scheme devised for solving the wave packet propagation problem is applied to stationary reactive-scattering problems. In order to relate the stationary problem to the time-dependent problem, a class of Schrödinger equation with an inhomogeneous wave source term is introduced. By using the equivalence between the stationary scattering eigenstate and the equilibrium state of the inhomogeneous Schrödinger equation, the scattering eigenstates can be computed by integrating the inhomogeneous Schrödinger equation with the FFT-SI scheme. A Gaussian wave source is proposed as an efficient wave source exhibiting rapid relaxation toward the eigenstate. Our method is tested by a one-dimensional example which has an analytical solution, and great numerical accuracy is confirmed. It is further examined by an example of time-dependent scattering and by a two-dimensional example of chaotic tunnel-scattering.The FFT-symplectic integrator (SI) scheme devised for solving the wave packet propagation problem is applied to stationary reactive-scattering problems. In order to relate the stationary problem to the time-dependent problem, a class of Schrödinger equation with an inhomogeneous wave source term is introduced. By using the equivalence between the stationary scattering eigenstate and the equilibrium state of the inhomogeneous Schrödinger equation, the scattering eigenstates can be computed by integrating the inhomogeneous Schrödinger equation with the FFT-SI scheme. A Gaussian wave source is proposed as an efficient wave source exhibiting rapid relaxation toward the eigenstate. Our method is tested by a one-dimensional example which has an analytical solution, and great numerical accuracy is confirmed. It is further examined by an example of time-dependent scattering and by a two-dimensional example of chaotic tunnel-scattering.

The Journal of Chemical PhysicsMar 15, 199720 citations

The One-Dimensional Schrödinger Equation

Jerry Goodisman

Springer eBooksJan 1, 19770 citations

Simplified Solving Procedure for the One-Dimensional Schrodinger Equation with Asymmetrical Boundary Conditions

H. Puszkarski

We propose a new simplified procedure for finding the analytical solutions of the stationary one-dimensional Schrödinger equation, with asymmetric boundary conditions imposed on the equation.The essence of the method consists in expressing the general solution explicitly in terms of the boundary parameters in a form which, by itself, satisfies one of the boundary conditions involved; then, the other boundary condition gives straightforwardly the characteristic equation.This method may turn out to be beneficial with regard to the recently growing interest in one-dimensional quantum systems.We propose a new simplified procedure for finding the analytical solutions of the stationary one-dimensional Schrödinger equation, with asymmetric boundary conditions imposed on the equation.The essence of the method consists in expressing the general solution explicitly in terms of the boundary parameters in a form which, by itself, satisfies one of the boundary conditions involved; then, the other boundary condition gives straightforwardly the characteristic equation.This method may turn out to be beneficial with regard to the recently growing interest in one-dimensional quantum systems.

Acta Physica Polonica ASep 1, 19990 citations

Schrödinger Equation for One-Dimensional Systems

Luigi E. Picasso

UNITEXT for physicsOct 26, 20150 citations

Asymptotes of non-stationary solutions of the Schrödinger equation for a particle interacting with a one-dimensional $δ$ -potentialAsymptotes of non-stationary solutions of the Schrödinger equation for a particle interacting with a one-dimensional $δ$-potential

Nikolay Leonidovich Chuprikov

It is shown that non-stationary solutions of the Schrödinger equation, which describes the quantum dynamics of a particle in the field of a one-dimensional delta potential (1DDP), are divided into two classes: some define pure states that have no free dynamics as $t\to\mp\infty$ ; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schrödinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.It is shown that non-stationary solutions of the Schrödinger equation, which describes the quantum dynamics of a particle in the field of a one-dimensional delta potential (1DDP), are divided into two classes: some define pure states that have no free dynamics as $t\to\mp\infty$; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schrödinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.

arXiv:2305.15381May 19, 20230 citations

Asymptotes of non-stationary solutions of the Schrödinger equation for a particle interacting with a one-dimensional

δ

-potential

t\to\mp\infty

; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schrödinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.