A NONADIABATIC QUANTUM MECHANICAL MONTE CARLO METHOD

2002 ◽  
Vol 13 (07) ◽  
pp. 909-915
Author(s):  
A. M. MAZZONE
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Efficient Solution ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Classical Dynamics ◽  
Hartree Fock

The problem addressed by this study is an efficient solution of the multi-particle time-dependent Schrödinger equation to be used under nonadiabatic conditions. To this purpose a solution combining classical dynamics for the nuclei and a quantum mechanical Monte Carlo method for the electrons is suggested as a practically feasible approach. As a show-case example, the method is applied to the evaluation of the ground state of H, He, H2 and H3 whose energy and structure is also obtained from stationary Hartree–Fock calculations.

scholarly journals Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

2018 ◽  
Vol 4 (1) ◽  
pp. 47-55
Author(s):  
Timothy Brian Huber
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Mechanical System ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

A quantum method for dynamic nonlinear programming technique using Schrödinger equation and Monte Carlo approach

2018 ◽  
Vol 32 (30) ◽  
pp. 1850374 ◽  
Author(s):  
Amandeep Kaur ◽  
Satnam Kaur ◽  
Gaurav Dhiman
Keyword(s):  
Monte Carlo ◽  
Quantum Computing ◽  
Nonlinear Programming ◽  
Monte Carlo Method ◽  
Schrödinger Equation ◽  
Nonlinear Problem ◽  
Schrodinger Equation ◽  
Programming Technique ◽  
Quantum Method ◽  
Speed Up

The power of quantum computing may allow for solving the problems which are not practically feasible on classical computers and suggest a considerable speed up to the best known classical approaches. In this paper, we present the contemporary quantum behaved approach which is based on Schrödinger equation and Monte Carlo method. The three basic steps of proposed technique are also mathematically modeled and discussed for effective movement of particles. The performance of the proposed approach is tested for solving the dynamic nonlinear problem. Experimental results reveal the supremacy of proposed approach for solving the nonlinear problem as compared to other approaches.

Quantum Dynamics Using The Time-Dependent Schrödinger Equation

2006 ◽  
Author(s):  
Abraham Nitzan
Keyword(s):  
Schrödinger Equation ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Full Description ◽  
Quantum Mechanical ◽  
Condensed Phases ◽  
Full Picture ◽  
Linear Differential ◽  
External Forces

This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.

The X (X = F, Cl, I) effect on the photoassociation of H+X→HX

2015 ◽  
Vol 14 (08) ◽  
pp. 1550062
Author(s):  
Wei Gao ◽  
Bin-Bin Wang ◽  
Yong-Chang Han ◽  
Shu-Lin Cong
Keyword(s):  
Schrödinger Equation ◽  
Vibrational Level ◽  
Ground State ◽  
Schrodinger Equation ◽  
Vibrational State ◽  
Time Dependent ◽  
Overlap Integral ◽  
Target State ◽  
Multiphoton Transition ◽  
Franck Condon

This work explores the vibrational state-selective photoassociation (PA) in the ground state of the HX (X = F, Cl, I) molecule by solving the time-dependent Schrödinger equation. For the three systems, the vibrational level of [Formula: see text] is set to be the target state and the PA probability of the target state is calculated and compared by considering different initial collision momentums. It is found that the PA probabilities are in accordance with Franck–Condon overlap integral for the HI and HCl systems, but it is not the case for the HF system. Moreover, for the HF system, it is shown that the PA probability of the target state is largest and the multiphoton transition is more likely to occur.

Coherent state theory of large amplitude collective motion

1976 ◽  
Vol 54 (19) ◽  
pp. 1941-1968 ◽  
Author(s):  
D. J. Rowe ◽  
R. Bassermann
Keyword(s):  
Schrödinger Equation ◽  
Large Amplitude ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Collective Motion ◽  
State Theory ◽  
Time Dependent ◽  
Hartree Fock ◽  
Starting Point ◽  
Reduced Representations

A theory of large amplitude collective motion of a many-particle system is presented, which is relevant, for example, to nuclear fission. The theory is a combination of techniques used in many areas of physics and mathematics. The starting point is the application of the time-dependent Schrödinger equation to generate invariant subspaces of the Hamiltonian in the Hartree–Fock approximation. This is a generalization of the group-theoretical device of generating orbits of a group in the construction of reduced representations. It is shown how solutions of the time-dependent Schrödinger equation can be expressed as instantaneous stationary states of a constrained static Hamiltonian. Thus contact is made with the traditional cranking models and constrained Hartree–Fock theories of large amplitude collective motion. The collective motion is quantized using the Hill–Wheeler–Griffin method of generator coordinates in a basis of generalized coherent states. One is thereby able to exploit much of the theory of harmonic oscillator coherent states, which have been so successfully used in the quantum theory of the laser. The resulting Schrödinger equation for the collective dynamics is expressed both in the Bargmann representation and in the more familiar Schrödinger representation. It is shown that solution of the Schrödinger equation in the small amplitude harmonic approximation reproduces the well-known RPA result. A pilot calculation for 28Si shows that application in large amplitude is also feasible.

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