Derivation of a time dependent Schrödinger equation as the quantum mechanical Landau–Lifshitz–Bloch equation

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.

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Quantum Dynamics Using The Time-Dependent Schrödinger Equation

Chemical Dynamics in Condensed Phases ◽

10.1093/oso/9780198529798.003.0007 ◽

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.

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EXACT NUMERICAL SOLUTION OF SCHRÖDINGER'S EQUATION FOR A PARTICLE IN AN INTERACTION POTENTIAL OF GENERAL SHAPE

International Journal of Modern Physics B ◽

10.1142/s0217979202014802 ◽

2002 ◽

Vol 16 (27) ◽

pp. 4081-4092 ◽

Cited By ~ 40

Author(s):

M. RIETH ◽

W. SCHOMMERS ◽

S. BASKOUTAS

Keyword(s):

Schrödinger Equation ◽

Reference System ◽

Schrodinger Equation ◽

Time Dependent ◽

Quantum Mechanical ◽

General Shape ◽

Stationary Schrödinger Equation ◽

Standard Methods ◽

Exact Numerical Solution

A method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows one to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods will be discussed critically.

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A NONADIABATIC QUANTUM MECHANICAL MONTE CARLO METHOD

International Journal of Modern Physics C ◽

10.1142/s0129183102004595 ◽

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.

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A note on asymptotic helix and quantum mechanical structure

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203212345 ◽

2003 ◽

Vol 2003 (48) ◽

pp. 3031-3039

Author(s):

Partha Guha

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Linear Time ◽

Time Dependent ◽

Quantum Mechanical ◽

Mechanical Structure

Using the formulation of a moving curve, we demonstrate that an asymptotic helix goes over to the linear time-dependent Schrödinger equation as shown by Dmitriyev (2002).

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Introduction

10.1093/oso/9780198805014.003.0001 ◽

2018 ◽

Author(s):

Niels Engholm Henriksen ◽

Flemming Yssing Hansen

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Degrees Of Freedom ◽

State Level ◽

Boltzmann Distribution ◽

Theoretical Description ◽

Time Dependent ◽

Nuclear Dynamics ◽

Molecular Reaction ◽

Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

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