scholarly journals The time-dependent Schrödinger equation with piecewise constant potentials

2018 ◽  
Vol 31 (1) ◽  
pp. 57-83
Author(s):  
NATALIE E. SHEILS ◽  
BERNARD DECONINCK
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Spatial Dimension ◽  
Time Dependent ◽  
Interface Problems ◽  
Solvable Models ◽  
Transform Method ◽  
Piecewise Constant ◽  
Constant Potential ◽  
Time Dependent Problem

The linear Schrödinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with physically important models. In examples such as ‘particle in a box’ and tunnelling, attention is restricted to the time-independent Schrödinger equation. This paper combines the unified transform method and recent insights for interface problems to present fully explicit solutions for the time-dependent problem.

Introduction

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.

Numerical Solution of the Time-Dependent Schrödinger Equation and Application toH+-H

1979 ◽  
Vol 43 (7) ◽  
pp. 512-515 ◽  
Author(s):  
Vida Maruhn-Rezwani ◽  
Norbert Grün ◽  
Werner Scheid
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
Time Dependent
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