scholarly journals Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip

2010 ◽  
Vol 234 (3) ◽  
pp. 777-793 ◽  
Author(s):  
Jicheng Jin ◽  
Xiaonan Wu
Keyword(s):  
Finite Element ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Two Dimensional ◽  
Finite Element Scheme ◽  
Element Scheme

Two-Grid Finite-Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation

2013 ◽  
Vol 5 (2) ◽  
pp. 180-193 ◽  
Author(s):  
Hongmei Zhang ◽  
Jicheng Jin ◽  
Jianyun Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Two Dimensional ◽  
Optimal Convergence ◽  
Standard Finite Element Method ◽  
Finite Element Schemes ◽  
Element Method

AbstractIn this paper, we construct semi-discrete two-grid finite element schemes and full-discrete two-grid finite element schemes for the two-dimensional time-dependent Schrödinger equation. The semi-discrete schemes are proved to be convergent with an optimal convergence order and the full-discrete schemes, verified by a numerical example, work well and are more efficient than the standard finite element method.

QUANTUM DYNAMICS OF WAVE PACKET IN HARMONIC POTENTIAL

2005 ◽  
Vol 19 (24) ◽  
pp. 3745-3754
Author(s):  
ZHAN-NING HU ◽  
CHANG SUB KIM
Keyword(s):  
Schrödinger Equation ◽  
Wave Packet ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Analytic Solution ◽  
Nonlinear Differential Equations ◽  
Time Dependent ◽  
Dimensional System ◽  
Two Dimensional ◽  
Vibrational States

In this paper, the analytic solution of the time-dependent Schrödinger equation is obtained for the wave packet in two-dimensional oscillator potential. The quantum dynamics of the wave packet is investigated based on this analytic solution. To our knowledge, this is the first time we solve, analytically and exactly this kind of time-dependent Schrödinger equation in a two-dimensional system, in which the Gaussian parameters satisfy the coupled nonlinear differential equations. The coherent states and their rotations of the system are discussed in detail. We find also that this analytic solution includes four kinds of modes of the evolutions for the wave packets: rigid, rotational, vibrational states and a combination of the rotation and vibration without spreading.

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