Berry phase in a quantum system with time-dependent boundary conditions

2014 ◽  
Vol 28 (15) ◽  
pp. 1450122
Author(s):  
An-Ling Wang ◽  
Fu-Ping Liu ◽  
Zhao-Xian Yu
Keyword(s):  
Vector Field ◽  
Boundary Conditions ◽  
Quantum System ◽  
Moving Boundary ◽  
Berry Phase ◽  
Quantum Particle ◽  
Time Dependent ◽  
Potential Well ◽  
Particle Behavior ◽  
Topological Term

We discuss the problem of the quantum particle behavior in an infinitely deep elliptical potential well with a moving boundary conditions. We show that this quantum system is equivalent to a particle interacting with a vector field. By introducing the squeezing transformations and making use of the removability of the topological term, the exact Berry phase in a cyclic evolution is calculated.

Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon a step potential and a square potential well

2016 ◽  
Vol 94 (1) ◽  
pp. 9-14 ◽  
Author(s):  
Mark R.A. Shegelski ◽  
Kevin Malmgren ◽  
Logan Salayka-Ladouceur
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Time Dependent ◽  
Potential Well ◽  
Step Potential ◽  
Exact Analytical Expression ◽  
Square Well ◽  
Transmission And Reflection

We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.

Nodal Integral and Finite Difference Solution of One-Dimensional Stefan Problem

2003 ◽  
Vol 125 (3) ◽  
pp. 523-527 ◽  
Author(s):  
James Caldwell ◽  
Svetislav Savovic´ ◽  
Yuen-Yick Kwan
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Moving Boundary ◽  
Finite Difference Methods ◽  
Melting Process ◽  
Time Dependent ◽  
One Dimensional ◽  
Stefan Problems ◽  
Finite Difference Solution ◽  
Very High

The nodal integral and finite difference methods are useful in the solution of one-dimensional Stefan problems describing the melting process. However, very few explicit analytical solutions are available in the literature for such problems, particularly with time-dependent boundary conditions. Benchmark cases are presented involving two test examples with the aim of producing very high accuracy when validated against the exact solutions. Test example 1 (time-independent boundary conditions) is followed by the more difficult test example 2 (time-dependent boundary conditions). As a result, the temperature distribution, position of the moving boundary and the velocity are evaluated and the results are validated.

scholarly journals Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method

2009 ◽  
Vol 13 (4) ◽  
pp. 165-174 ◽  
Author(s):  
Svetislav Savovic ◽  
James Caldwell
Keyword(s):  
Temperature Distribution ◽  
Boundary Conditions ◽  
Stefan Problem ◽  
Finite Differences ◽  
Moving Boundary ◽  
Grid Method ◽  
Time Dependent ◽  
Time Step ◽  
Space Grid ◽  
Variable Space

The variable space grid method based on finite differences is applied to the one-dimensional Stefan problem with time-dependent boundary conditions describing the solidification/melting process. The temperature distribution, the position of the moving boundary and its velocity are evaluated in terms of finite differences. It is found that the computational results obtained by the variable space grid method exhibit good agreement with the exact solution. Also the present results for temperature distribution are found to be more accurate compared to those obtained previously by the variable time step method.

Axisymmetric flow near the critical point on the moving boundary

2010 ◽  
Vol 7 ◽  
pp. 182-190
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh. Nasibullaeva
Keyword(s):  
Fluid Flow ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Axial Velocity ◽  
Moving Boundary ◽  
Axisymmetric Flow ◽  
Boundary Motion ◽  
Newton Raphson ◽  
Raphson Method

In this paper the investigation of the axisymmetric flow of a liquid with a boundary perpendicular to the flow is considered. Analytical equations are derived for the radial and axial velocity and pressure components of fluid flow in a pipe of finite length with a movable right boundary, and boundary conditions on the moving boundary are also defined. A numerical solution of the problem on a finite-difference grid by the iterative Newton-Raphson method for various velocities of the boundary motion is obtained.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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