Finite elements in action modeling quantum mechanics and electrodynamics in nanoscale systems L. Ramdas Ram-Mohan

Format:

Book

Author/Creator:

Ram-Mohan, L. Ramdas, author.

Language:

English

Subjects (All):

Finite element method.

Physical Description:

1 online resource

Place of Publication:

Oxford University Press, 2026.

Oxford Oxford University Press [2026]

Summary:

The focus of this textbook is the elucidation of the interplay between the principle of stationary action and Schr?odinger's equation, and its solution using the finite element method (FEM), a method of solving differential equations, in systems whose dimensions are on the order of nanometers. The treatment of the dynamics of electrons in such systems deserves a quantum mechanical description and typical applications at the nanoscale also require the modelling of electrodynamic fields. Aimed at graduate students and researchers in nanoscale systems, materials growth, optoelectronics, engineering, physics, and chemistry, as well as electrical engineers, mechanical engineers, computational scientists, and quantum computer developers, this book explores the development of variational methods and their implementation for several physical examples in the framework of the FEM and addresses issues that are very common in modelling nanoscale systems

Contents:

Cover

Title page

Copyright page

Preface

Contents

Part I The Action Integral in Quantum Mechanics

1 Schrödinger's equation and the action

1.1 Schrödinger's equation

1.2 Boundary conditions

1.3 The action integral and the variational method

1.4 Final comments

1.5 Problems

References

2 Action, FEM, and BCs

2.1 Action and FEM

2.2 Implementing Gauss quadrature over elements

2.3 Linear interpolation and element matrix calculations

2.3.1 Applying Dirichlet BCs to the action

2.4 Galerkin's method and Dirichlet BCs

2.5 Applying Neumann BCs

2.5.1 The action integral and Neumann BCs

2.5.2 FEM and the action with Neumann BC

2.5.3 Galerkin's method and the Neumann BC

2.6 Hermite interpolation and BCs

2.6.1 Hermite interpolation and action: Dirichlet BC

2.6.2 Hermite interpolation and action: Neumann BC

2.6.3 Galerkin's method and Hermite interpolation

2.7 Schrödinger's equation and 1D tunneling

2.7.1 Tunneling BCs

2.7.2 The Schrödinger action with surface terms

2.7.3 FEM and tunneling: linear interpolation

2.7.4 Galerkin's method and tunneling

2.7.5 Hermite interpolation and tunneling

2.8 Reconstruction of the solution from the nodal values

2.9 Final comments

2.10 Problems

3 Element geometries for 2D and 3D

3.1 Quadrilateral elements

3.2 Polar coordinates in 2D

3.2.1 The line integral and the signs of dl and

3.2.2 Integration by parts in 2D and 3D

3.3 Triangular elements

3.4 Finite elements in 3D

3.4.1 Brick elements in 3D

3.4.2 Tetrahedral elements in 3D

3.5 Problems

4 Boundary conditions at material interfaces

4.1 Interface boundary conditions in 1D

4.2 Interface boundary conditions in 2D

4.3 Interface smoothing using Fermi functions

4.4 Problems

5 Accidental degeneracy in cubic semiconductor quantum dots

5.1 Introduction

5.1.1 The Mendeleev Periodic Table of Elements and accidental degeneracy

5.1.2 The cubic quantum dot

5.2 Degeneracy and point group symmetry

5.3 FEM applied to cubic QDs

5.4 Results and discussions

5.5 Conclusions

Part II Quantum Scattering

6 Quantum scattering in 1D revisited

6.1 Quantum scattering in 1D

6.2 Sources and absorbers: boundary conditions

6.2.1 No-reflection condition at the stealth interface

6.2.2 Schrödinger's equation with a source term

6.3 Finite element analysis for scattering

6.4 Concluding remarks

6.5 Problems

7 2D quantum waveguides

7.1 Introduction

7.2 Action integral for 2D waveguides

7.3 Discretization of the action in FEM

7.4 From nodal variables to modal amplitudes

7.5 Surface terms

7.6 Applying BCs on the nodal/modal variables

7.7 Probability currents from nodal solutions

7.8 Landauer conductance

7.9 Further considerations

8 Quantum scattering in 2D waveguides

8.1 Introduction

Notes:

Includes bibliographical references

Online resource and publisher information; title from PDF title page (viewed on January 22, 2026)

Other Format:

Print version:

ISBN:

9780198975649

0198975643

OCLC:

1569786120

Publisher Number:

CIPO000342067

Access Restriction:

Restricted for use by site license