Finite Elements in Action Modeling Quantum Mechanics and Electrodynamics in Nanoscale Systems

The central focus of this textbook is the elucidation of the interplay between the principle of stationary action and Schrödinger's equation, and its solution using the finite element method (FEM), a method of solving differential equations, in physical 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 modeling of electrodynamic fields. For instance, nanoscale semiconductor laser design requires the interplay between electrons and photons to be modeled simultaneously. 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 modeling nanoscale systems.

1 - Schrödinger's equation and the action
2 - Action, FEM and BCs
3 - Element geometries for 2D and 3D
4 - Boundary conditions at material interfaces
5 - Accidental degeneracy in cubic semiconductor quantum dots
6 - Quantum scattering in 1D revisited
7 - 2D quantum waveguides
8 - Quantum scattering in 2D waveguides
9 - Open domain quantum scattering with sources and absorbers
10 - Wavefunction engineering of semiconductor nanostructures
11 - Schrödinger-Poisson self-consistency in layered semiconductor nanostructures
12 - The Extraordinary Magneto-Resistance effect in metal- semiconductor structures
13 - Read-head design based on the EMR effect
14 - Fields in electromagnetic waveguides
15 - Modeling photonic crystals with Hermite FEM
16 - Cavity Electrodynamics and symmetries
17 - Dimensional continuation of EM singularities in structures with re-entrant geometry
18 - The gauge degree of freedom in Electrodynamics
A - Derivation of shape functions using group theory
B - Shape functions for 1D, 2D, and 3D finite elements
C - Hermite Least Squares Data Fitting

L. Ramdas Ram-Mohan graduated from St. Stephen's College at Delhi University with a B.Sc. (Honors) in 1964 and did his graduate work at Purdue University in theoretical high energy physics earning his MS and PhD in 1971. After post-doctoral assignments at the Freie Universität in Berlin and Delhi University, he joined Purdue University from 1975 to 1978. He has been at Worcester Polytechnic Institute since 1978 as Assistant, Associate, and was made Full Professor in 1985. His interests have ranged from quantum field theory, linear and nonlinear optical properties of semiconductor heterostructures, to high-accuracy finite element methods for scattering theory and electrodynamics, band structures of solids, and wavefunction engineering of quantum well lasers.