Finite Element and Boundary Element Applications in Quantum Mechanics

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ISBN: 9780198525226 | 0198525222
Cover: Paperback
Copyright: 11/7/2002

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Starting from a clear, concise introduction, the powerful finite element and boundary element methods of engineering are developed for application to quantum mechanics. The reader is led through illustrative examples displaying the strengths of these methods using application to fundamental quantum mechanical problems and to the design/simulation of quantum nanoscale devices.

PART I INTRODUCTION TO THE FEM

Introduction

(28)

Basic concepts of quantum mechanics

(6)

Schrodinger's equation

(4)

Postulates of quantum mechanics

(2)

Principle of stationary action

(8)

The action integral

(4)

Examples

(3)

Finite elements

(4)

Historical comments

(1)

Problems

(8)

References

(3)

Simple quantum systems

(32)

The simple harmonic oscillator

(8)

The hydrogen atom

(6)

The Rayleigh--Ritz variational method

(5)

Programming considerations

(7)

Problems

(6)

References

(3)

Interpolation polynomials in one dimension

(20)

Introduction

(1)

Lagrange interpolation polynomials

(2)

Hermite interpolation polynomials

(4)

Transition elements

(1)

Low order interpolation polynomials

(1)

Low order Lagrange interpolation

(1)

Low order Hermite interpolation

(1)

Interpolation polynomials in Mathematica

(4)

Lagrange interpolation

(2)

Hermite interpolation

(2)

Infinite elements

(1)

Simple quantum systems revisited

(3)

Problems

(3)

References

(2)

Adaptive FEM

(26)

Introduction

(1)

Error in interpolation

(2)

Error in the discretized action

(5)

h-convergence

(2)

p-convergence

(3)

The action in adaptive calculations

(13)

An ordinary differential equation

(4)

The H atom again

(6)

Adaptive p-refinement

102

(2)

Concluding remarks

104

(5)

References

105

(4)

PART II APPLICATIONS IN 1D

Quantum mechanical tunneling

109

(36)

Introduction

109

(1)

Mixed BCs: redefining the action

110

(3)

The Galerkin method

113

(2)

Tunneling calculations in the FEM

115

(8)

Evaluation of the residual

115

(3)

Applying mixed BCs

118

(5)

Comparing Galerkin FEM with WKB

123

(2)

Quantum states in asymmetric wells

125

(12)

Problems

137

(8)

References

141

(4)

Schrodinger--Poisson self-consistency

145

(26)

Introduction

145

(4)

Schrodinger and Poisson equations

149

(2)

Source terms

151

(2)

The Fermi energy and charge neutrality

153

(2)

The Galerkin finite element approach

155

(10)

Boundary conditions

155

(4)

The iteration procedure

159

(2)

Numerical issues

161

(3)

Essential and natural boundary conditions

164

(1)

Further developments

165

(2)

Problems

167

(4)

References

169

(2)

Landau states in a magnetic field

171

(24)

Introduction

171

(6)

Landau levels

171

(3)

Density of states

174

(3)

Heterostructures in a B-field

177

(4)

Faraday configuration

177

(2)

Voigt configuration

179

(2)

Comparison with experiments

181

(7)

Interband transitions

181

(2)

Energy dependence on the orbit center

183

(2)

Level mixing in superlattices with small band offsets

185

(1)

Density of states in the Voigt geometry

186

(2)

Voigt geometry and a semiclassical model

188

(4)

Landau orbit theory

188

(3)

Envelope functions and the FEM in k-space

191

(1)

Problems

192

(3)

References

194

(1)

Wavefunction engineering

195

(22)

Introduction

195

(2)

k.P theory of band structure

197

(3)

Designing mid-infrared lasers

200

(10)

The type-II W-laser

200

(6)

The interband cascade laser

206

(4)

Concluding comments

210

(7)

References

211

(6)

PART III 2D APPLICATIONS OF THE FEM

2D elements and shape functions

217

(24)

Introduction

217

(1)

Rectangular elements

218

(4)

Lagrange elements

218

(3)

Hermite elements

221

(1)

Triangular elements

222

(6)

Defining curved edges

228

(5)

An element on a parametric curve

228

(3)

Parametric form of 2D surfaces

231

(2)

The action in 2D problems

233

(3)

Gauss integration in two dimensions

236

(5)

References

238

(3)

Mesh generation

241

(16)

Meshing simple regions

241

(6)

Distortion of regular regions

242

(5)

Using orthogonal curved coordinates

247

(1)

Regions of arbitrary shape

247

(10)

Delaunay meshing

247

(1)

Advancing front algorithms

248

(2)

The algebraic integer method

250

(6)

References

256

(1)

Applications in atomic physics

257

(18)

The H atom in a magnetic field

257

(12)

Schrodinger's equation and the action

258

(1)

Applying the FEM

259

(7)

Magnetic fields

266

(3)

Ground state energy in helium

269

(2)

Other results

271

(4)

References

273

(2)

Quantum wires

275

(24)

Introduction

275

(1)

Quantum wires and the FEM

276

(6)

Symmetry properties of the square wire

282

(3)

The checkerboard superlattice

285

(4)

Optical nonlinearity in the CBSL

289

(3)

Quantum wires of any cross-section

292

(7)

References

295

(4)

Quantum waveguides

299

(28)

Quantization of resistance

300

(3)

The straight waveguide

303

(8)

Quantum bound states in waveguides

311

(2)

The quantum interference transistor

313

(2)

``Stealth'' elements and absorbing BC

315

(7)

The Ginzburg-Landau equation

322

(5)

References

323

(4)

Time-dependent problems

327

(36)

Introduction

327

(1)

Standard approaches to time evolution

327

(4)

Schrodinger's equation and the method of finite differences

327

(3)

The finite difference method for the wave equation

330

(1)

A transfer matrix for time evolution

331

(4)

Lanczos reduction of transfer matrices

335

(3)

Instability with initial conditions

338

(9)

Comparing IVBC and two-point BCs

338

(9)

The variational approach

347

(8)

A variational difficulty

347

(1)

Variations using adjoint functions

348

(2)

Adjoint variations for the wave equation

350

(2)

Connection with quantum field theory

352

(3)

Concluding remarks

355

(8)

References

356

(7)

PART IV SPARSE MATRIX APPLICATIONS

Matrix solvers and related issues

363

(36)

Introduction

363

(1)

Bandwidth reduction

363

(3)

Solution of linear equations

366

(6)

Gauss elimination

366

(1)

The conjugate gradient method

367

(5)

The standard eigenvalue problem

372

(1)

The generalized eigenvalue problem

373

(26)

Sturm sequence check

376

(2)

Inverse vector iteration

378

(2)

The subspace vectors

380

(1)

The Rayleigh quotient

381

(2)

Subspace iteration

383

(2)

The Davidson algorithm

385

(3)

Least square residual minimization

388

(1)

The Lanczos method

388

(5)

References

393

(6)

PART V BOUNDARY ELEMENTS

The boundary element method

399

(36)

Introduction

399

(1)

The boundary integral

400

(2)

An analytical approach

402

(6)

A Dirichlet problem

402

(4)

A Neumann problem

406

(2)

Infinite domain Green's function

408

(6)

Numerical issues

414

(9)

Evaluation of the element integrals

414

(1)

Applying boundary conditions

415

(2)

Boundary condition at the corner node

417

(3)

Setting up the matrix equation

420

(2)

Construction of interior solution

422

(1)

A worked example

423

(2)

Two sum rules

425

(2)

Comparing the BEM with the FEM

427

(1)

Problems

428

(7)

References

432

(3)

The BEM and surface plasmons

435

(36)

Introduction

435

(2)

Multiregion BEM: two regions

437

(3)

Linear interpolation

437

(2)

Hermite interpolation

439

(1)

Bulk and surface plasmons

440

(21)

Bulk plasma oscillations

441

(2)

Surface plasmons at a single planar interface

443

(4)

Surface plasmons for slab geometry

447

(4)

Surface plasmons in a cylindrical wire

451

(4)

Two metallic wires

455

(3)

Metal wire on a substrate

458

(2)

Plasmons in other confining geometries

460

(1)

Surface-enhanced Raman scattering

461

(1)

Problems

462

(9)

References

468

(3)

The BEM and quantum applications

471

(112)

Introduction

471

(1)

2D electron waveguides

471

(16)

Implementing boundary conditions

477

(6)

Multiregion waveguide problems

483

(1)

Multiple ports and transmission

484

(3)

The BEM and 2D scattering

487

(4)

Eigenvalue problems and the BEM

491

(7)

Quantum wires

491

(4)

Hearing the shape of a drum

495

(3)

Concluding remarks on the BEM

498

(3)

Problems

501

(10)

References

505

(6)

PART VI APPENDICES

A Gauss quadrature

511

(16)

A.1 Introduction

511

(1)

A.2 Gauss--Legendre quadrature

511

(2)

A.3 Gauss--Legendre base points and weights

513

(4)

A.4 An algorithm for adaptive quadrature

517

(2)

A.5 Other Gauss formulas

519

(1)

A.6 The Cauchy principal value of an integral

520

(2)

A.7 Properties of Legendre functions

522

(2)

A.8 Problems

524

(1)

References

525

(2)

B Generalized functions

527

(14)

B.1 The Dirac δ-function

527

(3)

B.2 The δ-function as the limit of a ``normal'' function

530

(2)

B.3 δ-functions in three dimensions

532

(1)

B.4 Other generalized functions

533

(1)

B.4.1 The step-function θ(x)

533

(1)

B.4.2 The sign-function ε(x)

534

(1)

B.4.3 The Plemelj formula

535

(1)

B.4.4 An integral representation for θ(z)

536

(1)

B.5 Problems

536

(2)

References

538

(3)

C Green's functions

541

(40)

C.1 Introduction

541

(2)

C.2 Properties of Green's functions

543

(4)

C.3 Sturm--Liouville differential operators

547

(7)

C.4 Green's functions in electrostatics

554

(3)

C.5 Boundary integral solutions: a comment

557

(8)

C.6 Green's functions in electrodynamics

565

(5)

C.7 The wave equation in one dimension

570

(3)

C.8 The wave equation in two dimensions

573

(1)

C.9 Green's functions and integral equations

574

(2)

C.10 Problems

576

(3)

References

579

(2)

D Physical constants

581

(2)

Author index

583

(14)

Subject index

597

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Finite Element and Boundary Element Applications in Quantum Mechanics

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Finite Element and Boundary Element Applications in Quantum Mechanics

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