Finite element and boundary element applications in quantum mechanics

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
1 Introduction	p. 3
1.1 Basic concepts of quantum mechanics	p. 4
1.1.1 Schrodinger's equation	p. 4
1.1.2 Postulates of quantum mechanics	p. 8
1.2 Principle of stationary action	p. 10
1.2.1 The action integral	p. 11
1.2.2 Examples	p. 15
1.3 Finite elements	p. 18
1.4 Historical comments	p. 22
1.5 Problems	p. 23
References	p. 28
2 Simple quantum systems	p. 31
2.1 The simple harmonic oscillator	p. 31
2.2 The hydrogen atom	p. 39
2.3 The Rayleigh-Ritz variational method	p. 45
2.4 Programming considerations	p. 50
2.5 Problems	p. 57
References	p. 60
3 Interpolation polynomials in one dimension	p. 63
3.1 Introduction	p. 63
3.2 Lagrange interpolation polynomials	p. 64
3.3 Hermite interpolation polynomials	p. 66
3.4 Transition elements	p. 70
3.5 Low order interpolation polynomials	p. 71
3.5.1 Low order Lagrange interpolation	p. 71
3.5.2 Low order Hermite interpolation	p. 72
3.6 Interpolation polynomials in Mathematica	p. 72
3.6.1 Lagrange interpolation	p. 72
3.6.2 Hermite interpolation	p. 74
3.7 Infinite elements	p. 76
3.8 Simple quantum systems revisited	p. 77
3.9 Problems	p. 80
References	p. 81
4 Adaptive FEM	p. 83
4.1 Introduction	p. 83
4.2 Error in interpolation	p. 84
4.3 Error in the discretized action	p. 86
4.3.1 h-convergence	p. 86
4.3.2 p-convergence	p. 88
4.4 The action in adaptive calculations	p. 91
4.4.1 An ordinary differential equation	p. 92
4.4.2 The H atom again	p. 96
4.4.3 Adaptive p-refinement	p. 102
4.5 Concluding remarks	p. 104
References	p. 105
Part II Applications in 1D
5 Quantum mechanical tunneling	p. 109
5.1 Introduction	p. 109
5.2 Mixed BCs: redefining the action	p. 110
5.3 The Galerkin method	p. 113
5.4 Tunneling calculations in the FEM	p. 115
5.4.1 Evaluation of the residual	p. 115
5.4.2 Applying mixed BCs	p. 118
5.5 Comparing Galerkin FEM with WKB	p. 123
5.6 Quantum states in asymmetric wells	p. 125
5.7 Problems	p. 137
References	p. 141
6 Schrodinger-Poisson self-consistency	p. 145
6.1 Introduction	p. 145
6.2 Schrodinger and Poisson equations	p. 149
6.3 Source terms	p. 151
6.4 The Fermi energy and charge neutrality	p. 153
6.5 The Galerkin finite element approach	p. 155
6.5.1 Boundary conditions	p. 155
6.5.2 The iteration procedure	p. 159
6.5.3 Numerical issues	p. 161
6.5.4 Essential and natural boundary conditions	p. 164
6.6 Further developments	p. 165
6.7 Problems	p. 167
References	p. 169
7 Landau states in a magnetic field	p. 171
7.1 Introduction	p. 171
7.1.1 Landau levels	p. 171
7.1.2 Density of states	p. 174
7.2 Heterostructures in a B-field	p. 177
7.2.1 Faraday configuration	p. 177
7.2.2 Voigt configuration	p. 179
7.3 Comparison with experiments	p. 181
7.3.1 Interband transitions	p. 181
7.3.2 Energy dependence on the orbit center	p. 183
7.3.3 Level mixing in superlattices with small band offsets	p. 185
7.3.4 Density of states in the Voigt geometry	p. 186
7.4 Voigt geometry and a semiclassical model	p. 188
7.4.1 Landau orbit theory	p. 188
7.4.2 Envelope functions and the FEM in k-space	p. 191
7.5 Problems	p. 192
References	p. 194
8 Wavefunction engineering	p. 195
8.1 Introduction	p. 195
8.2 k P theory of band structure	p. 197
8.3 Designing mid-infrared lasers	p. 200
8.3.1 The type-II W-laser	p. 200
8.3.2 The interband cascade laser	p. 206
8.4 Concluding comments	p. 210
References	p. 211
Part III 2D Applications of the FEM
9 2D elements and shape functions	p. 217
9.1 Introduction	p. 217
9.2 Rectangular elements	p. 218
9.2.1 Lagrange elements	p. 218
9.2.2 Hermite elements	p. 221
9.3 Triangular elements	p. 222
9.4 Defining curved edges	p. 228
9.4.1 An element on a parametric curve	p. 228
9.4.2 Parametric form of 2D surfaces	p. 231
9.5 The action in 2D problems	p. 233
9.6 Gauss integration in two dimensions	p. 236
References	p. 238
10 Mesh generation	p. 241
10.1 Meshing simple regions	p. 241
10.1.1 Distortion of regular regions	p. 242
10.1.2 Using orthogonal curved coordinates	p. 247
10.2 Regions of arbitrary shape	p. 247
10.2.1 Delaunay meshing	p. 247
10.2.2 Advancing front algorithms	p. 248
10.2.3 The algebraic integer method	p. 250
References	p. 256
11 Applications in atomic physics	p. 257
11.1 The H atom in a magnetic field	p. 257
11.1.1 Schrodinger's equation and the action	p. 258
11.1.2 Applying the FEM	p. 259
11.1.3 Magnetic fields	p. 266
11.2 Ground state energy in helium	p. 269
11.3 Other results	p. 271
References	p. 273
12 Quantum wires	p. 275
12.1 Introduction	p. 275
12.2 Quantum wires and the FEM	p. 276
12.3 Symmetry properties of the square wire	p. 282
12.4 The checkerboard superlattice	p. 285
12.5 Optical nonlinearity in the CBSL	p. 289
12.6 Quantum wires of any cross-section	p. 292
References	p. 295
13 Quantum waveguides	p. 299
13.1 Quantization of resistance	p. 300
13.2 The straight waveguide	p. 303
13.3 Quantum bound states in waveguides	p. 311
13.4 The quantum interference transistor	p. 313
13.5 "Stealth" elements and absorbing BC	p. 315
13.6 The Ginzburg-Landau equation	p. 322
References	p. 323
14 Time-dependent problems	p. 327
14.1 Introduction	p. 327
14.2 Standard approaches to time evolution	p. 327
14.2.1 Schrodinger's equation and the method of finite differences	p. 327
14.2.2 The finite difference method for the wave equation	p. 330
14.3 A transfer matrix for time evolution	p. 331
14.4 Lanczos reduction of transfer matrices	p. 335
14.5 Instability with initial conditions	p. 338
14.5.1 Comparing IVBC and two-point BCs	p. 338
14.6 The variational approach	p. 347
14.6.1 A variational difficulty	p. 347
14.6.2 Variations using adjoint functions	p. 348
14.6.3 Adjoint variations for the wave equation	p. 350
14.6.4 Connection with quantum field theory	p. 352
14.7 Concluding remarks	p. 355
References	p. 356
Part IV Sparse Matrix Applications
15 Matrix solvers and related issues	p. 363
15.1 Introduction	p. 363
15.2 Bandwidth reduction	p. 363
15.3 Solution of linear equations	p. 366
15.3.1 Gauss elimination	p. 366
15.3.2 The conjugate gradient method	p. 367
15.4 The standard eigenvalue problem	p. 372
15.5 The generalized eigenvalue problem	p. 373
15.5.1 Sturm sequence check	p. 376
15.5.2 Inverse vector iteration	p. 378
15.5.3 The subspace vectors	p. 380
15.5.4 The Rayleigh quotient	p. 381
15.5.5 Subspace iteration	p. 383
15.5.6 The Davidson algorithm	p. 385
15.5.7 Least square residual minimization	p. 388
15.5.8 The Lanczos method	p. 388
References	p. 393
Part V Boundary Elements
16 The boundary element method	p. 399
16.1 Introduction	p. 399
16.2 The boundary integral	p. 400
16.3 An analytical approach	p. 402
16.3.1 A Dirichlet problem	p. 402
16.3.2 A Neumann problem	p. 406
16.4 Infinite domain Green's function	p. 408
16.5 Numerical issues	p. 414
16.5.1 Evaluation of the element integrals	p. 414
16.5.2 Applying boundary conditions	p. 415
16.5.3 Boundary condition at the corner node	p. 417
16.5.4 Setting up the matrix equation	p. 420
16.5.5 Construction of interior solution	p. 422
16.6 A worked example	p. 423
16.7 Two sum rules	p. 425
16.8 Comparing the BEM with the FEM	p. 427
16.9 Problems	p. 428
References	p. 432
17 The BEM and surface plasmons	p. 435
17.1 Introduction	p. 435
17.2 Multiregion BEM: two regions	p. 437
17.2.1 Linear interpolation	p. 437
17.2.2 Hermite interpolation	p. 439
17.3 Bulk and surface plasmons	p. 440
17.3.1 Bulk plasma oscillations	p. 441
17.3.2 Surface plasmons at a single planar interface	p. 443
17.3.3 Surface plasmons for slab geometry	p. 447
17.3.4 Surface plasmons in a cylindrical wire	p. 451
17.3.5 Two metallic wires	p. 455
17.3.6 Metal wire on a substrate	p. 458
17.3.7 Plasmons in other confining geometries	p. 460
17.4 Surface-enhanced Raman scattering	p. 461
17.5 Problems	p. 462
References	p. 468
18 The BEM and quantum applications	p. 471
18.1 Introduction	p. 471
18.2 2D electron waveguides	p. 471
18.2.1 Implementing boundary conditions	p. 477
18.2.2 Multiregion waveguide problems	p. 483
18.2.3 Multiple ports and transmission	p. 484
18.3 The BEM and 2D scattering	p. 487
18.4 Eigenvalue problems and the BEM	p. 491
18.4.1 Quantum wires	p. 491
18.4.2 Hearing the shape of a drum	p. 495
18.5 Concluding remarks on the BEM	p. 498
18.6 Problems	p. 501
References	p. 505
Part VI Appendices
A Gauss quadrature	p. 511
A.1 Introduction	p. 511
A.2 Gauss-Legendre quadrature	p. 511
A.3 Gauss-Legendre base points and weights	p. 513
A.4 An algorithm for adaptive quadrature	p. 517
A.5 Other Gauss formulas	p. 519
A.6 The Cauchy principal value of an integral	p. 520
A.7 Properties of Legendre functions	p. 522
A.8 Problems	p. 524
References	p. 525
B Generalized functions	p. 527
B.1 The Dirac [delta]-function	p. 527
B.2 The [delta]-function as the limit of a "normal" function	p. 530
B.3 [delta]-functions in three dimensions	p. 532
B.4 Other generalized functions	p. 533
B.4.1 The step-function [theta](x)	p. 533
B.4.2 The sign-function [varepsilon](x)	p. 534
B.4.3 The Plemelj formula	p. 535
B.4.4 An integral representation for [theta](z)	p. 536
B.5 Problems	p. 536
References	p. 538
C Green's functions	p. 541
C.1 Introduction	p. 541
C.2 Properties of Green's functions	p. 543
C.3 Sturm-Liouville differential operators	p. 547
C.4 Green's functions in electrostatics	p. 554
C.5 Boundary integral solutions: a comment	p. 557
C.6 Green's functions in electrodynamics	p. 565
C.7 The wave equation in one dimension	p. 570
C.8 The wave equation in two dimensions	p. 573
C.9 Green's functions and integral equations	p. 574
C.10 Problems	p. 576
References	p. 579
D Physical constants	p. 581
Author index	p. 583
Subject index	p. 597

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