Title:
Finite element and boundary element applications in quantum mechanics
Personal Author:
Series:
Oxford text in applied and engineering mathematics ; 5
Publication Information:
Oxford : Oxford University Press, 2002
ISBN:
9780198525219
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
---|---|---|---|---|---|
Searching... | 30000010133462 | QC174.17.F54 R35 2002 | Open Access Book | Book | Searching... |
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Summary
Summary
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.
Table of Contents
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 |