## Group Theory in Solid State Physics and Photonics: Problem Solving with MathematicaWhile group theory and its application to solid state physics is well established, this textbook raises two completely new aspects. First, it provides a better understanding by focusing on problem solving and making extensive use of Mathematica tools to visualize the concepts. Second, it offers a new tool for the photonics community by transferring the concepts of group theory and its application to photonic crystals. Clearly divided into three parts, the first provides the basics of group theory. Even at this stage, the authors go beyond the widely used standard examples to show the broad field of applications. Part II is devoted to applications in condensed matter physics, i.e. the electronic structure of materials. Combining the application of the computer algebra system Mathematica with pen and paper derivations leads to a better and faster understanding. The exhaustive discussion shows that the basics of group theory can also be applied to a totally different field, as seen in Part III. Here, photonic applications are discussed in parallel to the electronic case, with the focus on photonic crystals in two and three dimensions, as well as being partially expanded to other problems in the field of photonics. The authors have developed Mathematica package GTPack which is available for download from the book's homepage. Analytic considerations, numerical calculations and visualization are carried out using the same software. While the use of the Mathematica tools are demonstrated on elementary examples, they can equally be applied to more complicated tasks resulting from the reader's own research. |

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### Contents

A | 1 |

Part One Basics of Group Theory 9 | 11 |

Basics Abstract Group Theory | 33 |

Discrete Symmetry Groups in SolidState Physics and Photonics | 51 |

B 3 | 56 |

Representation Theory | 83 |

Symmetry and Representation Theory in kSpace | 133 |

Generalization to Include the Spin | 177 |

1 | 200 |

Solution of MAXWELLs Equations | 253 |

2 | 266 |

TwoDimensional Photonic Crystals | 269 |

ThreeDimensional Photonic Crystals | 287 |

Group Theory of Vibrational Problems | 301 |

Landau Theory of Phase Transitions of the Second Kind | 319 |

Appendix B Remarks on Databases | 337 |

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### Common terms and phrases

angle applied atoms band structure basis functions BRILLOUIN zone character table classes CLEBSCH-GORDAN Co(k coefficients command constructed coordinate cosets crystal field cubic database decomposition Definition denotes density direct product representation discussed double group eigenvalue electronic energy example given GOVerbose group G group theory GTCharacterTable GTInstallgroup GTIrep GTPack Hamiltonian IC2a IC2x icº improper rotations Inſ3 Inſt2 installed invariant subgroup inversion irreducible representations KGaA lattice vectors linear magnetic master equation Mathematica matrix elements matrix representation modes molecule Mulliken nanotubes notation parameters PAULI equation permittivity photonic band photonic band structure photonic crystal plane waves point group quaternion real-space representation matrices right cosets rotation axis rotation matrix SCHRöDINGER equation Section shown in Figure SHUBNIKOV space group spherical harmonics spin spin–orbit coupling symmetry elements symmetry group symmorphic space groups tight-binding Hamiltonian tion transformation translation unit cell verified WILEY-VCH Verlag