## The Theory of Groups and Quantum MechanicsThis landmark among mathematics texts applies group theory to quantum mechanics, first covering unitary geometry, quantum theory, groups and their representations, then applications themselves — rotation, Lorentz, permutation groups, symmetric permutation groups, and the algebra of symmetric transformations. |

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### Contents

UNITARY GEOMETRY i | 1 |

Linear Correspondences Matrix Calculus | 5 |

The Dual Vector Space | 12 |

Unitary Geometry and Hermitian Forms | 15 |

Transformation to Principal Axes | 21 |

Infinitesimal Unitary Transformations | 27 |

Remarks on odimensional Space | 31 |

It QUANTUM THEORY | 41 |

Rotation and Lorentz Groups | 140 |

Character of a Representation | 150 |

Schurs Lemma and Bumsides Theorem | 152 |

Orthogonality Properties of Group Characters | 157 |

APPLICATION or THB THEORY OF GROUPS TO QUANTUM MECHANICS | 185 |

B The Lorentz Group | 210 |

Energy and Momentum Remarks on the Interchange of Past | 218 |

Electron in Spherically Symmetric Field | 227 |

The de Brogiie Waves of a Particle | 48 |

Schrodingers Wave Equation The Harmonic Oscillator | 54 |

Spherical Harmonics | 60 |

Electron in Spherically Symmetric Field Directional Quan tization | 63 |

Collision Phenomena | 70 |

The Conceptual Structure of Quantum Mechanics | 74 |

The Dynamical Law Transition Probabilities | 80 |

Perturbation Theory | 86 |

The Problem of Several Bodies Product Space | 89 |

Commutation Rules Canonical Transformations | 93 |

Motion of a Particle in an Electromagnetic Field Zeeman Effect and Stark Effect | 98 |

Atom in Interaction with Radiation | 102 |

GROUPS AND THEIR REPRESENTATIONS I IO 1 Transformation Groups | 110 |

Abstract Groups and their Realization | 113 |

Subgroups and Conjugate Classes | 116 |

Representation of Groups by Linear Transformations | 120 |

Formal Processes ClebschGordan Series | 123 |

The JordanHolder Theorem and its Analogues | 131 |

Unitary Representations | 136 |

The Permutation Group | 238 |

The Problem of Several Bodies and the Quantization | 246 |

Quantization of the MaxwellDirac Field Equations | 253 |

The Energy and Momentum Laws of Quantum Physics | 264 |

Quantum Kinematics | 272 |

THE SYMMETRIC PERMUTATION GROUP AND THE ALGKBRA OF SYM | 281 |

Invariant Subspaces in Group Space | 291 |

Fields and Algebras | 302 |

Constructive Reduction of an Algebra into Simple Matric | 309 |

The Characters of the Symmetric Group and Equivalence | 319 |

Relation between the Characters of the Symmetric Per | 326 |

Direct Product Subgroups | 332 |

Perturbation Theory for the Construction of Molecules | 339 |

The Symmetry Problem of Quantum Theory | 347 |

AriNDix | 393 |

399 | |

409 | |

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### Common terms and phrases

abstract group accordance algebra anti-symmetric applied associated assume atom azimuthal quantum belongs character characteristic numbers classical coefficients column commutation rules complete orthogonal system completely reduced components conjugate consequently consider consists constitute contained defined definite denote determined differential equations dimensionality electron energy levels equivalent expressed fact factor follows formula frequency function fundamental given group g group manifold Hence Hermitian form homogeneous idempotent idempotent elements infinitesimal integral introducing invariant sub-space irreducible representations linear correspondence linear transformation linearly independent mathematical matrix momentum multiplication n-dimensional non-vanishing normal co-ordinate system obtained operator orthogonal particle permutation permutation group perturbation photon point-field polynomial problem quantum mechanics quantum theory regular representation replacing repre respect rotation group rows s-axis satisfies the equation scalar sentation solution spin sub-group system space tensors of order theorem tion transition unit unitary representation unitary space unitary transformation vanish variables vector space wave