Symmetry And Quantum Mechanics (chapman & Hall/crc Monographs And Research Notes In Mathematics)

Structured as a dialogue between a mathematician and a physicist, **Symmetry and Quantum Mechanics** unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry. Aimed at advanced undergraduate and beginning graduate students in mathematics with only a minimal background in physics, this title is also useful to physicists seeking a mathematical introduction to the subject. Part I focuses on spin, and covers such topics as Lie groups and algebras, while part II offers an account of position and momentum in the context of the representation theory of the Heisenberg group, along the way providing an informal discussion of fundamental concepts from analysis such as self-adjoint operators on Hilbert space and the Stone-von Neumann Theorem. Mathematical theory is applied to physical examples such as spin-precession in a magnetic field, the harmonic oscillator, the infinite spherical well, and the hydrogen atom. Content: Physical Space. Modeling space Real linear operators and matrix groups SO(3) is the group of rotations Spinor Space Angular momentum in classical mechanics Modeling spin Complex linear operators and matrix groups The geometry of SU(2). The tangent space to the circle U(1) = S1 The tangent space to the sphere SU(2) = S3 The exponential of a matrix. SU(2) is the universal cover of SO(3) Back to spinor space Observables and Uncertainty Spin observables The Lie algebra su(2) Commutation relations and uncertainty Some related Lie algebras Warm-up: the Lie algebra u(1) The Lie algebra sl2(C) The Lie algebra u(2) The Lie algebra gl2(C) Dynamics Time-independent external fields Time-dependent external fields The energy-time uncertainty principle Conserved quantities. Higher Spin. Group representations. Representations of SU(2). Lie algebra representations. Representations of su(2)C = sl2(C). Spin-s particles. Representations of SO(3). The so(3)-action Comments about analysis. Multiple Particles. Tensor products of representations. The Clebsch-Gordan problem. Identical particles-spin only. A One-dimensional World. Position. Momentum The Heisenberg Lie algebra and Lie group The meaning of the Heisenberg group action Time-evolution The free particle The infinite square well The simple harmonic oscillator A Three-dimensional World Position Linear momentum The Heisenberg group H3 and its algebra h3 Angular momentum The Lie group G = H3 o SO(3) and its Lie algebra g Time-evolution The free particle The three-dimensional harmonic oscillator Central potentials The infinite spherical well Two-particle systems The Coulomb potential Particles with spin The hydrogen atom Identical particles Towards a Relativistic Theory Galilean relativity Special relativity SL2(C) is the universal cover of SO+(1, 3) The Dirac equation Appendices Linear algebra Vector spaces and linear transformations Inner product spaces and adjoints Multivariable calculus Analysis Hilbert spaces and adjoints Some big theorems Solutions to selected exercises Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Contents -- Author Biography -- Preface -- Plan of the Book -- List of Figures -- Part I: Spin -- 1 Physical Space -- 1.1 Modeling space -- 1.2 Real linear operators and matrix groups -- 1.3 SO(3) is the group of rotations -- 2 Spinor Space -- 2.1 Angular momentum in classical mechanics -- 2.2 Modeling spin -- 2.3 Complex linear operators and matrix groups -- 2.4 The geometry of SU(2) -- 2.4.1 The tangent space to the circle U(1) = S(sup[1]) -- 2.4.2 The tangent space to the sphere SU(2) =S(sup[3])