<p>Preface</p>
<p>I An invitation to quantum mechanics<br>1 Motivation<br>1.1 Classical mechanics<br>1.2 Relativity theory<br>1.3 Statistical mechanics and thermodynamics<br>1.4 Hamiltonian mechanics<br>1.5 Quantum mechanics<br>1.6 Quantum field theory<br>1.7 The Schrödinger picture<br>1.8 The Heisenberg picture<br>1.9 Outline of the book<br>2 The simplest quantum system<br>2.1 Matrices, relativity and quantum theory<br>2.2 Continuous motions and matrix groups<br>2.3 Infinitesimal motions and matrix Lie algebras<br>2.4 Uniform motions and the matrix exponential<br>2.5 Volume preservation and special linear groups<br>2.6 The vector product, quaternions, and SL(2,C)<br>2.7 The Hamiltonian form of a Lie algebra <br>2.8 Atomic energy levels and unitary groups <br>2.9 Qubits and Bloch sphere <br>2.10 Polarized light and beam transformations <br>2.11 Spin and spin coherent states <br>2.12 Particles and detection probabilities <br>2.13 Photons on demand <br>2.14 Unitary representations of SU(2) <br>3 The symmetries of the universe <br>3.1 Rotations and SO(n) <br>3.2 3-dimensional rotations and SO(3) <br>3.3 Rotations and quaternions <br>3.4 Rotations and SU(2) <br>3.5 Angular velocity <br>3.6 Rigid motions and Euclidean groups <br>3.7 Connected subgroups of SL(2,R) <br>3.8 Connected subgroups of SL(3,R) <br>3.9 Classical mechanics and Heisenberg groups <br>3.10 Angular momentum, isospin, quarks <br>3.11 Connected subgroups of SL(4,R) <br>3.12 The Galilean group <br>3.13 The Lorentz groups O(1, 3), SO(1, 3), SO(1, 3)0 <br>3.14 The Poincare group ISO(1, 3) <br>3.15 A Lorentz invariant measure <br>3.16 Kepler's laws, the hydrogen atom, and SO(4) <br>3.17 The periodic systemand the conformal group SO(2, 4) <br>3.18 The interacting boson model and U(6) <br>3.19 Casimirs <br>3.20 Unitary representations of the Poincaré group <br>3.21 Some representations of the Poincaré group <br>3.22 Elementary particles <br>3.23 The position operator <br>4 From the theoretical physics FAQ <br>4.1 To be done <br>4.2 Postulates for the formal core of quantum mechanics <br>4.3 Lie groups and Lie algebras <br>4.4 The Galilei group as contraction of the Poincare group <br>4.5 Representations of the Poincare group <br>4.6 Forms of relativistic dynamics <br>4.7 Is there a multiparticle relativistic quantum mechanics? <br>4.8 What is a photon? <br>4.9 Particle positions and the position operator <br>4.10 Localization and position operators <br>4.11 SO(3) = SU(2)/Z2 <br>5 Classical oscillating systems <br>5.1 Systems of damped oscillators <br>5.2 The classical anharmonic oscillator <br>5.3 Harmonic oscillators and linear field equations <br>5.4 Alpha rays <br>5.5 Beta rays <br>5.6 Light rays and gamma rays <br>6 Spectral analysis <br>6.1 The quantum spectrum <br>6.2 Probing the spectrum of a system <br>6.3 The early history of quantum mechanics <br>6.4 The spectrum of many-particle systems <br>6.5 Black body radiation<br>6.6 Derivation of Planckés law<br>6.7 Stefan´s law and Wien´s displacement law</p>
<p>II Statistical mechanics<br>7 Phenomenological thermodynamics <br>7.1 Standard thermodynamical systems <br>7.2 The laws of thermodynamics <br>7.3 Consequences of the first law <br>7.4 Consequences of the second law <br>7.5 The approach to equilibrium <br>7.6 Description levels <br>8 Quantities, states, and statistics <br>8.1 Quantities <br>8.2 Gibbs states <br>8.3 Kubo product and generating functional <br>8.4 Limit resolution and uncertainty <br>9 The laws of thermodynamics <br>9.1 The zeroth law: Thermal states <br>9.2 The equation of state <br>9.3 The first law: Energy balance <br>9.4 The second law: Extremal principles <br>9.5 The third law: Quantization <br>10 Models, statistics, and measurements <br>10.1 Description levels <br>10.2 Local, microlocal, and quantum equilibrium <br>10.3 Statistics and probability <br>10.4 Classical measurements <br>10.5 Quantum probability <br>10.6 Entropy and information theory <br>10.7 Subjective probability</p>
<p>III Lie algebras and Poisson algebras<br>11 Lie algebras<br>11.1 Basic definitions <br>11.2 Lie algebras from derivations <br>11.3 Linear groups and their Lie algebras <br>11.4 Classical Lie groups and their Lie algebras <br>11.5 Heisenberg algebras and Heisenberg groups <br>11.6 Lie-algebras <br>12 Mechanics in Poisson algebras <br>12.1 Poisson algebras <br>12.2 Rotating rigid bodies <br>12.3 Rotations and angular momentum <br>12.4 Classical rigid body dynamics <br>12.5 Lie-Poisson algebras <br>12.6 Classical symplectic mechanics <br>12.7 Molecular mechanics <br>12.8 An outlook to quantum field theory <br>13 Representation and classification <br>13.1 Poisson representations <br>13.2 Linear representations <br>13.3 Finite-dimensional representations <br>13.4 Representations of Lie groups <br>13.5 Finite-dimensional semisimple Lie algebras <br>13.6 Automorphisms and coadjoint orbits </p>
<p>IV Nonequilibrium thermodynamics <br>14 Markov Processes <br>14.1 Activities <br>14.2 Processes <br>14.3 Forward morphisms and quantum dynamical semigroups <br>14.4 Forward derivations <br>14.5 Single-time, autonomous Markov processes <br>15 Diffusion processes <br>15.1 Stochastic differential equations <br>15.2 Closed diffusion processes <br>15.3 Ornstein-Uhlenbeck processes <br>15.4 Linear processes with memory <br>15.5 Dissipative Hamiltonian Systems <br>16 Collective Processes <br>16.1 The master equation <br>16.2 Canonical form and thermodynamic limit <br>16.3 Stirred chemical reactions <br>16.4 Linear response theory <br>16.5 Open system <br>16.6 Some philosophical afterthoughts <br>V Mechanics and differential geometry <br>17 Fields, forms, and derivatives <br>17.1 Scalar fields and vector fields <br>17.2 Multilinear forms <br>17.3 Exterior calculus <br>17.4 Manifolds as differential geometries <br>17.5 Manifolds as topological spaces <br>17.6 Noncommutative geometry <br>17.7 Lie groups as manifolds <br>18 Conservative mechanics on manifolds <br>18.1 Poisson algebras from closed 2-forms <br>18.2 Conservative Hamiltonian dynamics <br>18.3 Constrained Hamiltonian dynamics <br>18.4 Lagrangian mechanics <br>19 Hamiltonian quantum mechanics<br>19.1 Quantum dynamics as symplectic motion <br>19.2 Quantum-classical dynamics <br>19.3 Deformation quantization<br>19.4 The Wigner transform</p>
<p>VI Representations and spectroscopy<br>20 Harmonic oscillators and coherent states<br>20.1 The classical harmonic oscillator<br>20.2 Quantizing the harmonic oscillator<br>20.3 Representations of the Heisenberg algebra <br>20.4 Bras and Kets<br>20.5 Boson Fock space<br>20.6 Bargmann.Fock representation<br>20.7 Coherent states for the harmonic oscillator <br>20.8 Monochromatic beams and coherent states <br>21 Spin and fermions <br>21.1 Fermion Fock space <br>21.2 Extension to many degrees of freedom <br>21.3 Exterior algebra representation <br>21.4 Spin and metaplectic representation <br>22 Highest weight representations <br>22.1 Triangular decompositions <br>22.2 Triangulated Lie algebras of rank and degree one <br>22.3 Unitary representations of SU(2) and SO(3) <br>22.4 Some unitary highest weight representations <br>23 Spectroscopy and spectra <br>23.1 Introduction and historical background <br>23.2 Spectra of systems of particles <br>23.3 Examples of spectra<br>23.4 Dynamical symmetries<br>23.5 The hydrogen atom<br>23.6 Chains of subalgebras</p>
<p>References</p>