Preface Acknowledgements 1 Synopsis Part one Fundamental principles 2 The mathernatieal structure of quantum mechanics 2.1 The Copenhagen quantum postulate 2.2 The superstructure of quantum mechanics 2.3 Degree of freedom space F 2.4 State space V(F) 2.4.1 Hilbert space 2.5 Operators O(F) 2.6 The process of measurement 2.7 The Schrodinger differential equation 2.8 Heisenberg operator approach 2.9 Dirac-Feynman path integral formulation 2.10 Three formulations of quantum mechanics 2.11 Quantum entity 2.12 Summary 3 Operators 3.1 Continuous degree of freedom 3.2 Basis states for state space 3.3 Hermitian operators 3.3.1 Eigenfunctions; completeness 3.3.2 Hamiltonian for a periodic degree of freedom 3.4 Position and momentum operators h and/ 3.4.1 Momentum operator 3.5 Weyl operators 3.6 Quantum numbers; commuting operator 3.7 Heisenberg commutation equation 3.8 Unitary representation of Heisenberg algebra 3.9 Density matrix: pure and mixed states 3.10 Self-adjoint operators 3.10.1 Momentum operator on finite interval 3.11 Self-adjoint domain 3.11.1 Real eigenvalues 3.12 Hamiltonian's self-adjoint extension 3.12.1 Delta function potential 3.13 Fermi pseudo-potential 3.14 Summary 4 The Feynman path integral 4.1 Probability amplitude and time evolution 4.2 Evolution kernel 4.3 Superposition: indeterminate paths 4.4 The Dirac-Feynman formula 4.5 The Lagrangian 4.5.1 Infinite divisibility of quantum paths 4.6 The Feynman path integral 4.7 Path integral for evolution kernel 4.8 Composition rule for probability amplitudes 4.9 Summary
5 Hamiltonian mechanics 5.1 Canonical equations 5.2 Symmetries and conservation laws 5.3 Euclidean Lagrangian and Hamiltonian 5.4 Phase space path integrals 5.5 Poisson bracket 5.6 Commutation equations 5.7 Dirac bracket and constrained quantization 5.7.1 Dirac bracket for two constraints 5.8 Free particle evolution kernel 5.9 Hamiltonian and path integral 5.10 Coherent states 5.11 Coherent state vector 5.12 Completeness equation: over-complete 5.13 Operators; normal ordering …… 6 Path integral quantization Part two Stochastic processes 7 Stochastic systems Part three Discrete degrees of freedom 8 Ising model 9 Ising model: magnetic field 10 Fermions Part four Quadratic path integrals 11 Simple harmonic oscillator 12 Gaussian path integrals Part five Action with acceleration 13 Acceleration Lagrangian 14 Pseudo-Hermitian Euclidean Hamiltonian 15 Non-Hermitian Hamiltonian: Jordan blocks Part six Nonlinear path integrals 16 The quartic potential: instantons 17 Compact degrees of freedom 18 Conclusions References Index
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