| Preface Outline of Boxes Historical Comments with Portraits Sources for Portraits of Physicists Permissions for Use of Figures 1 Introduction to Quantum Trajectories 1.1 Dynamics with Quantum Trajectories 1.2 Routes to Quantum Trajectories 1.3 The Quantum Trajectory Method 1.4 Derivative Evaluation on Unstructured Grids 1.5 Applications of the Quantum Trajectory Method 1.6 Beyond Bohm Trajectories: Adaptive Methods 1.7 Approximations to the Quantum Force 1.8 Propagation of Derivatives Along Quantum Trajectories... 1.9 Trajectories in Phase Space 1.10 Mixed Quantum-Classical Dynamics 1.11 Additional Topics in Quantum Hydrodynamics 1.12 Quantum Trajectories for Stationary States 1.13 Coping with Problems 1.14 Topics Not Covered 1.15 Reading Guide 2 The Bohmian Route to the Hydrodynamic Equations 2.1 Introduction 2.2 The Madelung-Bohm Derivation of the Hydrodynamic Equations 2.3 The Classical Hamilton-Jacobi Equation 2.4 The Field Equations of Classical Dynamics 2.5 The Quantum Potential 2.6 The Quantum Hamilton-Jacobi Equation 2.7 Pilot 'Naves, Hidden Variables, and Bohr 3 The Phase Space Route to the Hydrodynamic Equations 3.1 Introduction 3.2 Classical Trajectories and Distribution Function in Phase Space 3.3 The Wigner Function 3.4 Moments of the Wigner Function 3.5 Equations of Motion for the Moments 3.6 Moment Analysis for Classical Phase Space Distributions 3.7 Time Evolution of Classical and Quantum Moments 3.8 Comparison Between Liouville and Hydrodynamic Phase Spaces 3.9 Discussion 4 The Dynamics and Properties of Quantum Trajectories 4.1 Introduction 4.2 Equations of Motion for the Quantum Trajectories 4.3 Wave Function Synthesis Along a Quantum Trajectory 4.4 Bohm Trajectory Integral Versus Feynman Path Integral 4.5 Wave Function Propagation and the Jacobian 4.6 The Initial Value Representation for Quantum Trajectories... 4.7 The Trajectory Noncrossing Rules 4.8 Dynamics of Quantum Trajectories Near Wave Function Nodes 4.9 Chaotic Quantum Trajectories 4.10 Examples of Chaotic Quantum Trajectories 4.11 Chaos and the Role of Nodes in the Wave Function 4.12 Why Weren't Quantum Trajectories Computed 50 Years Ago? 5 Function and Derivative Approximation on Unstructured Grids 5.1 Introduction 5.2 Least Squares Fitting Algorithms 5.3 Dynamic Least Squares 5.4 Fitting with Distributed Approximating Functionals 5.5 Derivative Computation via Tessellation and Fitting 5.6 Finite Element Method for Derivative Computation 5.7 Summary 6 Applications of the Quantum Trajectory Method Corey J. Trahan 6.1 Introduction 6.2 The Free Wave Packet 6.3 The Anisotropic Harmonic Oscillator 6.4 The Downhill Ramp Potential 6,5 Scattering from the Eckart Barrier 6.6 Discussion 7 Adaptive methods for trajectory dynamics 8 Quantum trajectories for multidimensional dynamics 9 Approximations to the quantum force 10 Derivative propagation along quantum trajectories 11 Quantum trajectories in phase space 12 Mixed quantum-classical dynamics 13 Topics in quantum hydrodynamics : the stress tensor and vorticity 14 Quantum trajectories for stationary states 15 Challenges and opportunities Appendix. 1 Atomic units Appendix. 2 Example QTM program |
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