Title Computational physics : simulation of classical and quantum systems / Philipp O.J. Scherer

Published Berlin ; Heidelberg : Springer, ©2010

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Description 1 online resource (xv, 319 pages) Series 2010 Springer E-Books Contents Note continued: 6.1.3. Newton-Raphson Method -- 6.1.4. Secant Method -- 6.1.5. Roots of Vector Functions -- 6.2. Optimization Without Constraints -- 6.2.1. Steepest Descent Method -- 6.2.2. Conjugate Gradient Method -- 6.2.3. Newton-Raphson Method -- 6.2.4. Quasi-Newton Methods -- Problems -- 7. Fourier Transformation -- 7.1. Discrete Fourier Transformation -- 7.1.1. Trigonometric Interpolation -- 7.1.2. Real-Valued Functions -- 7.1.3. Approximate Continuous Fourier Transformation -- 7.2. Algorithms -- 7.2.1. Goertzel's Algorithm -- 7.2.2. Fast Fourier Transformation -- Problems -- 8. Random Numbers and Monte Carlo Methods -- 8.1. Some Basic Statistics -- 8.1.1. Probability Density and Cumulative Probability Distribution -- 8.1.2. Expectation Values and Moments -- 8.1.3. Multivariate Distributions -- 8.1.4. Central Limit Theorem -- 8.1.5. Example: Binomial Distribution -- 8.1.6. Average of Repeated Measurements -- 8.2. Random Numbers -- 8.2.1. Method by Marsaglia and Zamann -- 8.2.2. Random Numbers with Given Distribution -- 8.2.3. Examples -- 8.3. Monte Carlo Integration -- 8.3.1. Numerical Calculation of & pi; -- 8.3.2. Calculation of an Integral -- 8.3.3. More General Random Numbers -- 8.4. Monte Carlo Method for Thermodynamic Averages -- 8.4.1. Simple (Minded) Sampling -- 8.4.2. Importance Sampling -- 8.4.3. Metropolis Algorithm -- Problems -- 9. Eigenvalue Problems -- 9.1. Direct Solution -- 9.2. Jacobi Method -- 9.3. Tridiagonal Matrices -- 9.4. Reduction to a Tridiagonal Matrix -- 9.5. Large Matrices -- Problems -- 10. Data Fitting -- 10.1. Least Square Fit -- 10.1.1. Linear Least Square Fit -- 10.1.2. Least Square Fit Using Orthogonalization -- 10.2. Singular Value Decomposition -- Problems -- 11. Equations of Motion -- 11.1. State Vector of a Physical System Note continued: 11.2. Time Evolution of the State Vector -- 11.3. Explicit Forward Euler Method -- 11.4. Implicit Backward Euler Method -- 11.5. Improved Euler Methods -- 11.6. Taylor Series Methods -- 11.7. Runge-Kutta Methods -- 11.7.1. Second-Order Runge-Kutta Method -- 11.7.2. Third-Order Runge-Kutta Method -- 11.7.3. Fourth-Order Runge-Kutta Method -- 11.8. Quality Control and Adaptive Step-Size Control -- 11.9. Extrapolation Methods -- 11.10. Multistep Methods -- 11.10.1. Explicit Multistep Methods -- 11.10.2. Implicit Multistep Methods -- 11.10.3. Predictor-Corrector Methods -- 11.11. Verlet Methods -- 11.11.1. Liouville Equation -- 11.11.2. Split Operator Approximation -- 11.11.3. Position Verlet Method -- 11.11.4. Velocity Verlet Method -- 11.11.5. Standard Verlet Method -- 11.11.6. Error Accumulation for the Standard Verlet Method -- 11.11.7. Leap Frog Method -- Problems -- pt. II Simulation of Classical and Quantum Systems -- 12. Rotational Motion -- 12.1. Transformation to a Body Fixed Coordinate System -- 12.2. Properties of the Rotation Matrix -- 12.3. Properties of W, Connection with the Vector of Angular Velocity -- 12.4. Transformation Properties of the Angular Velocity -- 12.5. Momentum and Angular Momentum -- 12.6. Equations of Motion of a Rigid Body -- 12.7. Moments of Inertia -- 12.8. Equations of Motion for a Rotor -- 12.9. Explicit Solutions -- 12.10. Loss of Orthogonality -- 12.11. Implicit Method -- 12.12. Example: Free Symmetric Rotor -- 12.13. Kinetic Energy of a Rotor -- 12.14. Parametrization by Euler Angles -- 12.15. Cayley-Klein parameters, Quaternions, Euler Parameters -- 12.16. Solving the Equations of Motion with Quaternions -- Problems -- 13. Simulation of Thermodynamic Systems -- 13.1. Force Fields for Molecular Dynamics Simulations -- 13.1.1. Intramolecular Forces Note continued: 13.1.2. Intermolecular Forces -- 13.1.3. Approximate Separation of Rotation and Vibrations -- 13.2. Simulation of a van der Waals System -- 13.2.1. Integration of the Equations of Motion -- 13.2.2. Boundary Conditions and Average Pressure -- 13.2.3. Initial Conditions and Average Temperature -- 13.2.4. Analysis of the Results -- 13.3. Monte Carlo Simulation -- 13.3.1. One-Dimensional Ising Model -- 13.3.2. Two-Dimensional Ising Model -- Problems -- 14. Random Walk and Brownian Motion -- 14.1. Random Walk in One Dimension -- 14.1.1. Random Walk with Constant Step Size -- 14.2. Freely Jointed Chain -- 14.2.1. Basic Statistic Properties -- 14.2.2. Gyration Tensor -- 14.2.3. Hookean Spring Model -- 14.3. Langevin Dynamics -- Problems -- 15. Electrostatics -- 15.1. Poisson Equation -- 15.1.1. Homogeneous Dielectric Medium -- 15.1.2. Charged Sphere -- 15.1.3. Variable & epsilon; -- 15.1.4. Discontinous & epsilon; -- 15.1.5. Solvation Energy of a Charged Sphere -- 15.1.6. Shifted Grid Method -- 15.2. Poisson Boltzmann Equation for an Electrolyte -- 15.2.1. Discretization of the Linearized Poisson-Boltzmann Equation -- 15.3. Boundary Element Method for the Poisson Equation -- 15.3.1. Integral Equations for the Potential -- 15.3.2. Calculation of the Boundary Potential -- 15.4. Boundary Element Method for the Linearized Poisson-Boltzmann Equation -- 15.5. Electrostatic Interaction Energy (Onsager Model) -- 15.5.1. Example: Point Charge in a Spherical Cavity -- Problems -- 16. Waves -- 16.1. One-Dimensional Waves -- 16.2. Discretization of the Wave Equation -- 16.3. Boundary Values -- 16.4. Wave Equation as an Eigenvalue Problem -- 16.4.1. Eigenfunction Expansion -- 16.4.2. Application to the Discrete One-Dimensional Wave Equation -- 16.5. Numerical Integration of the Wave Equation -- 16.5.1. Simple Algorithm Note continued: 16.5.2. Stability Analysis -- 16.5.3. Alternative Algorithm with Explicit Velocities -- 16.5.4. Stability Analysis -- Problems -- 17. Diffusion -- 17.1. Basic Physics of Diffusion -- 17.2. Boundary Conditions -- 17.3. Numerical Integration of the Diffusion Equation -- 17.3.1. Forward Euler or Explicit Richardson Method -- 17.3.2. Stability Analysis -- 17.3.3. Implicit Backward Euler Algorithm -- 17.3.4. Crank-Nicolson Method -- 17.3.5. Error Order Analysis -- 17.3.6. Practical Considerations -- 17.3.7. Split Operator Method for d> 1 Dimensions -- Problems -- 18. Nonlinear Systems -- 18.1. Iterated Functions -- 18.1.1. Fixed Points and Stability -- 18.1.2. Ljapunow Exponent -- 18.1.3. Logistic Map -- 18.1.4. Fixed Points of the Logistic Map -- 18.1.5. Bifurcation Diagram -- 18.2. Population Dynamics -- 18.2.1. Equilibria and Stability -- 18.2.2. Continuous Logistic Model -- 18.3. Lotka-Volterra model -- 18.3.1. Stability Analysis -- 18.4. Functional Response -- 18.4.1. Holling-Tanner Model -- 18.5. Reaction-Diffusion Systems -- 18.5.1. General Properties of Reaction-Diffusion Systems -- 18.5.2. Chemical Reactions -- 18.5.3. Diffusive Population Dynamics -- 18.5.4. Stability Analysis -- 18.5.5. Lotka-Volterra Model with Diffusion -- Problems -- 19. Simple Quantum Systems -- 19.1. Quantum Particle in a Potential Well -- 19.2. Expansion in a Finite Basis -- 19.3. Time-Independent Problems -- 19.3.1. Simple Two-Level System -- 19.3.2. Three-State Model (Superexchange) -- 19.3.3. Ladder Model for Exponential Decay -- 19.4. Time-Dependent Models -- 19.4.1. Landau-Zener Model -- 19.4.2. Two-State System with Time-Dependent Perturbation -- 19.5. Description of a Two-State System with the Density Matrix Formalism -- 19.5.1. Density Matrix Formalism Note continued: 19.5.2. Analogy to Nuclear MagneticResonance -- 19.5.3. Relaxation Processes -- Bloch Equations -- Problems Summary This book encapsulates the coverage for a two-semester course in computational physics. The first part introduces the basic numerical methods while omitting mathematical proofs but demonstrating the algorithms by way of numerous computer experiments. The second part specializes in simulation of classical and quantum systems with instructive examples spanning many fields in physics, from a classical rotor to a quantum bit. All program examples are realized as Java applets ready to run in your browser and do not require any programming skills Bibliography Includes bibliographical references and index Notes Print version record Subject Mathematical physics. Physics -- Data processing Physique. Mathematical physics Physics -- Data processing Form Electronic book LC no. 2010937781 ISBN 9783642139901 3642139906 3642139892 9783642139895

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