Computational Physics: Problem Solving with Python

Citation

Landau, R. H., Páez, M. J., & Bordeianu, C. C. (2024). Computational Physics: Problem Solving with Python (4th ed.). Wiley-VCH.

Chapter Summary

The fourth edition has 27 chapters in four parts, about 540 pages of main text plus an appendix of codes and animations, references and an index. Chapter 1 explains the philosophy: computational physics is treated as a bridge between physics, applied mathematics and computer science, taught through a hands-on, inquiry-based cycle of a problem, a theory or model, an algorithm and an assessment of the results. Almost every chapter opens with a keynote "Problem" that leads into those steps. Python is the language throughout; earlier editions used Java, Fortran and C. The publisher says the first third of the book lays the fundamentals of scientific computing and the latter two-thirds cover more advanced topics, with new chapters in this edition on quantum computing, machine learning and general relativity. The entries below group chapters by the book's four parts and follow its contents list, including section headings.

Chapter 1: Introduction

Positions computational physics within science, describes the book's subjects, the roughly 60 video lecture modules (now mp4 videos and PDF slides) and the supplied codes, and surveys the Python ecosystem (Jupyter, NumPy, Matplotlib, Pandas, SymPy, VPython, with the Anaconda distribution recommended). The authors suggest roughly the first third of the book for a one-quarter course and the rest for a two-quarter course, and say the material could fill a year.

Chapters 2–3: Software Basics; Errors and Uncertainties

Chapter 2 covers computer number representations (IEEE floating-point numbers), a Python mini tutorial, a programming warm-up on over- and underflow and machine precision, and plotting with VPython and Matplotlib (2D, 3D surfaces and animations). Chapter 3 treats types of errors, subtractive cancellation, round-off accumulation, errors in power series and in Bessel functions computed by recursion.

Chapters 4–5: Monte Carlo Simulations; Differentiation and Integration

Monte Carlo begins with random numbers and goes to random walks (including in a brain and in protein folding), spontaneous decay and testing random distributions. Chapter 5 covers forward, central and extrapolated differences, the trapezoid and Simpson rules, Gaussian quadrature, Monte Carlo integration, variance reduction, importance sampling and von Neumann rejection.

Chapters 6–8: Trial-and-Error Searching and Data Fitting; Matrix Computing and N–D Searching; Differential Equations and Nonlinear Oscillations

Chapter 6 uses quantum bound states and a magnetization problem to introduce bisection and Newton–Raphson searching, then data fitting by Lagrange interpolation, cubic splines and least squares, ending with a nonlinear fit to a resonance. Chapter 7 treats matrices in Python and NumPy through masses on a string and spin states in hyperfine structure, with a section on vectorization. Chapter 8 reviews ordinary differential equations and the Euler, Runge–Kutta and predictor-corrector algorithms, then applies them to nonlinear oscillators, resonances, beats and friction.

Chapters 9–10: Fourier Analyses; Wavelet and Principal Components Analysis (Part II: Data Science)

Fourier series and transforms, discrete transforms and aliasing, noise filtering and the fast Fourier transform; then wave packets and the uncertainty principle, short-time Fourier transforms and wavelet transforms, and principal components analysis.

Chapters 11–12: Neural Networks and Machine Learning; Quantum Computing

Chapter 11 builds a simple neural network from a single neuron, then covers TensorFlow, scikit-learn, clustering, Keras and image processing with OpenCV. Chapter 12, written with G. He, introduces Dirac notation, qubits, entanglement, logic gates, programming with IBM Quantum and Qiskit, the quantum Fourier transform, Grover's search algorithm and Shor's factoring.

Chapters 13–16: ODE Applications; Fractals and Statistical Growth Models; Nonlinear Population Dynamics; Nonlinear Dynamics of Continuous Systems (Part III: Applications)

Eigenvalues for arbitrary potentials, classical chaotic scattering, projectile motion with drag and planetary orbits; fractals from the Sierpiński gasket to diffusion-limited aggregation; the logistic map, bifurcation diagrams and predator–prey models; and the chaotic pendulum, phase space and other chaotic systems such as the Lorenz attractor.

Chapters 17–20: Thermodynamics Simulations and Feynman Path Integrals; Molecular Dynamics Simulations; General Relativity; Integral Equations

The Ising chain, the Metropolis algorithm and path-integral quantum mechanics; molecular dynamics with Verlet algorithms; Einstein's field equations, light deflection, perihelion precession and wormholes; and nonlocal potentials and momentum-space scattering.

Chapters 21–24: PDE Review, Electrostatics and Relaxation; Heat Flow and Leapfrogging; String and Membrane Waves; Quantum Wave Packets and EM Waves (Part IV: PDE Applications)

Laplace's equation with finite differences and relaxation; the heat equation with leapfrog and Crank–Nicolson algorithms; string and membrane waves; and the time-dependent Schrödinger equation and finite-difference time-domain Maxwell solvers.

Chapters 25–27: Shock and Soliton Waves; Fluid Hydrodynamics; Finite Element Electrostatics

Burgers' equation, KdV solitons and the sine-Gordon equation; the Navier–Stokes equation in its vorticity form; and the finite element method for a two-plate electrostatic problem.

Key Concepts

Part I: Basics (Chapters 1–8)

• Floating-point arithmetic and error: IEEE numbers, round-off and subtractive cancellation.
• Monte Carlo methods: random walks, decay simulations and Monte Carlo integration.
• Searching and fitting: bisection, Newton–Raphson, splines and least squares.
• ODE algorithms: Euler, Runge–Kutta and predictor-corrector rules.

Part II: Data Science (Chapters 9–12)

• Fourier and wavelet analysis: discrete transforms, the FFT, wavelets and principal components.
• Neural networks and machine learning: TensorFlow, Keras and scikit-learn examples.
• Quantum computing: qubits, gates, Grover's and Shor's algorithms.

Parts III–IV: Applications and PDEs (Chapters 13–27)

• Nonlinear dynamics and chaos: the logistic map, chaotic pendulum and Lorenz attractor.
• Finite-difference and finite-element methods: relaxation, leapfrog, Crank–Nicolson and the finite element method.
• Wave phenomena: string waves, wave packets, shocks and solitons.

Critical Analysis

Strengths

• Learning by doing. Chapter 1 sets out an explicit cycle of problem, model, algorithm and assessment, and almost every chapter begins with a keynote problem. The authors provide programs for most problems and suggest that students write short lab reports covering the equations solved, the method, the code, a visualization and a critique.
• Range of applications. The contents run from errors and quadrature to path integrals, molecular dynamics, general relativity, fluids and finite elements, which gives a course many project choices.
• Companion material. About 60 video lecture modules with slides, the codes, and notebook versions of earlier editions accompany the book, according to Chapter 1 and the publisher.
• Current topics. The fourth edition adds chapters on machine learning, quantum computing and general relativity, which the publisher presents as an increased emphasis on data science.

Limitations

• A survey, not a foundations text. The authors say the book is not a scholarly exposition of the foundations of computational physics but a survey, pitched at the undergraduate level "except maybe for the latter parts of some chapters".
• Python is introduced briefly. Chapter 2 is the only dedicated programming chapter, and the authors point readers to external tutorials and books for learning Python, so beginners may need a separate Python resource.
• Some tooling is dated within the text. The book itself notes that the Visual (VPython) module is no longer supported and now survives only within Jupyter or Web VPython, which matters for the visualization sections of Chapter 2.
• Breadth limits depth. With 27 chapters in about 540 pages, many chapters (such as Chapters 18 and 19) run to fewer than 20 pages, so each application is an introduction rather than a full treatment.

Where It Fits in the Course

In the computational physics list, Landau, Páez and Bordeianu is the broad, application-rich textbook that teaches numerical methods through worked physics problems. Newman's Computational Physics is the nearest alternative, also built on worked examples and exercises in Python; Newman's contents, as recorded in the Princeton University Library catalogue, cover a similar core (accuracy, integrals and derivatives, equations, Fourier transforms, ODEs, PDEs, Monte Carlo) in fewer chapters, whereas Landau's adds the data-science chapters and a long list of application chapters. Kinder and Nelson's A Student's Guide to Python for Physical Modeling is a short practical guide for a student who has no programming experience and can be read before Chapter 2.

Press and colleagues' Numerical Recipes is a reference to numerical methods with complete routines in C++, which the authors themselves cite as the standard general text on numerical methods; Thijssen's Computational Physics is a graduate-level account of research methods in condensed matter physics and chemistry; and Taylor's An Introduction to Error Analysis treats the uncertainty of experimental data and complements Chapter 3. Landau's book sits between an introduction to the tools and the research-level methods in Thijssen.

How to Use This Book

The book is for physics, engineering and materials-science students who have completed introductory physics and calculus and who know some differential equations and linear algebra. Programming experience helps but is not required; the authors expect readers to be able to run and modify the sample codes.

A practical plan:

• Core (Chapters 1–8): Python, errors, Monte Carlo, differentiation and integration, searching and fitting, matrices and ODEs. Code every keynote problem.
• Data science (Chapters 9–12): Chapter 9 on Fourier analysis is the most broadly useful; Chapters 10–12 can be chosen by interest.
• Applications (Chapters 13–27): pick by project. Chapters 13 and 16 build directly on the ODE solvers of Chapter 8, and Chapters 22–25 follow the PDE review in Chapter 21.
• Alongside: run the supplied codes, then modify them; write a short lab report for each problem, and use the video lectures for preview or review.