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Computational Quantum Mechanics by Joshua Izaac and Jingbo Wang

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' Computational Quantum Mechanics written by Joshua Izaac , Department of Physics, The University of Western Australia, Perth, Australia and Jingbo Wang , Department of Physics, The University of Western Australia, Perth, Australia. Quantum mechanics undergraduate courses mostly focus on systems with known analytical solutions; the finite well, simple Harmonic, and spherical potentials. However, most problems in quantum mechanics cannot be solved analytically.
This textbook introduces the numerical techniques required to tackle problems in quantum mechanics, providing numerous examples en route. No programming knowledge is required an introduction to both Fortran and Python is included, with code examples throughout. With a hands-on approach, numerical techniques covered in this book include differentiation and integration, ordinary and differential equations, linear algebra, and the Fourier transform. By completion of this book, the reader will be armed to solve the Schro¨dinger equation for arbitrarily complex potentials, and for single and multi-electron systems.

Computational Quantum Mechanics written by Joshua Izaac and Jingbo Wang cover the following topics.

• I Scientific Programming: an Introduction for Physicists

• 1. Numbers and Precision
1.1 Fixed-Point Representation
1.2 Floating-Point Representation
1.3 Floating-Point Arithmetic
Exercises

• 2. Fortran
2.1 Variable Declaration
2.2 Operators and Control Statements
2.3 String Manipulation
2.4 Input and Output
2.5 Intrinsic Functions
2.6 Arrays
2.7 Procedures: Functions and Subroutines
2.8 Modules
2.9 Command-Line Arguments
Exercises

• 3. Python
3.1 Preliminaries: Tabs, Spaces, Indents, and Cases
3.2 Variables, Numbers, and Precision
3.3 Operators and Conditionals
3.4 String Manipulation
3.5 Data Structures
3.6 Loops and Flow Control
3.7 Input and Output
3.8 Functions
3.9 NumPy and Arrays
3.10 Command-Line Arguments
Exercises

• II Numerical Methods for Quantum Physics

• 4. Finding Roots
4.1 Big-O Notation
4.2 Convergence
4.3 Bisection Method
4.4 Newton-Raphson Method
4.5 Secant Method
4.6 False Position Method
Exercises 178

• 5. Differentiation and Initial Value Problems
5.1 Method of Finite Differences
5.2 The Euler Method(s)
5.3 Numerical Error
5.4 Stability
5.5 The Leap-Frog Method
5.6 Round-Off Error
5.7 Explicit Runge–Kutta Methods
5.8 Implicit Runge–Kutta Methods
5.9 RK4: The Fourth-Order Runge Kutta Method
Exercises

• 6. Numerical Integration
6.1 Trapezoidal Approximation
6.2 Midpoint Rule
6.3 Simpson’s Rule
6.4 Newton–Cotes Rules
6.7 Monte Carlo Methods
Exercises

• 7. The Eigenvalue Problem
7.1 Eigenvalues and Eigenvectors
7.2 Power Iteration
7.3 Krylov Subspace Techniques
7.4 Stability and the Condition Number
7.5 Fortran: Using LAPACK
7.6 Python: Using SciPy
Exercises

• 8. The Fourier Transform
8.1 Approximating the Fourier Transform
8.2 Fourier Differentiation
8.3 The Discrete Fourier Transform
8.4 Errors: Aliasing and Leaking
8.5 The Fast Fourier Transform
8.6 Fortran: Using FFTW
8.7 Python: Using SciPy
Exercises 354

• III Solving the Schrödinger Equation

• 9. One Dimension
9.1 The Schr¨odinger Equation
9.2 The Time-Independent Schr¨odinger Equation
9.3 Boundary Value Problems . 361
9.4 Shooting method for the Schr¨odinger Equation
9.5 The Numerov–Cooley Shooting Method
9.6 The Direct Matrix Method
Exercises

• 10. Higher Dimensions and Basic Techniques
10.1 The Two-Dimensional Direct Matrix Method
10.2 Basis Diagonalization
10.3 The Variational Principle
Exercises

• 11. Time Propagation
11.1 The Unitary Propagation Operator
11.2 Euler Methods
11.3 Split Operators
11.4 Direct Time Discretisation
11.5 The Chebyshev Expansion
11.6 The Nyquist Condition
Exercises
12 Central Potentials
12.1 Spherical Coordinates
12.3 The Coulomb Potential
12.4 The Hydrogen Atom
Exercises

• 13. Multi-electron Systems
13.1 The Multi-electron Schr¨odinger Equation
13.2 The Hartree Approximation
13.3 The Central Field Approximation
13.4 Modelling Lithium
13.5 The Hartree–Fock Method
13.6 Quantum Dots (and Atoms in Flatland)
Exercises
Index
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