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El. knyga: Computational Quantum Mechanics

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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 Schrodinger equation for arbitrarily complex potentials, and for single and multi-electron systems.
I Scientific Programming: an Introduction for Physicists 1(162)
1 Numbers and Precision
3(14)
1.1 Fixed-Point Representation
4(1)
1.2 Floating-Point Representation
5(5)
1.3 Floating-Point Arithmetic
10(5)
Exercises
15(2)
2 Fortran
17(66)
2.1 Variable Declaration
18(5)
2.2 Operators and Control Statements
23(6)
2.3 String Manipulation
29(5)
2.4 Input and Output
34(4)
2.5 Intrinsic Functions
38(2)
2.6 Arrays
40(9)
2.7 Procedures: Functions and Subroutines
49(15)
2.8 Modules
64(9)
2.9 Command-Line Arguments
73(5)
2.10 Timing Your Code
78(3)
Exercises
81(2)
3 Python
83(80)
3.1 Preliminaries: Tabs, Spaces, Indents, and Cases
85(3)
3.2 Variables, Numbers, and Precision
88(5)
3.3 Operators and Conditionals
93(7)
3.4 String Manipulation
100(6)
3.5 Data Structures
106(12)
3.6 Loops and Flow Control
118(8)
3.7 Input and Output
126(4)
3.8 Functions
130(6)
3.9 NumPy and Arrays
136(19)
3.10 Command-Line Arguments
155(3)
3.11 Timing Your Code
158(4)
Exercises
162(1)
II Numerical Methods for Quantum Physics 163(194)
4 Finding Roots
165(16)
4.1 Big-O Notation
165(2)
4.2 Convergence
167(1)
4.3 Bisection Method
168(3)
4.4 Newton-Raphson Method
171(2)
4.5 Secant Method
173(3)
4.6 False Position Method
176(2)
Exercises
178(3)
5 Differentiation and Initial Value Problems
181(48)
5.1 Method of Finite Differences
181(5)
5.2 The Euler Method(s)
186(10)
5.3 Numerical Error
196(1)
5.4 Stability
197(5)
5.5 The Leap-Frog Method
202(3)
5.6 Round-Off Error
205(4)
5.7 Explicit Runge-Kutta Methods
209(4)
5.8 Implicit Runge-Kutta Methods
213(5)
5.9 RK4: The Fourth-Order Runge Kutta Method
218(6)
Exercises
224(5)
6 Numerical Integration
229(36)
6.1 Trapezoidal Approximation
229(7)
6.2 Midpoint Rule
236(2)
6.3 Simpson's Rule
238(6)
6.4 Newton-Cotes Rules
244(2)
6.5 Gauss-Legendre Quadrature
246(6)
6.6 Other Common Gaussian Quadratures
252(2)
6.7 Monte Carlo Methods
254(5)
Exercises
259(6)
7 The Eigenvalue Problem
265(44)
7.1 Eigenvalues and Eigenvectors
265(4)
7.2 Power Iteration
269(9)
7.3 Krylov Subspace Techniques
278(6)
7.4 Stability and the Condition Number
284(2)
7.5 Fortran: Using LAPACK
286(12)
7.6 Python: Using SciPy
298(6)
Exercises
304(5)
8 The Fourier Transform
309(48)
8.1 Approximating the Fourier Transform
309(7)
8.2 Fourier Differentiation
316(3)
8.3 The Discrete Fourier Transform
319(15)
8.4 Errors: Aliasing and Leaking
334(3)
8.5 The Fast Fourier Transform
337(11)
8.6 Fortran: Using FFTW
348(3)
8.7 Python: Using SciPy
351(3)
Exercises
354(3)
III Solving the Schrodinger Equation 357(130)
9 One Dimension
359(28)
9.1 The Schrodinger Equation
359(1)
9.2 The Time-Independent Schrodinger Equation
360(1)
9.3 Boundary Value Problems
361(7)
9.4 Shooting method for the Schr8dinger Equation
368(9)
9.5 The Numerov-Cooley Shooting Method
377(3)
9.6 The Direct Matrix Method
380(3)
Exercises
383(4)
10 Higher Dimensions and Basic Techniques
387(28)
10.1 The Two-Dimensional Direct Matrix Method
387(10)
10.2 Basis Diagonalization
397(5)
10.3 The Variational Principle
402(5)
Exercises
407(8)
11 Time Propagation
415(24)
11.1 The Unitary Propagation Operator
416(2)
11.2 Euler Methods
418(2)
11.3 Split Operators
420(2)
11.4 Direct Time Discretisation
422(3)
11.5 The Chebyshev Expansion
425(3)
11.6 The Nyquist Condition
428(7)
Exercises
435(4)
12 Central Potentials
439(20)
12.1 Spherical Coordinates
439(4)
12.2 The Radial Equation
443(2)
12.3 The Coulomb Potential
445(2)
12.4 The Hydrogen Atom
447(8)
Exercises
455(4)
13 Multi-electron Systems
459(28)
13.1 The Multi-electron Schrodinger Equation
459(1)
13.2 The Hartree Approximation
460(5)
13.3 The Central Field Approximation
465(4)
13.4 Modelling Lithium
469(8)
13.5 The Hartree-Fock Method
477(3)
13.6 Quantum Dots (and Atoms in Flatland)
480(3)
Exercises
483(4)
Index 487
Dr. Josh Izaac holds a PhD in Quantum Information and Computation from The University of Western Australia. He has been involved in teaching methods of computational physics and quantum mechanics for the past five years, and currently works as a computational physicist developing photonics-based quantum computing software. A freelance science journalist, his work has also appeared in Science and Australian Geographic.





Professor Jingbo Wang currently leads an active research group at The University of Western Australia, working in the general area of quantum information and computation, in particular quantum walks, quantum simulation, and quantum algorithm development. This textbook evolved out of several lecture courses on Computational Physics, Computational Quantum Mechanics, and Scientific High Performance Computation, which Professor Wang has taught since 2002 at The University of Western Australia.
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