Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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9 Electrons in a periodic potential 66 9.1 Crystalline solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 9.1.1 Periodic Boundary Conditions . . . . . . . . . . . . . . . 67 9.1.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . 68 9.1.3 The empty potential . . . . . . . . . . . . . . . . . . . . . 68 9.1.4 Solution for the crystal potential . . . . . . . . . . . . . . 69 9.1.5 Plane-wave basis set . . . . . . . . . . . . . . . . . . . . . 70 9.2 Code: periodicwell . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9.2.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 73 10 Pseudopotentials 74 10.1 Three-dimensional crystals . . . . . . . . . . . . . . . . . . . . . . 74 10.2 Plane waves, core states, pseudopotentials . . . . . . . . . . . . . 75 10.3 Code: cohenbergstresser . . . . . . . . . . . . . . . . . . . . . . . 76 10.3.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 78 11 Exact diagonalization of quantum spin models 79 11.1 The Heisenberg model . . . . . . . . . . . . . . . . . . . . . . . . 79 11.2 Hilbert space in spin systems . . . . . . . . . . . . . . . . . . . . 80 11.3 Iterative diagonalization . . . . . . . . . . . . . . . . . . . . . . . 81 11.4 Code: heisenberg exact . . . . . . . . . . . . . . . . . . . . . . 82 11.4.1 Computer Laboratory . . . . . . . . . . . . . . . . . . . . 83 A From two-body to one-body problem 85 B Accidental degeneracy and dynamical symmetry 87 C Composition of angular momenta: the coupled representation 88 D Two-electron atoms 90 D.1 Perturbative Treatment for Helium atom . . . . . . . . . . . . . . 91 D.2 Variational treatment for Helium atom . . . . . . . . . . . . . . . 92 D.3 ”Exact” treatment for Helium atom . . . . . . . . . . . . . . . . 93 D.3.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 95 E Useful algorithms 96 E.1 Search of the zeros of a function . . . . . . . . . . . . . . . . . . 96 E.1.1 Bisection method . . . . . . . . . . . . . . . . . . . . . . . 96 E.1.2 Newton-Raphson method . . . . . . . . . . . . . . . . . . 97 E.1.3 Secant method . . . . . . . . . . . . . . . . . . . . . . . . 97 iii
Introduction The aim of these lecture notes is to provide an introduction to methods and techniques used in the numerical solution of simple (non-relativistic) quantummechanical problems, with special emphasis on atomic and condensed-matter physics. The practical sessions are meant to be a sort of “computational laboratory”, introducing the basic ingredients used in the calculation of materials properties at a much larger scale. The latter is a very important field of today’s computational physics, due to its technological interest and potential applications. The codes provided during the course are little more than templates. Students are expected to analyze them, to run them under various conditions, to examine their behavior as a function of input data, and most important, to interpret their output from a physical point of view. The students will be asked to extend or modify those codes, by adding or modifying some functionalities. For further insight on the theory of Quantum Mechanics, many excellent textbooks are available (e.g. Griffiths, Schiff, or the ever-green Dirac and Landau). For further insight on the properly computational aspects of this course, we refer to the specialized texts quotes in the Bibliography section, and in particular to the book of Thijssen. 0.1 About Software This course assumes some basic knowledge of how to write and execute simple programs, and how to plot their results. All that is needed is a fortran or C compiler and some visualization software. The target machine is a PC running Linux, but other operating systems can be used as well (including Mac OS-X and Windows), as long as the mentioned software is installed and working, and if you know how to use it in oractise. 0.1.1 Compilers In order to run a code written in any programming language, we must first translate it into machine language, i.e. a language that the computer can understand. The translation is done by an interpreter or by a compiler: the former translates and immediately executes each instruction, the latter takes the file, produces the so-called object code that together with other object codes and with libraries is finally assembled into an executable file. Python, Java (or at 1
Page 1 and 2: Lecture notes Numerical Methods in
Page 3: 3.3 Code: crossection . . . . . . .
Page 7 and 8: is an important and well-known libr
Page 9 and 10: Chapter 1 One-dimensional Schrödin
Page 11 and 12: Under the assumption of Eq.(1.8), E
Page 13 and 14: point its value is still ∼ 60% of
Page 15 and 16: an equation involving finite differ
Page 17 and 18: eigenvalue) to force the code to pe
Page 19 and 20: 1.3.3 Laboratory Here are a few hin
Page 21 and 22: The Hamiltonian can thus be written
Page 23 and 24: In the following we will use q 2 e
Page 25 and 26: The dominating term close to the nu
Page 27 and 28: whose goal is to give an estimate,
Page 29 and 30: Chapter 3 Scattering from a potenti
Page 31 and 32: 3.2 Scattering of H atoms from rare
Page 33 and 34: 1. Solution of the radial Schrödin
Page 35 and 36: • In the limit of a weak potentia
Page 37 and 38: By modifying ψ as ψ + δψ, the e
Page 39 and 40: This demonstrates Eq.(4.19), since
Page 41 and 42: that the same equations, for a fini
Page 43 and 44: eported here are immediately genera
Page 45 and 46: BLAS 5 (collected here: dgemm.f 6 )
Page 47 and 48: known as generalized eigenvalue pro
Page 49 and 50: The Hamiltonian operator for this p
Page 51 and 52: Chapter 6 Self-consistent Field A w
Page 53 and 54: (the variations of all other normal
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nucleus. Even in open-shell atoms,
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Chapter 7 The Hartree-Fock approxim
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Now that we have the expectation va
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number of possible Slater determina
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Let us now introduce a further vari
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The index R is a reminder that both
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Molecular orbitals theory can expla
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Verify how sensitive the result is
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potential, we can hope to obtain a
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with wave vector k will be purely k
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Fast Fourier Transform In the simpl
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truncate it, it is convenient to co
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A reciprocal lattice of vectors G m
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The origin of the coordinate system
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Chapter 11 Exact diagonalization of
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The only nonzero matrix elements fo
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The code requires the number N of s
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Appendix A From two-body to one-bod
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Appendix B Accidental degeneracy an
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and [J 2 1 , J z] = [J 2 2 , J z] =
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part and symmetric coordinate part)
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and the corresponding energy is E =
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D.3.1 Laboratory • observe the ef
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• Relative error: |a − b| < ɛa
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Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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