Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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Contents Introduction 1 0.1 About Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1.1 Compilers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1.2 Visualization Tools . . . . . . . . . . . . . . . . . . . . . . 2 0.1.3 Mathematical Libraries . . . . . . . . . . . . . . . . . . . 2 0.1.4 Pitfalls in C-Fortran interlanguage calls . . . . . . . . . . 3 0.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 One-dimensional Schrödinger equation 5 1.1 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.2 Exact solution . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Comparison with classical probability density . . . . . . . 8 1.2 Quantum mechanics and numerical codes: some observations . . 9 1.2.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 A pitfall: pathological asymptotic behavior . . . . . . . . 9 1.3 Numerov’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 Code: harmonic0 . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Code: harmonic1 . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Schrödinger equation for central potentials 16 2.1 Variable separation . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 Radial equation . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Coulomb potential . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Energy levels . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Radial wave functions . . . . . . . . . . . . . . . . . . . . 20 2.3 Code: hydrogen radial . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Logarithmic grid . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Improving convergence with perturbation theory . . . . . 22 2.3.3 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Scattering from a potential 25 3.1 Short reminder of the theory of scattering . . . . . . . . . . . . . 25 3.2 Scattering of H atoms from rare gases . . . . . . . . . . . . . . . 27 3.2.1 Derivation of Van der Waals interaction . . . . . . . . . . 27 i
3.3 Code: crossection . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 The Variational Method 32 4.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1.1 Demonstration of the variational principle . . . . . . . . . 32 4.1.2 Alternative demonstration of the variational principle . . 33 4.1.3 Ground state energy . . . . . . . . . . . . . . . . . . . . . 34 4.1.4 Variational method in practice . . . . . . . . . . . . . . . 35 4.2 Secular problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.1 Expansion into a basis set of orthonormal functions . . . 36 4.3 Plane-wave basis set . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 Code: pwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4.1 Diagonalization routines . . . . . . . . . . . . . . . . . . . 40 4.4.2 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 Non-orthonormal basis sets 42 5.1 Non-orthonormal basis set . . . . . . . . . . . . . . . . . . . . . . 42 5.1.1 Gaussian functions . . . . . . . . . . . . . . . . . . . . . . 44 5.1.2 Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Code: hydrogen gauss . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6 Self-consistent Field 47 6.1 The Hartree Approximation . . . . . . . . . . . . . . . . . . . . . 47 6.2 Hartree Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.2.1 Eigenvalues and Hartree energy . . . . . . . . . . . . . . . 49 6.3 Self-consistent potential . . . . . . . . . . . . . . . . . . . . . . . 50 6.3.1 Self-consistent potential in atoms . . . . . . . . . . . . . . 50 6.4 Code: helium hf radial . . . . . . . . . . . . . . . . . . . . . . . . 51 6.4.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7 The Hartree-Fock approximation 53 7.1 Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.1.1 Coulomb and exchange potentials . . . . . . . . . . . . . . 55 7.1.2 Correlation energy . . . . . . . . . . . . . . . . . . . . . . 56 7.1.3 The Helium atom . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Code: helium hf gauss . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2.1 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8 Molecules 60 8.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . 60 8.2 Potential Energy Surface . . . . . . . . . . . . . . . . . . . . . . . 61 8.3 Diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8.4 Code: h2 hf gauss . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8.4.1 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . 63 8.4.2 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ii
Page 1: Lecture notes Numerical Methods in
Page 5 and 6: Introduction The aim of these lectu
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
Page 55 and 56:
nucleus. Even in open-shell atoms,
Page 57 and 58:
Chapter 7 The Hartree-Fock approxim
Page 59 and 60:
Now that we have the expectation va
Page 61 and 62:
number of possible Slater determina
Page 63 and 64:
Let us now introduce a further vari
Page 65 and 66:
The index R is a reminder that both
Page 67 and 68:
Molecular orbitals theory can expla
Page 69 and 70:
Verify how sensitive the result is
Page 71 and 72:
potential, we can hope to obtain a
Page 73 and 74:
with wave vector k will be purely k
Page 75 and 76:
Fast Fourier Transform In the simpl
Page 77 and 78:
truncate it, it is convenient to co
Page 79 and 80:
A reciprocal lattice of vectors G m
Page 81 and 82:
The origin of the coordinate system
Page 83 and 84:
Chapter 11 Exact diagonalization of
Page 85 and 86:
The only nonzero matrix elements fo
Page 87 and 88:
The code requires the number N of s
Page 89 and 90:
Appendix A From two-body to one-bod
Page 91 and 92:
Appendix B Accidental degeneracy an
Page 93 and 94:
and [J 2 1 , J z] = [J 2 2 , J z] =
Page 95 and 96:
part and symmetric coordinate part)
Page 97 and 98:
and the corresponding energy is E =
Page 99 and 100:
D.3.1 Laboratory • observe the ef
Page 101:
• Relative error: |a − b| < ɛa
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Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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