Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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10.3.1 Laboratory • Verify which cutoff is needed to achieve converged results for E(k) • Reproduce the results in the original paper for Si, Ge, Sn. • Try to figure out what the charge density would be by plotting the sum of the four lowest wavefunctions squared at the Γ point. It is convenient to plot along the (110) plane (that is: one axis along (1,1,0), the other along (0,0,1) ). • In the zincblende lattice, the two atoms are not identical. Cohen and Bergstresser introduce a “symmetric” and an “antisymmetric” contribution, corresponding respectively to a cosine and a sine times the imaginary unit in the structure factor: 〈b i,k |V |b j,k 〉 = V s (G) cos(G · d) + iV a (G) sin(G · d). (10.16) What do you think is needed in order to extend the code to cover the case of Zincblende lattices? 78
Chapter 11 Exact diagonalization of quantum spin models Systems of interacting spins are used since many years to model magnetic phenomena. Their usefulness extends well beyond the field of magnetism, since many different physical phenomena can be mapped, exactly or approximately, onto spin systems. Many models exists and many techniques can be used to determine their properties under various conditions. In this chapter we will deal with the exact solution (i.e. finding the ground state) for the Heisenberg model, i.e. a quantum spin model in which spin centered at lattice sites interact via the exchange interaction. The hyper-simplified model we are going to study is sufficient to give an idea of the kind of problems one encounters when trying to solve many-body systems without resorting to mean-field approximations (i.e. reducing the many-body problem to that of a single spin under an effective field generated by the other spins). Moreover it allows to introduce two very important concepts in numerical analysis: iterative diagonalization and sparseness of a matrix. 11.1 The Heisenberg model Let us consider a set of atoms in a crystal, each atom having a magnetic moment, typically due to localized, partially populated states such as 3d states in transition metals and 4f states in rare earths. The energy of the crystal may in general depend upon the orientation of the magnetic moments. In many cases 1 these magnetic moments tend to spontaneously orient (at sufficiently low temperatures) along a given axis, in the same direction. This phenomenon is known as ferromagnetism. Other kinds of ordered structures are also known, and in particular antiferromagnetism: two or more sublattices of atoms are formed, having opposite magnetization. Well before the advent of quantum mechanics, it was realized that these phenomena could be quite well modeled by a system of interacting magnetic moments. The origin of the interaction was however mysterious, since direct dipole-dipole interaction is way too small to justify the 1 but not for our model: it can be demonstrated that the magnetization vanishes at T ≠ 0, for all 1-d models with short-range interactions only 79
Page 1 and 2:
Lecture notes Numerical Methods in
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3.3 Code: crossection . . . . . . .
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Introduction The aim of these lectu
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is an important and well-known libr
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Chapter 1 One-dimensional Schrödin
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Under the assumption of Eq.(1.8), E
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point its value is still ∼ 60% of
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an equation involving finite differ
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eigenvalue) to force the code to pe
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1.3.3 Laboratory Here are a few hin
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The Hamiltonian can thus be written
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In the following we will use q 2 e
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The dominating term close to the nu
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whose goal is to give an estimate,
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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: The origin of the coordinate system
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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