Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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For Hydrogen (and all one-electron atoms: He + , Li 2+ , etc.,) the two particles are a proton (or a heavier nucleus) and an electron, with a mass ratio equal to or larger than 1836. The reduced mass will be just a little bit smaller than the electron mass, m e = 0.911 × 10 −30 Kg. 86
Appendix B Accidental degeneracy and dynamical symmetry The energy levels of the Coulomb potential depend only upon the main quantum number n. We have thus a degeneracy on the energy levels with the same n and different l, in addition to the one due to the 2l + 1 possible values of the quantum number m The total degeneracy (not considering spin) for a given n is thus n−1 ∑ (2l + 1) = n 2 . (B.1) l=0 The degeneracy of the energies having different l and same n is present only if the interaction potential is Coulombian. Although it is known as accidental degeneracy, it is not really “accidental”. A degeneracy usually indicates the presence of a symmetry and thus of a conserved quantity. For instance, the degeneracy in m is related to spherical symmetry and to conservation of angular momentum. In classical mechanics, the equivalent of the accidental degeneracy is the conservation of the Runge-Lenz vector verified for a classical Hamiltonian M = p × L m − α r r (B.2) H = p2 2m − α r . (B.3) This is the case of the relative motion of two bodies interacting via gravitational forces. In this case, the orbits are ellipses, and they are always closed: the orientation of the ellipses does not change with time. The Runge-Lenz vector lies along the major ellipses axis. The corresponding quantum vector has a slightly different expression: M = 1 2m (p × L − L × p) − Zq2 e r r. (B.4) It is possible to show that M is orthogonal to L, and that [M, H] = 0, i.e. it is a conserved quantity. 87
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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
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3.2 Scattering of H atoms from rare
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1. Solution of the radial Schrödin
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• In the limit of a weak potentia
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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: Appendix A From two-body to one-bod
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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