Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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goal is to force solutions to be positive, since the solution at the left of the matching point is also positive. The value dx is arbitrary: the solution is anyway rescaled in order to be continuous at the matching point. The code stopst the same index icl corresponding to x c . We thus get a function ψ R (x) defined in [x c , x max ]. In general, the two parts of the wave function have different values in x c : ψ L (x c ) and ψ R (x c ). We first of all re-scale ψ R (x) by a factor ψ L (x c )/ψ R (x c ), so that the two functions match continuously in x c . Then, the whole function ψ(x) is renormalized in such a way that ∫ |ψ(x)| 2 dx = 1. Now comes the crucial point: the two parts of the function will have in general a discontinuity at the matching point ψ ′ R (x c) − ψ ′ L (x c). This difference should be zero for a good solution, but it will not in practice, unless we are really close to the good energy E = E n . The sign of the difference allows us to understand whether E is too high or too low, and thus to apply again the bisection method to improve the estimate of the energy. In order to calculate the discontinuity with good accuracy, we write the Taylor expansions: yi−1 L = yL i − y i ′L ∆x + 1 2 y′′L i (∆x) 2 + O[(∆x) 3 ] yi+1 R = yR i + y i ′R ∆x + 1 2 y′′R i (∆x) 2 + O[(∆x) 3 ] (1.34) For clarity, in the above equation i indicates the index icl. We sum the two Taylor expansions and obtain, noting that yi L = yi R = y i , and that y i ′′L = y i ′′R = y i ′′ = −g iy i as guaranteed by Numerov’s method: that is y L i−1 + y R i+1 = 2y i + (y ′R i − y ′L i )∆x − g i y i (∆x) 2 + O[(∆x) 3 ] (1.35) y i ′R − y i ′L = yL i−1 + yR i+1 − [2 − g i(∆x) 2 ]y i + O[(∆x) 2 ] (1.36) ∆x or else, by using the notations as in (1.31), y i ′R − y i ′L = yL i−1 + yR i+1 − (14 − 12f i)y i + O[(∆x) 2 ] (1.37) ∆x In this way the code calculated the discontinuity of the first derivative. If the sign of y i ′R −y i ′L is positive, the energy is too high (can you give an argument for this?) and thus we move to the lower half-interval; if negative, the energy is too low and we move to the upper half interval. As usual, convergence is declared when the size of the energy range has been reduced, by successive bisection, to less than a pre-determined tolerance threshold. During the procedure, the code prints on standard output a line for each iteration, containing: the iteration number; the number of nodes found (on the positive x axis only); if the number of nodes is the correct one, the discontinuity of the derivative y i ′R − y i ′L (zero if number of nodes not yet correct); the current estimate for the energy eigenvalue. At the end, the code writes the final wave function (this time, normalized to 1!) to the output file. 14
1.3.3 Laboratory Here are a few hints for “numerical experiments” to be performed in the computer lab (or afterward), using both codes: • Calculate and plot eigenfunctions for various values of n. It may be useful to plot, together with eigenfunctions or eigenfunctions squared, the classical probability density, contained in the fourth column of the output file. It will clearly show the classical inversion points. With gnuplot, e.g.: plot "filename" u 1:3 w l, "filename" u 1:4 w l (u = using, 1:3 = plot column3 vs column 1, w l = with lines; the second ”filename” can be replaced by ””). • Look at the wave functions obtained by specifying an energy value not corresponding to an eigenvalue. Notice the difference between the results of harmonic0 and harmonic1 in this case. • Look at what happens when the energy is close to but not exactly an eigenvalue. Again, compare the behavior of the two codes. • Examine the effects of the parameters xmax, mesh. For a given ∆x, how large can be the number of nodes? • Verify how close you go to the exact results (notice that there is a convergence threshold on the energy in the code). What are the factors that affect the accuracy of the results? Possible code modifications and extensions: • Modify the potential, keeping inversion symmetry. This will require very little changes to be done. You might for instance consider a “double-well” potential described by the form: [ (x ) 4 ( ) x 2 V (x) = ɛ − 2 + 1] , ɛ, δ > 0. (1.38) δ δ • Modify the potential, breaking inversion symmetry. You might consider for instance the Morse potential: [ ] V (x) = D e −2ax − 2e −ax + 1 , (1.39) widely used to model the potential energy of a diatomic molecule. Which changes are needed in order to adapt the algorithm to cover this case? 15
Page 1 and 2: Lecture notes Numerical Methods in
Page 3 and 4: 3.3 Code: crossection . . . . . . .
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: eigenvalue) to force the code to pe
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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