Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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1.3.1 Code: harmonic0 Code harmonic0.f90 1 (or harmonic0.c 2 ) solves the Schrödinger equation for the quantum harmonic oscillator, using the Numerov’s algorithm above described for integration, and searching eigenvalues using the “shooting method”. The code uses the adimensional units introduced in (1.4). The shooting method is quite similar to the bisection procedure for the search of the zero of a function. The code looks for a solution ψ n (x) with a pre-determined number n of nodes, at an energy E equal to the mid-point of the energy range [E min , E max ], i.e. E = (E max + E min )/2. The energy range must contain the desired eigenvalue E n . The wave function is integrated starting from x = 0 in the direction of positive x; tt the same time, the number of nodes (i.e. of changes of sign of the function) is counted. If the number of nodes is larger than n, E is too high; if the number of nodes is smaller than n, E is too low. We then choose the lower half-interval [E min , E max = E], or the upper halfinterval [E min = E, E max ], respectively, select a new trial eigenvalue E in the mid-point of the new interval, iterate the procedure. When the energy interval is smaller than a pre-determined threshold, we assume that convergence has been reached. For negative x the function is constructed using symmetry, since ψ n (−x) = (−1) n ψ n (x). This is of course possible only because V (−x) = V (x), otherwise integration would have been performed on the entire interval. The parity of the wave function determines the choice of the starting points for the recursion. For n odd, the two first points can be chosen as y 0 = 0 and an arbitrary finite value for y 1 . For n even, y 0 is arbitrary and finite, y 1 is determined by Numerov’s formula, Eq.(1.32), with f 1 = f −1 and y 1 = y −1 : The code prompts for some input data: y 1 = (12 − 10f 0)y 0 2f 1 . (1.33) • the limit x max for integration (typical values: 5 ÷ 10); • the number N of grid points (typical values range from hundreds to a few thousand); note that the grid point index actually runs from 0 to N, so that ∆x = x max /N; • the name of the file where output data is written; • the required number n of nodes (the code will stop if you give a negative number). Finally the code prompts for a trial energy. You should answer 0 in order to search for an eigenvalue with n nodes. The code will start iterating on the energy, printing on standard output (i.e. at the terminal): iteration number, number of nodes found (on the positive x axis only), the current energy eigenvalue estimate. It is however possible to specify an energy (not necessarily an 1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/harmonic0.f90 2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/harmonic0.c 12
eigenvalue) to force the code to perform an integration at fixed energy and see the resulting wave function. It is useful for testing purposes and to better understand how the eigenvalue search works (or doesn’t work). Note that in this case the required number of nodes will not be honored; however the integration will be different for odd or even number of nodes, because the parity of n determines how the first two grid points are chosen. The output file contains five columns: respectively, x, ψ(x), |ψ(x)| 2 , ρ cl (x) and V (x). ρ cl (x) is the classical probability density (normalized to 1) of the harmonic oscillator, given in Eq.(1.19). All these quantities can be plotted as a function of x using any plotting program, such as gnuplot, shortly described in the introduction. Note that the code will prompt for a new value of the number of nodes after each calculation of the wave function: answer -1 to stop the code. If you perform more than one calculation, the output file will contain the result for all of them in sequence. Also note that the wave function are written for the entire box, from −x max to x max . It will become quickly evident that the code “sort of” works: the results look good in the region where the wave function is not vanishingly small, but invariably, the pathological behavior described in Sec.(1.2.2) sets up and wave functions diverge at large |x|. As a consequence, it is impossible to normalize the ψ(x). The code definitely needs to be improved. The proper way to deal with such difficulty is to find an inherently stable algorithm. 1.3.2 Code: harmonic1 Code harmonic1.f90 3 (or harmonic1.c 4 ) is the improved version of harmonic0 that does not suffer from the problem of divergence at large x. Two integrations are performed: a forward recursion, starting from x = 0, and a backward one, starting from x max . The eigenvalue is fixed by the condition that the two parts of the function match with continuous first derivative (as required for a physical wave function, if the potential is finite). The matching point is chosen in correspondence of the classical inversion point, x cl , i.e. where V (x cl ) = E. Such point depends upon the trial energy E. For a function defined on a finite grid, the matching point is defined with an accuracy that is limited by the interval between grid points. In practice, one finds the index icl of the first grid point x c = icl∆x such that V (x c ) > E; the classical inversion point will be located somewhere between x c − ∆x and x c . The outward integration is performed until grid point icl, yielding a function ψ L (x) defined in [0, x c ]; the number n of changes of sign is counted in the same way as in harmonic0. If n is not correct the energy is adjusted (lowered if n too high, raised if n too low) as in harmonic0. We note that it is not needed to look for changes of sign beyond x c : in fact we know a priori that in the classically forbidden region there cannot be any nodes (no oscillations, just decaying solution). If the number of nodes is the expected one, the code starts to integrate inward from the rightmost points. Note the statement y(mesh) = dx: its only 3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/harmonic1.f90 4 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/harmonic1.c 13
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: an equation involving finite differ
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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