Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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3. Calculation of the spherical Bessel functions j l and n l . The analytical forms of these functions are known, but they become quickly unwieldy for high l. One can use recurrence relation. In the code the following simple recurrence is used: with the initial conditions z l+1 (x) = 2l + 1 x z l(x) − z l−1 (x), z = j, n (3.20) j −1 (x) = cos x x , j 0(x) = sin x x ; n −1(x) = sin x x , n 0(x) = − cos x x . (3.21) Note that this recurrence is unstable for large values of l: in addition to the good solution, it has a ”bad” divergent solution. This is not a serious problem in practice: the above recurrence should be sufficiently stable up to at least l = 20 or so, but you may want to verify this. 4. Sum the phase shifts as in Eq.(3.8) to obtain the total cross section and a graph of it as a function of the incident energy. The relevant range of energies is of the order of a few meV, something like 0.1 ≤ E ≤ 3 ÷ 4 meV. If you go too close to E = 0, you will run into numerical difficulties (the wave length of the wave function diverges). The angular momentum ranges from 0 to a value of l max to be determined. The code writes a file containing the total cross section and each angular momentum contribution as a function of the energy (beware: up to l max = 13; for larger l lines will be wrapped). 3.3.1 Laboratory • Verify the effect of all the parameters of the calculation: grid step for integration, r min , r 1 = r max , r 2 , l max . In particular the latter is important: you should start to find good results when l max ≥ 6. • Print, for some selecvted values of the energy, the wave function and its asymptotic behavior. You should verify that they match beyond r max . • Observe the contributions of the various l and the peaks for l = 4, 5, 6 (resonances). Make a plot of the effective potential as a function of l: can you see why there are resonances only for a few values of l? Possible code modifications and extensions: • Modify the code to calculate the cross section from a different potential, for instance, the following one: [ V (r) = −A exp −(r − r 0 ) 2] , r < r max ; V (r > r max ) = 0 What changes do you need to apply to the algorithm, in addition to changing the form of the potential? 30
• In the limit of a weak potential (such as e.g. the potential introduced above), the phase shifts δ l are well approximated by the Born approximation: δ l ≃ − 2m ∫ ∞ ¯h 2 k r 2 jl 2 (kr)V (r)dr, 0 (k = √ 2mE/¯h). Write a simple routine to calculate this integral, compare with numerical results. 31
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 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: 1. Solution of the radial Schrödin
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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