Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

More documents

Recommendations

Info

a tiny amount of the “bad” solution (due to numerical noise, for instance) is present, the integration algorithm will inexorably make it grow in the classically forbidden region. As the integration goes on, the “bad” solution will sooner or later dominate the “good” one and eventually produce crazy numbers (or crazy NaN’s: Not a Number). Thus a nice-looking wave function in the classically allowed region, smoothly decaying in the classically forbidden region, may suddenly start to diverge beyond some limit, unless some wise strategy is employed to prevent it. The second code presented at the end of this chapter implements such a strategy. 1.3 Numerov’s method Let us consider now the numerical solution of the (time-independent) Schrödinger equation in one dimension. The basic assumption is that the equation can be discretized, i.e. written on a suitable finite grid of points, and integrated, i.e. solved, the solution being also given on the grid of points. There are many big thick books on this subject, describing old and new methods, from the very simple to the very sophisticated, for all kinds of differential equations and all kinds of discretizations and integration algorithms. In the following, we will consider Numerov’s method (named after Russian astronomer Boris Vasilyevich Numerov) as an example of a simple yet powerful and accurate algorithm. Numerov’s method is useful to integrate second-order differential equations of the general form d 2 y = −g(x)y(x) + s(x) (1.21) dx2 where g(x) and s(x) are known functions. Initial conditions for second-order differential equations are typically given as y(x 0 ) = y 0 , y ′ (x 0 ) = y ′ 0. (1.22) The Schrödinger equation (1.1) has this form, with g(x) ≡ 2m [E − V (x)] and ¯h 2 s(x) = 0. We will see in the next chapter that also the radial Schrödinger equations in three dimensions for systems having spherical symmetry belongs to such class. Another important equation falling into this category is Poisson’s equation of electromagnetism, d 2 φ = −4πρ(x) (1.23) dx2 where ρ(x) is the charge density. In this case g(x) = 0 and s(x) = −4πρ(x). Let us consider a finite box containing the system: for instance, −x max ≤ x ≤ x max , with x max large enough for our solutions to decay to negligibly small values. Let us divide our finite box into N small intervals of equal size, ∆x wide. We call x i the points of the grid so obtained, y i = y(x i ) the values of the unknown function y(x) on grid points. In the same way we indicate by g i and s i the values of the (known) functions g(x) and s(x) in the same grid points. In order to obtain a discretized version of the differential equation (i.e. to obtain 10
an equation involving finite differences), we expand y(x) into a Taylor series around a point x n , up to fifth order: y n−1 = y n+1 = y n − y n∆x ′ + 1 2 y′′ n(∆x) 2 − 1 6 y′′′ n (∆x) 3 + 1 +O[(∆x) 6 ] y n + y n∆x ′ + 1 2 y′′ n(∆x) 2 + 1 6 y′′′ n (∆x) 3 + 1 +O[(∆x) 6 ]. If we sum the two equations, we obtain: 24 y′′′′ 24 y′′′′ n (∆x) 4 − 1 n (∆x) 4 + 1 120 y′′′′′ 120 y′′′′′ n (∆x) 5 n (∆x) 5 (1.24) y n+1 + y n−1 = 2y n + y ′′ n(∆x) 2 + 1 12 y′′′′ n (∆x) 4 + O[(∆x) 6 ]. (1.25) Eq.(1.21) tells us that y n ′′ = −g n y n + s n ≡ z n . (1.26) The quantity z n above is introduced to simplify the notations. The following relation holds: z n+1 + z n−1 = 2z n + z ′′ n(∆x) 2 + O[(∆x) 4 ] (1.27) (this is the simple formula for discretized second derivative, that can be obtained in a straightforward way by Taylor expansion up to third order) and thus y ′′′′ n ≡ z ′′ n = z n+1 + z n−1 − 2z n (∆x) 2 + O[(∆x) 2 ]. (1.28) By inserting back these results into Eq.(1.25) one finds y n+1 = 2y n − y n−1 + (−g n y n + s n )(∆x) 2 + 1 12 (−g n+1y n+1 + s n+1 − g n−1 y n−1 + s n−1 + 2g n y n − 2s n )(∆x) 2 +O[(∆x) 6 ] (1.29) and finally the Numerov’s formula [ ] [ ] ] (∆x) y n+1 1 + g 2 (∆x) n+1 = 2y n 1 − 5g 2 (∆x) n − y n−1 [1 + g 2 n−1 12 12 +(s n+1 + 10s n + s n−1 ) (∆x)2 12 + O[(∆x) 6 ] 12 (1.30) that allows to obtain y n+1 starting from y n and y n−1 , and recursively the function in the entire box, as long as the value of the function is known in the first two points (note the difference with “traditional” initial conditions, Eq.(1.22), in which the value at one point and the derivative in the same point is specified). It is of course possible to integrate both in the direction of positive x and in the direction of negative x. In the presence of inversion symmetry, it will be sufficient to integrate in just one direction. In our case—Schrödinger equation—the s n terms are absent. It is convenient to introduce an auxiliary array f n , defined as (∆x) 2 f n ≡ 1 + g n 12 , where g n = 2m ¯h 2 [E − V (x n)]. (1.31) Within such assumption Numerov’s formula can be written as y n+1 = (12 − 10f n)y n − f n−1 y n−1 f n+1 . (1.32) 11
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: point its value is still ∼ 60% of
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 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
show all

Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Create successful ePaper yourself

Delete template?

Save as template?