Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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with V 0 > 0, b < a. The matrix elements of the Hamiltonian are given by Eq.(4.47) for the kinetic part. by Eq.(4.48) for the potential. The latter can be explicitly calculated: 〈b i |V (x)|b j 〉 = − 1 a = − V 0 a = V 0 a ∫ b/2 V 0 e −i(k i−k j )x dx (4.51) −b/2 e −i(k i−k j )x b/2 −i(k i − k j ) ∣ −b/2 The case k i = k j must be separately treated, yielding V 0 (4.52) sin (b(k i − k j )/2) , k i ≠ k j . (4.53) (k i − k j )/2 Ṽ (0) = V 0b a . (4.54) Code pwell.f90 1 (or pwell.c 2 ) generates the k i , fills the matrix H ij and diagonalizes it. The code uses units in which ¯h 2 /2m = 1 (e.g. atomic Rydberg units). Input data are: width (b) and depth (V 0 ) of the potential well, width of the box (a), number of plane waves (2N + 1). On output, the code prints the three lowest energy levels; moreover it writes to file gs-wfc.out the wave function of the ground state. 4.4.1 Diagonalization routines The practical solution of the secular equation, Eq.(4.32), is not done by naively calculating the determinant and finding its roots! Various well-established, robust and fast diagonalization algorithms are known. Typically they are based on the reduction of the original matrix to Hessenberg or tridiagonal form via successive transformations. All such algorithms require the entire matrix (or at least half, exploiting hermiticity) to be available in memory at the same time, plus some work arrays. The time spent in the diagonalization is practically independent on the content of the matrix and it is invariably of the order of O(N 3 ) floating-point operations for a N × N matrix, even if eigenvalues only and not eigenvectors are desired. Matrix diagonalization used to be a major bottleneck in computation, due to its memory and time requirements. With modern computers, diagonalization of 1000 × 1000 matrix is done in less than no time. Still, memory growing as N 2 and time as N 3 are still a serious obstacle towards larger N. At the end of these lecture notes alternative approaches will be mentioned. The computer implementation of diagonalization algorithms is also rather well-established. In our code we use subroutine dsyev.f 3 from the linear algebra library LAPACK 4 . Several subroutines from the basic linear algebra library 1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/pwell.f90 2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/pwell.c 3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/Lapack/dsyev.f 4 http://www.netlib.org/lapack/ 40
BLAS 5 (collected here: dgemm.f 6 ) are also required. dsyev implements reduction to tri-diagonal form for a real symmetric matrix (d=double precision, sy=symmetric matrix, ev=calculate eigenvalues and eigenvectors). The usage of dsyev requires either linking to a pre-compiled LAPACK and BLAS libraries, or compilation of the Fortran version and subsequent linking. Instructions on the correct way to call dsyev are contained in the header of the subroutine. Beware: most diagonalization routines overwrite the matrix! For the C version of the code, it may be necessary to add an underscore (as in dsyev () ) in the calling program. Moreover, the tricks explained in Sec.0.1.4 are used to define matrices and to pass arguments to BLAS and LAPACK routines. 4.4.2 Laboratory • Observe how the results converge with respect to the number of plane waves, verify the form of the wave function. Verify the energy you get for a known case. You may use for instance the following case: for V 0 = 1, b = 2, the exact result is E = −0.4538. You may (and should) also verify the limit V 0 → ∞ (what are the energy levels?). • Observe how the results converge with respect to a. Note that for values of a not so big with respect to b, the energy calculated with the variational method is lower than the exact value. Why is it so? • Plot the ground-state wavefunction. You can also modify the code to write excited states. Do they look like what you expect? Possible code modifications and extensions: • Modify the code, adapting it to a potential well having a Gaussian form (whose Fourier transform can be analytically calculated: what is the Fourier transform of a Gaussian function?) For the same ”width”, which problem converges more quickly: the square well or the Gaussian well? • We know that for a symmetric potential, i.e. V (−x) = V (x), the solutions have a well-defined parity, alternating even and odd parity (ground state even, first excited state odd, and so on). Exploit this property to reduce the problem into two subproblems, one for even states and one for odd states. Use sine and cosine functions, obtained by suitable combinations of plane waves as above. Beware the correct normalization and the k n = 0 term! Why is this convenient? What is gained in computational terms? 5 http://www.netlib.org/blas/ 6 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/Blas/dgemm.f 41
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 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: eported here are immediately genera
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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