Numerical Methods in Quantum Mechanics

 

Numerical Methods in Quantum Mechanics

Corso di Laurea Magistrale Interateneo (Master) in Physics

Academic Year 2020-2021

Teacher: Paolo Giannozzi, room L1-2-BE, Department of Mathematics, Computer Science and Physics, Via delle Scienze 206, Udine
tel. +39 0432 558216, fax +39 0432 558222
web: http://www.fisica.uniud.it/~giannozz, e-mail: paolo . giannozzi at uniud . it
Office hours: during the semester when possible, I'll be in the Physics building Monday and Wednesday from 9:30 to 11:00. Send me an e-mail or phone me to get an appointment (in Trieste or in Udine) at a different time when not possible.

Course Description

Goals: this course provides an introduction to numerical methods and techniques useful for the numerical solution of quantum mechanical problems, especially in atomic and condensed-matter physics. The course is organized as a series of theoretical lessons in which the physical problems and the numerical concepts needed for their resolution are presented, followed by practical sessions in which examples of implementatation for specific simple problems are presented. The student will learn to use the concepts and to practise scientific programming by modifying and extending the examples presented during the course.

Syllabus: Schroedinger equations in one dimension: techniques for numerical solutions. Solution of the Schroedinger equations for a potential with spherical symmetry. Scattering from a potential. Variational method: expansion on a basis of functions, secular problem, eigenvalues and eigenvectors. Examples: gaussian basis, plane-wave basis. Many-electron systems: Hartree-Fock equations, self-consistent field, exchange interaction. Numerical solution of Hartree-Fock equations in atoms with radial integration and on a gaussian basis set. Introduction to numerical solution of electronic states in molecules. Electronic states in solids: solution of the Schroedinger equation for periodic potentials. Introduction to exact diagonalization of spin systems. Introduction to Density-Functional Theory.

Bibliography
Lecture Notes (updated): Introduction, Ch.1, Ch.2, Ch.3, Ch.4, Ch.5, Ch.6, Ch.7, Ch.8, Ch.9, Ch.10, Ch.11, Appendix, all.
See also: J. M. Thijssen, Computational Physics, Cambridge University Press, Cambridge, 1999, Ch.2-4, 5, 6.1-6.4, 6.7.
A rather detailed introduction to Density-Functional techniques can be found in the first chapters of the lecture notes of my (now defunct) course on Numerical Methods in Electronic Structure.

Requirements: basic knowledge of Quantum Mechanics, of Fortran or C programming, of an operating system (preferrably Linux).

Exam: personal project consisting in the numerical solution of a problem, followed by oral examination (typically consisting in the discussion of a subject, different from the one of the personal project, chosen by the student). Contact me a few weeks before the exam to receive the personal project (if not yet assigned) and to set a date. A short written report on the personal project and the related code(s) should be provided no later than the day before the exam.

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Schedule

The course starts on March 8th. Classes are held online on Teams (Link on , https://corsi.units.it/didattica-a-distanza, send me an email if you cannot find it) until further notice

• Monday 11-13, Aula C
• Wednesday 11-13, Lab. T21

Deviation from the above schedule are marked in boldface in the detailed table of arguments (subject to changes>/i>) below.

N.	Date	Subject
1.	8 March	One-dimensional Schroedinger equation: Reminder: harmonic oscillator, analytical solution. Discretization, Numerov algorithm, numerical stability, eigenvalue search using stable outwards and inwards integrations. (Notes: Ch.1)
2.	10 March practical session	Numerical solution of the one-dimensional Schroedinger equation: examples for the harmonic oscillator (code harmonic0: fortran, C; code harmonic1: fortran, C).
3.	15 March	Three-dimensional Schroedinger equation: Central potentials, variable separation, logarithmic grids, perturbative estimate to accelerate eigenvalue convergence. (Notes: Ch.2). A glimpse on true three-dimensional problems on a grid. (Notes: appendix A)
4.	17 March practical session	Numerical solution for spherically symmetric potentials: example for Hydrogen atom (code hydrogen_radial: fortran, C)
5.	22 March	Scattering from a potential: cross section, phase shifts, resonances. (Notes: Ch.3; Thijssen: Ch.2)
6.	24 March pratical session	Calculation of cross sections: numerical solution for Lennard-Jones potential (code crossection: fortran, C).
7.	29 March	Variational method: Schroedinger equation as minimum problem, expansion on a basis of functions, secular problem, introduction to diagonalization algorithms. (Notes: Ch.4; Thijssen: Ch.3)
8.	31 March pratical session	Variational method using an orthonormal basis set: example of a potential well in plane waves (code pwell: fortran, C).
9.	12 April	Non-orthonormal basis sets: gaussian functions. (Notes: Ch.5)
10.	14 April practical session	Variational method with gaussian basis set: solution for Hydrogen atom (code hydrogen_gauss: fortran, C)
11.	19 April	The Hartree-Fock method: Slater determinants, Hartree-Fock equations, self-consistent field. (Notes: Ch.6; Thijssen: Ch.4.1-4.5)
12.	21 April practical session	He atom in Hartree-Fock approximation: solution with radial integration and self-consistency (code helium_hf_radial: fortran, C).
13.	26 April	Molecules: Born-Oppenheimer approximation, potential energy surface, diatomic molecules. introduction to numerical solution for molecules. (Notes: Ch.7; Thijssen: Ch.4.6-4.8)
14.	28 April practical session	Molecules with gaussian basis: solution of Hartree-Fock equations on a gaussian basis for a H2 molecule (code h2_hf_gauss: fortran, C).
15.	3 May	Electronic states in crystals: Bloch theorem, band structure. (Notes: Ch.8; Thijssen: Ch.4.6-4.8)
16.	5 May practical session	Periodic potentials: numerical solution with plane waves of the Kronig-Penney model (code periodicwell: fortran, C; example of usage of FFT: fortran, C).
17.	10 May	Electronic states in crystals II: three-dimensional case, methods of solution, plane wave basis set, introduction to the concept of pseudopotential. (Notes: Ch.9; Thijssen: Ch.6.1-6.4, 6.7)
18.	12 May pratical session	Pseudopotentials: solution of the Cohen-Bergstresser model for Silicon (code cohenbergstresser: fortran, C).
19.	17 May	Spin systems Introduction to spin systems: Heisenberg model, exact diagonalization, iterative methods for diagonalization, sparseness. (Notes: Ch.10)
20.	19 May practical session	Exact Diagonalization Solution of the Heisenberg model with Lanczos chains (code heisenberg_exact: fortran, C).
21.	24 May	Density-Functional Theory Hohenberg-Kohn theorem, Kohn-Sham equations (Notes: Ch.11)
22.	26 May semi-practical session	DFT with plane waves and pseudopotentials Fast Fourier-Trasform and iterative techniques; sample code ah (fortran, C) solving Si with Appelbaum-Hamann pseudopotentials).
23.	31 May	Density-Functional Theory II
24.	7 June	Assignment of exam problems

Last modified 23 May 2021