Academic Year 2023-2024
PhD course: Mathematical Methods for Many-Body Quantum Systems, Milan, Fall 2023
After a very brief review of the mathematical framework of quantum mechanics, I will discuss second quantization methods for the analysis of many-body quantum systems. Topics include Fock space, creation and annihilation operators, coherent states, Bogoliubov transformations, quasifree states, Bose-Einstein condensation, and if time permits touch also upon aspects of variational approximations such as Hartree-Fock theory and the BCS theory of superconductivity.
Place: Aula dottorato, first floor, Via Saldini 50.
- Wednesday, 8 November 2023, 13:30 (sharp) – 16:05
Hilbert spaces, densely defined operators, compact operators, trace-class and Hilbert-Schmidt operators, integral kernels
- Wednesday, 15 November 2023, 13:30 – 16:05
spectrum and Weyl criterion, confined systems, thermodynamic limit, tensor product of Hilbert spaces, unitary representation of the symmetric group, symmetric and antisymmetric tensor product, Slater determinants, Hamiltonians with pair interaction
- Wednesday, 22 November 2023, 12:45 – 15:25
Fock space, creation and annihilation operators, canonical (anti)commutation relations
- Wednesday, 29 November 2023, 13:30 – 16:05
second quantization, reduced density matrix, pairing density, generalized 1-pdm, generalized creation/annihilation operators, Bogoliubov transformations, quasifree states
- Wednesday, 13 December 2023, 13:30 – 16:05
generalized Hartree-Fock theory and BCS theory
Evaluation: Oral exam, pass/fail. 3 CFU, in any form that is compatible with your study plan.
Literature:
- Jan Philip Solovej: Many Body Quantum Mechanics, Draft of Lecture Notes of March 5, 2014
- Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.
Metodi matematici della meccanica quantistica, Milan, Fall 2023
The first part will be similar to the course of Advanced Mathematical Physics I taught 2017 in Copenhagen (see below for lecture notes). I will introduce von Neumann's framework of unbounded operators in separable Hilbert spaces as a rigorous base for quantum mechanics. We will then discuss Schrödinger operators (H = - ∆ + V) in this framework and analyze physical questions such as existence of solutions to the time-dependent Schrödinger equation, symmetries, time-dependent scattering theory, exponential localization of bound states, stability of the essential spectrum, as well as perturbation theory and its convergence.
Lecture calendar and exercise sheets:
- Wednesday, 27 September: the formalism of quantum mechanics, Banach spaces, Lp-spaces, bounded operators, Hilbert spaces, Fréchet-Riesz representation theorem, closed graph theorem, densely defined operators
- Friday, 29 September: resolvent, spectrum, analytic functions, operator-valued analytic functions, Neumann series
- Wednesday, 4 October: operator adjoint, symmetric and self-adjoint operators, Hellinger-Töplitz theorem, momentum operator, generalized theorem of the bounded inverse, self-adjointness criterion
- Friday, 6 October: exercise solutions and discussion
- Wednesday, 11 October: self-adjointness criterion (cnt.), Kato-Rellich theorem, Schrödinger equation, essential self-adjointness, Schrödinger operators, unitary equivalence, Fourier transform
- Friday, 13 October: multiplication operators, Weyl criterion, Laplace operators, Sobolev inequalities
- Exercises: Sheet 2, with short solution
- Wednesday, 18 October: uniform boundedness principle, strong convergence, strongly continuous unitary groups, generator of a SCUG
- Friday, 20 October: exercise solutions and discussion
- Wednesday, 25 October: existence of SCUGs, commuting operators, conserved quantities, symmetries, Quantum Noether theorem
- Friday, 27 October: scattering theory I: in- and out going states, wave operators, bound states, scattering states
- Wednesday, 8 November: scattering theory II: S-matrix, asymptotic completeness in absence of bound states
- Friday, 10 November: exercise solutions, review on residues, review on Riemann-Lebesgue lemma
- Wednesday, 15 November: stationary scattering theory, IMS formula, localization of bound states
- Friday, 17 November: Sobolev lemma, regularity of bound states, compact operators
- Exercises: Sheet 4 to be handed in Wednesday, November 22
- Wednesday, 22 November: Hilbert-Schmidt operators, relative compactness
- Friday, 24 November: exercise solutions
- Wednesday, 29 November: stability of the essential spectrum, functional calculus I
- Friday, 1 December: functional calculus II
- Exercises: Sheet 5 to be handed in Wednesday, December 13 NEW
- Wednesday, 6 December: NO LECTURE
- Friday, 8 December: NO LECTURE
- Wednesday, 13 December: spectral theorem, perturbation theory I
- Friday, 15 December: exercise solutions
- Exercises: Sheet 6 to be handed in Wednesday, January 10
- Wednesday, 20 December: NO LECTURE
- Friday, 22 December: NO LECTURE
- Wednesday, 10 January: perturbation theory II
- Friday, 12 January: exercise solutions
- Wednesday, 17 January: ??
- Friday, 19 January: ??
Passing criteria:
Reach approximately 50% of the points averaged over the homework assignments, to be handed in every two weeks for correction. Oral exam.
Literature:
- Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version
Contains essentially all topics of the lecture, but in a different arrangement. Some details are missing.
- Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version
Nice selection of material, but not all proofs are given completely. Good overview.
- Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.
Recommended as a reference work, not as a text book. Concise but rather dense.
- Elliott H. Lieb, Michael Loss: Analysis. Errata
A functional analysis book with a constructive point of view. Contains many useful explicit estimates.
If you want to read up on quantum mechanics from the physicist point of view (which is not the focus of this course):
- Gordon Baym: Lectures on Quantum Mechanics
- Leslie E. Ballentine: Quantum Mechanics, A Modern Development
- Steven Weinberg: Lectures on Quantum Mechanics
Fisica Matematica 3 (Meccanica Quantistica), Milan, Spring 2024
TBA
Academic Year 2022-2023
Sistemi Hamiltoniani laboratorio, Milan, Fall 2022
There are partial notes to be expanded over time.
Matematica generale (Biologia), Milan, Fall 2022
Informazioni su Ariel.
Meccanica Analitica (Fisica), Milan, Fall 2022
Informazioni su Ariel.
Metodi e modelli matematici per le applicazioni laboratorio, Milan, Fall 2022
Informazioni su Ariel.
Academic Year 2021-2022
Fisica Matematica 3 (Meccanica Statistica), Milan, Spring 2022
Videoregistrazione su Ariel. Riassunti delle lezioni:
- Lunedì, 11 aprile: cos'è la meccanica statistica, la teoria microscopica, l'ipotesi ergodica
- Martedì, 12 aprile: l'insieme microcanonico, le quantità termodinamiche, il gas ideale (con compito)
- Mercoledì, 13 aprile: la correzione di Gibbs 1/N!, la mistura di due gas ideali
- Giovedì, 21 aprile: la mistura di due gas diversi (soluzione compito), Cosa significa reversibile?, Quanto è marcato il picco nelle distribuzione delle quantità macroscopiche?, l'insieme canonico (continua)
- Martedì, 26 aprile: l'insieme canonico, applicazioni: la statistica di Maxwell-Boltzmann e la formula ipsometrica
- Mercoledì, 27 aprile: la statistica di Maxwell-Boltzmann (situazione generale), le quantità termodinamiche per l'insieme canonico (e la trasformata di Legendre)
- Giovedì, 5 maggio: l'insieme gran canonico
- Lunedì, 9 maggio: il limite termodinamico, interazioni stabili (con compito)
- Martedì, 10 maggio: il limite termodinamico per l'insieme microcanonico configurazionale ed esteso, monotonia di entropia ed energia
- Mercoledì, 11 maggio: sistemi in vicinanza, limite su cubi diadici
- Giovedì, 12 maggio: convessità dell'energia, stime superiori per l'energia
- Lunedì, 16 maggio: estensione continua, limite nel senso di Fisher
- Martedì, 17 maggio: limite termodinamico dell'entropia
- Mercoledì, 18 maggio: dall'insieme configurazionale all'insieme microcanonico
- Giovedì, 19 maggio: soluzioni compiti, nuovi compiti, introduzione transizioni di fase, teoria di Van der Waals
- Lunedì, 23 maggio: modello di Ising, rottura di simmetria, parametro d'ordine
- Martedì, 24 maggio: condizioni al contorno, teoria rigorosa del modello di Ising, soluzione Ising in D=1
- Mercoledì, 25 maggio: stati in volume infinito, disequazioni di correlazione (GKS)
- Giovedì, 26 maggio: disequazioni di correlazione (FKG)
- Lunedì, 30 maggio: invarianza translazionale, diagramma di fase, criteri per non-unicità
- Martedì, 31 maggio: definizione temperatura critica, equivalenza definizione probabilistica ed analitica di transizione di fase, rottura di simmetria per d=2, argomento di Peierls
- Mercoledì, 1 giugno: sviluppo per temperatura alta
- Lunedì, 6 giungo: Prof. Bambusi
- Martedì, 7 giungo: Prof. Bambusi
- Mercoledì, 8 giugno: la meccanica statistica quantistica, la condensazione di Bose-Einstein
- Giovedì, 9 giugno: soluzioni per gli esercizi del 19/5, nuovi compiti
- Martedì, 14 giugno: esempio: legge di Planck/Stefan-Boltzmann, riscaldamento globale (non rilevante per l'esame), soluzioni esercizi della settimana scorsa
- Soluzione dell'esercizio 8 del 24/6/2022
- Correzione per la dimostrazione della condizione di stabilità del 9/5/2022
- Esame del 23/6/2022 con soluzione
- Esame del 12/7/2022 con bozzetto della soluzione
- Esame del 20/9/2022
- Esame del 22/11/2022
- Esame del 26/1/2023
- Esame del 22/2/2023
Libri ed altre risorse:
- Teitel, Stephen: Statistical Mechanics I (Lecture notes 2021), University of Rochester
- Porta, Marcello: Mathematical aspects of classical statistical mechanics (Lecture notes 2015), University of Zurich -> Details -> Script
- Gros, Claudius: Thermodynamik & Statistische Mechanik (Lecture notes 2017/2018), University of Frankfurt
- Schwabl, Franz: Statistical Mechanics, Springer 2006, ISBN 978-3-642-06887-4
- Ruelle, David: Statistical Mechanics - Rigorous Results, World Scientific 1999, ISBN 978-9-810-23862-9
- Kiessling, Michael: On Ruelle’s construction of the thermodynamic limit for the classical microcanonical entropy, arXiv:0810.2084, J. Stat. Phys. 134, pp.19-25 (2009)
- Luca Peliti: Statistical Mechanics in a Nutshell, Princeton University Press 2011, ISBN 978-0-691-20177-1
- Sascha Friedli and Yvan Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press 2017, ISBN 978-1-107-18482-4
Sistemi Hamiltoniani laboratorio, Milan, Fall 2021
Informazioni su Ariel.
Academic Year 2020-2021
Sistemi Hamiltoniani laboratorio, Milan 2020/2021
This course deals with Hamiltonian mechanics and their perturbation theory. Recordings for my part of the theory lectures (corresponding to Chapter 2 "Canonical Transformations" of the lecture notes) can be found on Ariel (link below).
For materials and recording of the associated programming course, please see the official Ariel page. Every section on Ariel contains a subsection "Lezioni" and "Laboratorio"; for the programming course go to "Laboratorio". Here is the overview of the topics I discussed up to now:
- Lab 1: Introduction to Linux, Installing Mizar, Introduction to C
- Lab 2: Working with arrays, memcpy, and a first idea of numerics: solving a variational problem
- Lab 3: Mizar and the forward Euler method
- Lab 4: Symplectic integrators: the Leapfrog method
- Lab 5: analysis of harmonic oscillators with linear coupling, Poincare sections
- Lab 6: Henon-Heiles model, algorithm for Poincare sections [Henon1982], analytic solution for linearly coupled HOs
- Lab 7: Splitting methods from Leapfrog to SABA3
- Lab 8: SABA3 with corrector and "smarter" splitting; introduction to construction of first integrals
- Lab 9: conclusion numerics, algebraic computation with polynomials in one variable
- Lab 10: algebraic computation with polynomials in multiple variables, Poisson bracket in complex representation
- Lab 11: iterative construction of first integrals for the Hénon-Heiles model
- Lab 12: analysis of algebraically constructed truncated first integrals along numerical trajectories
- Lab 13: plotting level sets of truncated first integrals
- Lab 14: comparison of level sets to Poincare sections
Matematica del continuo (Sicurezza dei sistemi e delle reti informatiche), Milan 2020/2021
Motivazione per la matematica: "This equation will change how you see the world (the logistic map)".
Insegnerò solo il secondo semestre, da aprile 2021. La prima parte (fino a fine marzo) è tenuta da Prof. Maggis. Per informazioni guardate la pagina Ariel del corso.
I teach only the second semester, from April 2021. The first part (until end of march) is taught by Prof. Maggis. For more information, please look at the Ariel page of the course.
Youmath.it: Essercizi aggiuntivi e lezioni.
Academic Year 2019-2020
Matematica del continuo (Sicurezza dei sistemi e delle reti informatiche), Milan 2019/2020
L'insegnamento fornisce gli strumenti base dell'Analisi Matematica, sia dal punto di vista teorico che pratico, indispensabili per poter seguire con profitto un corso universitario di carattere scientifico. Le conoscenze proposte sono propedeutiche ad altri corsi base del CdS.
Descrizione sulla pagina dell'Università: Matematica del continuo. Tutte le informazioni su Ariel. Per ora (per il corona virus) le lezioni sono su Zoom in orario normale (martedì 10:30-13:30, mercoledì 8:30-11:30).
Riassunto di quasi tutto: riassunto per stampare. Attenzione: non contiene le lezioni di Lorenzo Luperi Baglini (in particolare il capitolo sul integrale definito), quelli si trovano solo su Ariel. Invece, tutti i riassunti dell'elenco seguente sono contenuti nel file mdc2up.pdf.
Riassunti (anche su Ariel):
- Martedì, 10 marzo: pagine 91-97 del libro
- Mercoledì, 11 marzo: pagine 97-101 del libro
- Martedì, 17 marzo: pagine 101-106 del libro
- Mercoledì, 18 marzo: soluzioni degli esercizi di 11 marzo, pagine 106-108 e 112-114 del libro
- Martedì, 24 marzo: soluzioni degli esercizi di 18 marzo, pagine 108-111 e 114-115 del libro
- Mercoledì, 25 marzo: pagine 115-125 del libro
- Martedì, 31 marzo: soluzioni degli esercizi di 25 marzo, pagine 124-127 del libro
- Mercoledì, 1 aprile: pagine 128-143 del libro (+ congettura 3n+1, non rilevante per l'esame)
- Martedì, 7 aprile: soluzioni degli esercizi di 1 aprile, pagine 144-147 del libro
- Mercoledì, 8 aprile: pagine 147-154 del libro
- Lezioni 21, 22, 28, 29 aprile su Ariel (docente: Lorenzo Luperi Baglini)
- Martedì, 5 maggio: soluzioni degli esercizi, pagina 213 del libro
- Mercoledì, 6 maggio: pagine 214, 216, 221-225 del libro (+ frattale di Collatz e di Mandelbrot, non rilevante per l'esame). Per la simulazione del Mandelbrot insieme -> Sezione "Software", mandelbrot.tar.gz
- Martedì, 12 maggio: pagine 225-232 del libro
- Mercoledì, 13 maggio: soluzioni Esercizio VI, pagine 232-235 del libro
- Martedì, 19 maggio: soluzioni Esercizio VII, pagine 238-239 del libro
- Mercoledì, 20 maggio: pagine 247-248, 250-251, 253 del libro; per il Criterio dell'integrale per le serie: Youmath.it
- Martedì, 26 maggio: soluzioni Esercizio VIII, equazioni differenziali e la matematica della pandemia (non obbligatorio), per l'algoritmo di Euler (non obbligatorio): Khan Academy. Per la simulazione -> Sezione "Software", corona.zip
- Mercoledì, 27 maggio: tempo per domande sul file EserciziVari.pdf (pagina di Slido). Correzione problema 63(2) rispetto alla lezione; aggiunto problema 60(2).
Esercizi (anche su Ariel):
- Limiti di funzioni, 11 marzo
- Continuità, 18 marzo
- Continuità e Derivate, 25 marzo
- Derivate e Studio di funzioni, 1 aprile
- Funzioni convesse e concave, la retta tangente, l'Hôpital, 9 aprile
- Esercizi 22 aprile e 29 aprile su Ariel (docente: Lorenzo Luperi Baglini)
- Integrali, funzione lipschitziane, 6 maggio
- Integrazione per parti e per sostituzione, 13 maggio. Correzioni dei volontari su Ariel.
- Integrali impropri, critero dell'integrale per serie, formula di Taylor, 13 maggio
Correggo le prime tre soluzioni che arrivano nella mia email niels.benedikter "[a t]" unimi.it dopo lunedı̀ 25 maggio, 12:00. Soluzione generale alla lezione di martedı̀ 26 maggio.
Before 2019
Teaching assistant, IST Austria 2018
Teaching assistant for the course "Stability of Matter in Quantum Mechanics" with Prof. Robert Seiringer.
Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. This course provides a self-contained description of research on the stability of matter problem. It introduces the necessary aspects of functional analysis as well as the quantum mechanical background. The topics covered include Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, and gravitational stability of stars.
Advanced Mathematical Physics, Copenhagen 2017
Lecture Notes:
Lectures Niels:
- April 24: quantum mechanics, basic functional analysis, definition of spectrum
- April 25: resolvent identities, Neumann series, analyticity of the resolvent, adjoint operator, self-adjointness
- April 28: Hellinger-Töplitz, Schwartz space, basic criterion for self-adjointness, Kato-Rellich, uniqueness for the time-dependent Schrödinger equation
- May 2: self-adjoint extensions, Fourier transform, Sobolev spaces, Weyl criterion, multiplication operators, the Laplacian
- May 5: Schrödinger operators with Coulomb potential, Sobolev inequalities, strongly continuous unitary groups
- May 8: the generator of a strongly continuous unitary group and its properties
- May 9: translations & rotations, existence of SCUGs, commuting operators, Noether theorem, scattering theory: the free evolution
- May 15: Wave operators
- May 16: Absence of bound states, asymptotic completeness
- May 19: Stationary scattering theory (heuristic), exponential localization of eigenstates
Lectures Jérémy:
- The Spectral Theorem
- Self-Adjoint Extensions
- Quadratic forms
- Rollnik potentials
- Spectral Analysis of (some) Schrödinger Operators I
- Spectral Analysis of (some) Schrödinger Operators II
- Hartree-Fock theory
Assignments (to be handed in at the beginning of the Friday seminar!):
- Assignment 1 (Deadline: May 5)
- Assignment 2 (Deadline: May 19)
- Assignment 3 (Deadline: June 2)
- Assignment 4 (Deadline: June 23, strictly before the seminar!)
Summer School on Current Topics in Mathematic Physics at the University of Zurich, Switzerland, July 17 - July 21:
Seminar: All talks have to be about 40 minutes long (not more!). Schedule is subject to changes depending on our progression in the lecture. The summary is due on Monday after the seminar. Please remember to provide a list of references. Contact Jérémy or me at least two weeks before your talk for a briefing. Topics and summaries:
- May 5: Compact operators
- May 5: Fredholm alternative
- May 26: Solutions for Problems 3-5 of Assigment 2
- May 26: Trotter product formula & BCH formula
- June 2: Uniqueness of the ground state (positivity improving operators)
- June 2: Lieb's estimate on maximum ionization
- June 9: Solution of Assigment 3, Problems 3 and 4
- June 9: Tensor products; fermions and bosons
- June 16: Fock space & creation/annihilation operators
- June 16: Perturbation theory
- June 23: Derivation of the Van-der-Waals force
- June 23: Discussion of Assigment 4
Criteria for passing the course:
Reach approximately 50% of the points averaged over the four assignments.
Give a seminar talk and produce a summary of your talk for the other participants.
Literature: The lecture notes should be self-contained for most of the course. If you are looking for additional reading, here are some recommendations.
- Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version
Contains essentially all topics of the lecture, but in a different arrangement. Sometimes details are missing.
- Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version
Nice selection of material, but sometimes sketchy. Good overview.
- Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.
Recommended as a reference work, not as a text book. Concise but rather dense.
- Elliott H. Lieb, Michael Loss: Analysis. Errata
A functional analysis book aimed at applications in quantum mechanics. Contains many useful explicit estimates.
If you want to read up on quantum mechanics from the physicist point of view (which is not the focus of this course):
- Gordon Baym: Lectures on Quantum Mechanics
- Leslie E. Ballentine: Quantum Mechanics, A Modern Development
- Steven Weinberg: Lectures on Quantum Mechanics
Advanced Mathematical Physics, Copenhagen 2016
Thank you all for following the course! New materials for the 2017 course will appear above.
Teaching Assistance
Teaching assistant, Bonn 2011-2014
I assisted Prof. Benjamin Schlein at the University of Bonn for the courses
- Analysis 2, spring 2013
- Analysis 1, fall 2012
- Vorkurs Mathematik, fall 2012
- Functional Analysis and Partial Differential Equations, fall 2011.
and for a Summer School at the University of Heidelberg
- Mathematical Physics, Analysis and Stochastics, 21.-26. July 2014.