Physics

ALESSANDRO TETA
GIULIA BASTI

General Objectives:
To acquire knowledge of fundamental topics in Mathematical Physics and the related mathematical methods.


Specific Objectives:

Knowledge and Understanding:

By the end of the course, students will be familiar with the fundamentals of operator theory in Hilbert spaces, the mathematical structure of quantum mechanics, and the basic methods for studying problems in quantum mechanics from the perspective of mathematical physics.



Applying knowledge and understanding:

Students who pass the exam will be able to: i) apply the results of operator theory, ii) construct self-adjoint Hamiltonians and study their spectra, iii) rigorously approach a scattering problem, iv) understand the mathematical methods required for the study of the dynamics of many-body systems, with particular regard to the phenomenon of condensation.



Critical Thinking and Judgment:

Students who pass the exam will be able to recognize a mathematical-physical approach and analyze similarities and differences compared to the typical approach in Theoretical Physics.



Communication Skills:

Students who pass the exam will have developed the ability to communicate concepts, ideas, and methodologies in mathematical physics.

Learning Skills:

The knowledge acquired will enable students to pursue further study—either independently or through other courses—on more specialized aspects of the methods of mathematical physics.

Acquire a basic understanding of the typical approach in Mathematical Physics.
In particular, gain a deeper understanding of operator theory in Hilbert spaces, comprehend the mathematical structure of quantum mechanics, and be able to undertake a rigorous study of relevant applications, such as scattering theory and Bose-Einstein condensation.

Good knowledge of the mathematical concepts discussed in the courses:
Analisi, Geometria, Analisi Vettoriale, Modelli e Metodi Matematici della Fisica.
Good knowledge of the basic notions of Quantum Mechanics

a) Linear Operators in Hilbert Spaces

Review of bounded and unbounded operators, adjoint, symmetric, and self-adjoint operators. Self-adjointness criterion, Kato-Rellich perturbation theorem, applications to the Hamiltonians of the free particle and the hydrogen atom. Definition of a resolvent set, spectrum, and resolvent operator. Real spectrum of a self-adjoint operator. Isometric and unitary operators. Formulation of the spectral theorem (without proof). Unitary groups. Classification of the spectrum.

b) Rules of Quantum Mechanics

Axiomatic formulation of the rules. Overview and comments on interpretation issues.

c) Review of the Free Particle

Characterization of the spectrum. Computationof the resolvent, boundary values on the spectrum. Unitary group and its asymptotic behavior for large times.

d) Point Interactions

Construction of the Hamiltonian with point interaction in d = 3. Self-adjointness and spectrum.

Optional: Hamiltonian with N point interactions; study of the eigenvalue problem for N = 2. Generalities on the Efimov effect for ultracold atoms. Approach using point interactions in the Born–Oppenheimer approximation. Overview of the proof of the Efimov effect.

e) Scattering Theory

Asymptotically free states. Wave operators and their properties. Definition of asymptotic completeness. Scattering operator and its properties. Scattering into cones theorem. Eigenfunction Expansion theorem (without proof). Proof of asymptotic completeness and representation of the scattering operator. Perturbative solution of the Lippmann-Schwinger equation. Born approximation.

f) Many-body systems

Tensor product of Hilbert spaces. Hamiltonian of many-body systems. Identical particles in quantum mechanics: bosons and fermions. Mixed states. Reduced density matrices. Statistical ensembles in quantum mechanics (overview). Second quantization: Fock space, creation and annihilation operators.

g) Bose-Einstein condensation

Condensation in an ideal bosonic gas. Condensation in interacting systems: definition. Scaling limits. Condensation in the mean-field limit.

h) Bogoliubov theory

Weyl operators, Bogoliubov transformations. Bogoliubov theory: heuristics in the thermodynamic limit and derivation of the Lee-Huang-Yang formula for the ground state. Rigorous Bogoliubov theory in the mean-field limit.

Optional: Derivation of the Hartree equation for mean-field dynamics. Bogoliubov theory in the Gross-Pitaevskii regime.

i) Some relevant aspects of the theory from a foundational perspective

One or more optional topics, presented in form of seminars, to be chosen from among:
Dirac’s article (1925) and the algebraic formulation of quantum mechanics;
the classical limit, decoherence, and trajectories in a cloud chamber (Mott, 1929);
the incompleteness argument in the article by Einstein, Podolsky, and Rosen (1935)

Teta, A Mathematical Primer on Quantum Mechanics, Springer 2018
Lecture notes prepared by the teachers on:
point interactions,
scattering theory,
many-body systems, and Bose-Einstein condensation

Teschl G., Mathematical Methods in quantum Mechanics, AMS 2009
Hall B.C., Quantum Theory for Mathematicians, Springer, 2013
E.H.Lieb and R.Seiringer. The Stability of Matter in Quantum Mechanics. Cambridge, 2009.
R. Seiringer. Cold Quantum Gases and Bose-Einstein Condensation. Lecture notes from the school ``Quantum Theory from Small to Large Scales'', August 2-27, 2010

Lectures (70%) on the theoretical concepts with classroom exercises (30%)

Attendance at lessons is essential for a good understanding of the course

The exam consists of an interview on the most relevant topics
presented in the course. To pass the exam, the student must be able to
present arguments and repeat calculations discussed and explained
during the course. The student will be asked to apply the
methods learned during the course to exercises or to examples and
situations similar to those that were discussed in the course.
The evaluation takes into account:
- correctness and completeness of the concepts discussed by the student;
- clarity and rigor of presentation;
- analytical development of the theory;
- problem-solving skills (method and results).

Definition of a self-adjoint operator, self-adjointness of the Hamiltonian of the hydrogen atom, properties of wave operators, definition of Bose-Einstein condensation for interacting systems.