### Sciences

## Subject: ADVANCED QUANTUM MECHANICS (A.A. 2021/2022)

### master degree course in PHYSICS – FISICA

Course year | 2 |
---|---|

CFU | 6 |

Teaching units |
Unit Advanced Quantum Mechanics
Theory and Foundations of Physics (lesson)
- TAF: Compulsory subjects, characteristic of the class SSD: FIS/02 CFU: 6
Marco GIBERTINI |

Exam type | oral |

Evaluation | final vote |

Teaching language | Italiano |

### Teachers

### Overview

This course offers the basic concepts and tools in quantum mechanics at a higher level with respect to the courses of the 3-year lower degree in Physics.

At the end of the course the student will master advanced topics in quantum mechanics that are useful in different fields of modern physics: from quantum dynamics, to the use of symmetry and group theory, the path-integral method, scattering theory, the formalism of second quantization for many-body systems and its application to the electromagnetic field.

### Admission requirements

Knowledge of the contents of the courses of Quantum Mechanics and Mathematical Methods of Physics at the level of the undergraduate three-year Physics course.

### Course contents

This course takes place during the first (fall) semester and comprises 48 hours of face-to-face lectures (6 ECTS).

The hour repartition provided below is purely indicative, as it can vary during the course depending on the feedback and interests of students.

Review of the postulates of Quantum Mechanics and Quantum Dynamics (6 hours): vectors versus rays, operators and observables, measurement and probabilities, continuum states, unitary evolution operator and Hamiltonian, Schrödinger, Heisenberg, and Dirac (interaction) pictures, adiabatic theorem and Berry phase.

Symmetry and Invariance (8 hours): elements of group theory, symmetry group, Wigner theorem, unitary and antiunitary operators, invariance and time evolution, projective representations, superselection rules, continuous symmetries, generators, Lie algebra, central charges, simply connected groups, universal covering group, applications (time- and space translations, Galilean invariance, rotational invariance, parity and time reversal).

From the propagator to the path-integral (8 hours): propagator and connection with Green's functions, free particle propagator, propagator as a sum over paths, factorization of quantum fluctuations, quadratic Lagrangians, Gelfand-Yaglom approach, propagator for the harmonic oscillator, Van Vleck determinant, perturbation expansion, Feynman diagrams, perturbation theory for the resolvent, propagator in imaginary time, Feynman-Kac formula, connection with the partition function

Scattering theory (12 hours): pseudo-stationary states, scattering (S) matrix, scattering states, Lippmann-Schwinger equation, transition (T) matrix, optical theorem, scattering in 1D, scattering phase shifts, potential scattering in 3D, scattering cross-section, scattering amplitude, Born approximation, central potential, partial-wave expansion, low-energy limit, spherical well/barrier, physical and unphysical poles of the scattering matrix, Levinson theorem, Breit-Wigner resonances, scattering of particles with spin, identical particles

Second quantization (6 hours): identical particles and indistinguishability, permutation symmetry, N-particle Hilbert spaces for bosons and fermions, occupation representation, Fock space, creation and annihilation operators, canonical quantisation and (non relativistic) quantum field theory

From classical to quantum electromagnetic fields (8 hours): longitudinal and transverse fields, gauge transformations, momentum and energy of particles and fields, quantum particle in a classical field, quantization of the electromagnetic field, photons, helicity, coherent states and their properties, Casimir effect, radiation-matter interaction, transition rates, spontaneous and stimulated emission, absorption, dipole approximation, Jaynes-Cummings model

### Teaching methods

Face-to-face lectures (*) using the blackboard. Active participation is strongly encouraged through questions. Students are offered the possibility to test their knowledge and understanding through the solution of exercises provided by the teacher during the semester. (*) depending on the evolution of the Covid19 pandemic, lectures might be online, through video recordings available on the platform Dolly and live streaming sessions of questions/discussion.

### Assessment methods

Oral exam on the topics of the course. The evaluation mark will be communicated immediately after the end of the exam. Oral exams might need to be performed in person or remotely (online) depending on the evolution of the Covid19 pandemic.

### Learning outcomes

Knowledge and Understanding:

At the end of the course the student will master advanced topics in quantum mechanics, ranging from quantum dynamics (evolution operator and its expansion), to the use of symmetry and group theory, the path-integral method, scattering theory, the formalism of second quantization for many-body systems (bosons and fermions) and its application to the electromagnetic field.

Applying Knowledge and Understanding:

The gained knowledge must allow the reading of more advanced textbooks and to solve basic problems in quantum physics with the appropriate mathematical tools.

Making judgements:

The student should own, at the end of the course, the ability to use and to choose the methods to solve problems in atomic physics, in molecular physics, in nuclear physics and condensed matter science with the appropriate tools (exact methods or perturbation theory, operator algebra, group theory,...).

Communicating Skills:

To learn and to present the description of physical problems with the language and the methods of quantum mechanics.

Learning Skills:

The student, with this course, will be able to read, discuss and more deeply understand the subjects in theoretical physics in different fields, as required by an activity of consulting, research and technological applications in the field of modern physics.

### Readings

- Lecture Notes provided by the teacher for the whole course.

Suggested books:

- J.J. Sakurai - Modern Quantum Mechanics, Addison-Wesley,1994.

- S. Weinberg, Lectures on Quantum Mechanics, Cambridge University Press, 2015, 2nd ed.

- L. D. Landau and E. M. Lifshitz, Quantum mechanics : non-relativistic theory, Butterworth-Heinemann, 1981, 3rd ed.

- Ramamurti Shankar, Principles of Quantum Mechanics, Springer 1994, 2nd ed.

- L. S. Schulman - Techniques and applications of path integration, Dover, 2005.

- S. Weinberg, The Quantum Theory of Fields (vol I), Cambridge University Press, 2005.

- J. F. Cornwell - Group theory in Physics. An Introduction, Academic Press, 1997.