(Advanced Quantum Physics)

M.Sc in Physics and M.Sc. in Mathematical Engineering
Department of Physics and Astronomy "Galileo Galilei"
University di Padua, a.y. 2023-2024

Lecturer: Prof. Luca Salasnich


The lecturer can be contacted via e-mail:

The video lectures can be found (free of charges for University students) in the Federica Web Learning platform.


The course consists of 48 hours: 36 hours of theory and 12 hours of exercises. The exam will be a colloquium at the blackboard with chalk, where the student will discuss two topics chosen on the spot by the examination board. Please, do the exam ONLY IF you really know ALL THE TOPICS of the course. The student MUST really know the meaning of the symbols she/he wrote at the blackbord: it is not enough to memorize the formulas and the text of book and lecture notes. It is necessary to really know the physical and mathematical meaning of the formulas. The student MUST KNOW VERY WELL differential and integral calculus. Please, do not say that you must pass the exam because otherwhise you will lose your fellowship. Please, do not stress the examiners with protests and complaints. The topics of the course are:
Second quantization of the electromagnetic field (6+2 hours)
Properties of the classical electromagnetic field in the vacuum. Coulomb Gauge. Expansion in plane waves of the vector potential. Quantum oscillators and quantization of the electromagnetic field. Fock states and coherent states of the electromagnetic field. Electromagnetic field at finite temperature. Casimir effect.
Electromagnetic transitions (8+4 hours)
An atom in the presence of the electromagnetic field. Fermi golden rule. Dipole approximation. Absorption, stimulated and spontaneus emission of radiation: Einstein coefficients. Selection rules. Lifetime of atomic states and linewidths. Population inversion and laser light.
The Spin of the Electron (4+2 hours)
Klein-Gordon and Dirac equations. The Pauli equation and the spin. Dirac equation with a central potential. Relativistic hydrogen atom and fine splitting.
Atoms with many electrons and quantum many-body theory (8+6 hours)
Identical particles. Bosons and Bose-Einstein condensation. Fermions and Pauli exclusion principle. Variational principle. Hartree variational method for bosons and the Gross-Pitaevskii equation. Hartree-Fock variational method for fermions. Density functional theory: theorems of Hoemberg-Kohn, density functional of Thomas-Fermi-Dirac-Von Weizsacker and density functional of Khom-Sham.
Quantum Schrodinger field (6+2 hours)
Field operators for bosons and fermions. Fock and coherent states of the bosonic field operator. Schrodinger field at finite temperature. Matter field for interacting bosons and fermions. Bosons in a double-well potential and the two-site Bose-Hubbard model.

Main book
[T] Luca Salasnich, Quantum Physics of Light and Matter (Springer, 2017).

Other books
[AT1] F. Mandl and G. Shaw, Quantum Field Theory (Wiley, New York, 1984).
[AT2] B.H. Bransden and C.J. Joachain, Physics of Atoms and Molecules, 2nd edition (Prentice Hall, Upper Saddle River, 2003).

Books for further reading
[A1] M.O. Scully and M.S. Zubairy, Quantum Optics (Cambridge Univ. Press, Cambridge, 1997).
[A2] R.W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples (Oxford Univ. Press, Oxford, 2006).
[A3] K. Huang, Statistical Mechanics (Wiley, New York, 1987).
[A4] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).
[A5] E. Lipparini, Modern Many-Particle Physics: Atomic Gases, Quantum Dots and Quantum Fluids (World Scientific, Singapore, 2003).
[A6] A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge Univ. Press, Cambridge, 2006).
[A7] M. Le Bellac, A Short Introduction to Quantum Information and Quantum Computation (Cambridge Univ. Press, Cambridge, 2006).
[A8] L. Maccone, L. Salasnich, Meccanica Quantistica, Caos e Sistemi Complessi (Carocci, Roma, 2008).

Lesson 1 Second Quantization of Light
unit 1.1 Electromagnetic waves
unit 1.2 Quantization of the electromagnetic field
Lesson 2 Quantum Properties of Light
unit 2.1 Fock states vs Coherent states
unit 2.2 Gas of photons and Planck law
unit 2.3 Casimir effect
Lesson 3 Matter-Radiation Interaction
unit 3.1 Minimal coupling and dipole approximation
unit 3.2 Fermi golden rule and spontaneous emission
unit 3.3 Absorption and stimulated emission
Lesson 4 Electromagnetic Transitions
unit 4.1 Selection rules and Einstein equations
unit 4.2 Line width of electromagnetic transitions
Lesson 5 Relativistic Wave Equations
unit 5.1 Klein-Gordon and Dirac equations
unit 5.2 Pauli equation and the spin
unit 5.3 Energy spectrum of the relativistic hydrogen atom
Lesson 6 Quantum Many-Body Systems
unit 6.1 Identical particles, bosons and fermions
unit 6.2 Uniform Fermi gas of non-interacting electrons
unit 6.3 Hartree-Fock method for interacting bosons and fermions
Lesson 7 Superfluids
unit 7.1 Gross-Pitaevskii equation
unit 7.2 Equations of superfluid hydrodynamics
Lesson 8 Density Functional Theory
unit 8.1 Thomas-Fermi density functional
unit 8.2 Hoemberg-Kohn theorem and Kohn-Sham density functional
Lesson 9 Second Quantization of Matter
unit 9.1 Schrodinger field for bosons and fermions
unit 9.2 Statistical physics of non-interacting bosons and fermions
unit 9.3 Hamiltonian in second quantization with interaction
Lesson 10 Bosons in a Double-Well Potential
unit 10.1 Two-site Bose-Hubbard Hamiltonian
unit 10.2 Josephson equations