Quantum Phisycs 2
A.Y. 2024/2025
Learning objectives
This is an advanced quantum mechanics course that builds upon the
introductory course of the prevous semester, and specifically introduces
three-dimensional systems (in particular the hydrogen atom) and a
variety of theoretical developments, including the theory of angular
momentum, spin, path-integral methods, perturbation theory, scattering
theory, identical particles, and entanglement.
introductory course of the prevous semester, and specifically introduces
three-dimensional systems (in particular the hydrogen atom) and a
variety of theoretical developments, including the theory of angular
momentum, spin, path-integral methods, perturbation theory, scattering
theory, identical particles, and entanglement.
Expected learning outcomes
At the end of this course the student
1. will know how to deal with the Schroedinger equation for intera
cting two-particles systems (including the case of identical particles) 2. will be able to solve for the spectrum of the Hamiltonian for central problems using spherical coordinates
3. will be able to determine the spectrum of the hydrogen atom
4. will be able to determine the spectrum of the orbital angular momentum
operator and of intrinsic angular momentum (spin) operators, and will
be able to add angular momenta
5. will be capable of connecting classical and quantum equations of motion, using either the WKB approximation or a path-integral approach
6. will be able to compute time-independent perturbations to the spectrum of a known Hamiltonian
7. will be able to calculate a transition amplitude using time-dependent perturbation theory
8. will be able to compute a cross section in terms of an amplitude
9. will be able to write down the wave function for a system of identical particles
10. will be able to determine the density matrix for a statistical ensemble and use it to calculate expectation values.
1. will know how to deal with the Schroedinger equation for intera
cting two-particles systems (including the case of identical particles) 2. will be able to solve for the spectrum of the Hamiltonian for central problems using spherical coordinates
3. will be able to determine the spectrum of the hydrogen atom
4. will be able to determine the spectrum of the orbital angular momentum
operator and of intrinsic angular momentum (spin) operators, and will
be able to add angular momenta
5. will be capable of connecting classical and quantum equations of motion, using either the WKB approximation or a path-integral approach
6. will be able to compute time-independent perturbations to the spectrum of a known Hamiltonian
7. will be able to calculate a transition amplitude using time-dependent perturbation theory
8. will be able to compute a cross section in terms of an amplitude
9. will be able to write down the wave function for a system of identical particles
10. will be able to determine the density matrix for a statistical ensemble and use it to calculate expectation values.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
CORSO A
Responsible
Lesson period
First semester
Course syllabus
A. Quantum systems in more than one dimension
1. Direct product spaces
2. Separable potentials
3. The two-body problem and central potentials
B. Angular momentum
1. Angular momentum and rotations
2. The angular momentum operator and its spectrum
3. Spin
4. Addition of angular momenta
C. Three-dimensional problems
1. The radial Schrödinger equation
2. The isotropic harmonic oscillator
3. The Coulomb potential and the hydrogen atom
D. The semiclassical limit of quantum mechanics
1. The action in quantum mechanics
2. The Lagrangian approach to quantum mechanics: the path integral
3. The semiclassical (or WKB) approximation
E. Perturbation theory
1. Time-independent perturbations
2. Time-dependent perturbations and the interaction representation
3. Introduction to scattering theory
F. Identical particles
1. Systems of many identical particles
2. Bose and Fermi statistics
3. The spin-statistics theorem
G. Entanglement
1. Density matrix, entanglement, partial measurements
2. The Einstein-Podolsky-Rosen paradox and local realism
3. Bell inequalities and the measurement problem
1. Direct product spaces
2. Separable potentials
3. The two-body problem and central potentials
B. Angular momentum
1. Angular momentum and rotations
2. The angular momentum operator and its spectrum
3. Spin
4. Addition of angular momenta
C. Three-dimensional problems
1. The radial Schrödinger equation
2. The isotropic harmonic oscillator
3. The Coulomb potential and the hydrogen atom
D. The semiclassical limit of quantum mechanics
1. The action in quantum mechanics
2. The Lagrangian approach to quantum mechanics: the path integral
3. The semiclassical (or WKB) approximation
E. Perturbation theory
1. Time-independent perturbations
2. Time-dependent perturbations and the interaction representation
3. Introduction to scattering theory
F. Identical particles
1. Systems of many identical particles
2. Bose and Fermi statistics
3. The spin-statistics theorem
G. Entanglement
1. Density matrix, entanglement, partial measurements
2. The Einstein-Podolsky-Rosen paradox and local realism
3. Bell inequalities and the measurement problem
Prerequisites for admission
Basics of quantum physics and quantum mechanics. Quantum mechanics in
one space dimension. Calculus in more than one dimension. Basics of complex analysis.
one space dimension. Calculus in more than one dimension. Basics of complex analysis.
Teaching methods
The course consists of lectures (40 hours) and recitations (36 hours). All lectures are done on the blackboard and involve the presentation of theoretical and methodological arguments. Of the recitations, 10 hours are devoted to explaining standard applications, while 26 hours are devoted to the discussion of problem sets that have been assigned at the end of each lecture, directly involving students in the solution. The lecture record and all problems (with solution hints) are published on the instructors' website along the way. A teaching assistant will be available and will take care of extra exercise sessions including a simulation of the written test.
Teaching Resources
J.J. Sakurai, Modern Quantum Mechanics , Pearson (general reference)
A. Berera and L. Del Debbio, Quantum Mechanics, Cambridge U. P. (useful for explicit computations)
S. Weinberg, Lectures on Quantum Mechanics; Cambridge U.P. (for deepening and special topics)
K. Gottfried and T.M. Yan, Quantum Mechanics: Fundamentals; Springer (for deepening and special topics)
J. Binney and D. Skinner, The Physics of Quantum Mechanics; Oxford U.P. (for deepening and special topics)
Collections of problems and exercises
G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary)
L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementary)
E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary and intermediate)
A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and avanced)
K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and avanced)
V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics; Oxford U.P. (700 problems, mostly intermediate and advanced)
A. Berera and L. Del Debbio, Quantum Mechanics, Cambridge U. P. (useful for explicit computations)
S. Weinberg, Lectures on Quantum Mechanics; Cambridge U.P. (for deepening and special topics)
K. Gottfried and T.M. Yan, Quantum Mechanics: Fundamentals; Springer (for deepening and special topics)
J. Binney and D. Skinner, The Physics of Quantum Mechanics; Oxford U.P. (for deepening and special topics)
Collections of problems and exercises
G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary)
L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementary)
E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary and intermediate)
A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and avanced)
K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and avanced)
V. Galitski, B. Karnakov, V. Kogan e V. Galitski, Exploring Quantum Mechanics; Oxford U.P. (700 problems, mostly intermediate and advanced)
Assessment methods and Criteria
The final exam is a three-hour long written test that requires solving a number of quantum physics problems of increasing degree of complexity, that cover the main topics of the syllabus, and answering a simple theory question. All written tests of the last several years are available (with solutions) from the instructors' website. The final grade is the weighted average of the grades obtained in the intermediate test (at the end of quantum physics I) and in the final exam.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 8
Practicals: 36 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Forte Stefano, Röntsch Raoul Horst
CORSO B
Responsible
Lesson period
First semester
Course syllabus
A. Quantum systems in more than one dimension
1. Direct product spaces
2. Separable potentials
3. The two-body problem and central potentials
B. Angular momentum
1. Angular momentum and rotations
2. The angular momentum operator and its spectrum
3. Spin
4. Addition of angular momenta
C. Three-dimensional problems
1. The radial Schrödinger equation
2. The isotropic harmonic oscillator
3. The Coulomb potential and the hydrogen atom
D. The semiclassical limit of quantum mechanics
1. The action in quantum mechanics
2. The Lagrangian approach to quantum mechanics: the path integral
3. The semiclassical (or WKB) approximation
E. Perturbation theory
1. Time-independent perturbations
2. Time-dependent perturbations and the interaction representation
3. Introduction to scattering theory
F. Identical particles
1. Systems of many identical particles
2. Bose and Fermi statistics
3. The spin-statistics theorem
G. Entanglement
1. Density matrix, entanglement, partial measurements
2. The Einstein-Podolsky-Rosen paradox and local realism
3. Bell inequalities and the measurement problem
1. Direct product spaces
2. Separable potentials
3. The two-body problem and central potentials
B. Angular momentum
1. Angular momentum and rotations
2. The angular momentum operator and its spectrum
3. Spin
4. Addition of angular momenta
C. Three-dimensional problems
1. The radial Schrödinger equation
2. The isotropic harmonic oscillator
3. The Coulomb potential and the hydrogen atom
D. The semiclassical limit of quantum mechanics
1. The action in quantum mechanics
2. The Lagrangian approach to quantum mechanics: the path integral
3. The semiclassical (or WKB) approximation
E. Perturbation theory
1. Time-independent perturbations
2. Time-dependent perturbations and the interaction representation
3. Introduction to scattering theory
F. Identical particles
1. Systems of many identical particles
2. Bose and Fermi statistics
3. The spin-statistics theorem
G. Entanglement
1. Density matrix, entanglement, partial measurements
2. The Einstein-Podolsky-Rosen paradox and local realism
3. Bell inequalities and the measurement problem
Prerequisites for admission
Basics of quantum physics and quantum mechanics. Quantum mechanics in one space dimension. Basics of complex analysis.
Teaching methods
The course consists of lectures (50 hours) and exercises (26 hours). All lectures are done on the blackboard and involve the
presentation of theoretical and methodological arguments. The 26 hours of exercises are devoted to the discussion of problem sets that have been assigned at the end of each lecture, directly involving students in the solution. A teaching assistant, in common with Course A, will be available and will take care of extra exercise sessions including a simulation of the written test.
presentation of theoretical and methodological arguments. The 26 hours of exercises are devoted to the discussion of problem sets that have been assigned at the end of each lecture, directly involving students in the solution. A teaching assistant, in common with Course A, will be available and will take care of extra exercise sessions including a simulation of the written test.
Teaching Resources
J.J. Sakurai, Modern Quantum Mechanics , Pearson (general reference);
Stefano Forte e Luca Rottoli, Fisica Quantistica; Zanichelli;
G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary)
L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementary)
E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary and intermediate)
A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and avanced)
K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and avanced)
Stefano Forte e Luca Rottoli, Fisica Quantistica; Zanichelli;
G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli (elementary)
L. Angelini, Meccanica quantistica: problemi scelti; Springer (elementary)
E. d'Emilio, L. E. Picasso, Problemi di meccanica quantistica; ETS (elementary and intermediate)
A. Z. Capri, Problems and Solutions in Nonrelativistic Quantum Mechanics; World Scientific (elementary, intermediate and avanced)
K. Tamvakis, Problems and Solutions in Quantum Mechanics; Cambridge U.P. (intermediate and avanced)
Assessment methods and Criteria
The final exam is a three-hour long written test that requires solving a number of quantum physics problems of increasing degree of complexity, that cover the main topics of the syllabus. All written tests of the last several years are available. The final grade is the weighted average of the grades obtained in the intermediate test (at the end of quantum physics I) and in the final exam.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 8
Practicals: 36 hours
Lessons: 40 hours
Lessons: 40 hours
Professor:
Vicini Alessandro
Educational website(s)
Professor(s)