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Advanced methods for physics and Quantum physics
01TUSOD
A.A. 2022/23
Course Language
Inglese
Degree programme(s)
Course structure
| Teaching | Hours |
|---|---|
| Lezioni | 60 |
Lecturers
| Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
|---|
Co-lectures
Context
| SSD | CFU | Activities | Area context |
|---|
2022/23
The course is aimed at providing the students with some important notions of "classical" (in the sense of non-quantum mechanical) physics that are necessary for the continuation of their studies.
In particular, the course will allow the student to fill the conceptual gap between the formalism of classical mechanics taught in the basic courses of Physics 1 and the formalism of quantum mechanics, by describing the approaches to classical mechanics alternative to Newton's - in other words, the Lagrangian and Hamiltonian approaches.
Similarly, the course will cover special relativity and some complements of electromagnetism that are not included in the program of the basics Physics courses but are needed, for instance, in the description of atomic and nuclear physics and solid state physics.
The striking technological advancements of these years, as well as our response to the future challenges (protection of the environment, waste reduction, optimal exploitation of renewable energies and alternative routes of energy production, and so on) are essentially based on a worldwide research on physical phenomena that are very far from our everyday's experience, occur in "extreme" conditions (e.g. very small length scales, very high energies) and whose description is not possible without quantum mechanics and relativity. New solutions are always found by pushing forward the boundary of our present knowledge, and a Physical Engineer, who will work at the very forefront of technology, is expected to master today’s physical concepts prior to develop their practical applications.
However, modern Physics is not disconnected from its ancestors, i.e. classical electromagnetism and Newtonian mechanics. Understanding these connections is fundamental to really grasp the novelty and the power of modern Physics theories and models, to understand the respective domains of validity, and to trace back the genesis and the evolution of today’s physical conceptions.
The course of “Advanced Methods for Physics” has exactly this purpose.
Its first part will allow the student to fill the conceptual gap between the formalism of classical mechanics taught in the basic courses of Physics 1 and the formalism of quantum mechanics, by describing the approaches to classical mechanics alternative to Newton's - in other words, the ones due to Euler-Lagrange, Hamilton and Hamilton-Jacobi.
Its second part will briefly describe how and why Special Relativity has been formulated, and explain its tremendous conceptual impact. The incredibly rich consequences of a very general, but radical, relativity principle will be shown. Then, kinematics, dynamics and electromagnetism will be re-analyzed within the formalism of Special Relativity.
The two parts will be connected by some concepts of classical electromagnetism that are necessary for the subsequent treatment of special relativity, and concern, in particular, the potential equations, gauge transformations, retarded potentials.
Examples and exercises will be suggested and analyzed during the lectures, in order to deepen the understanding of abstract concepts and stimulate the development of problem-solving skills that are fundamental for a Physical Engineer.
At the end of the course, the students should:
- become familiar with the fundamentals of analytical mechanics (Lagrangian and Hamiltonian approach), understand the classical roots of the formalism they are going to use in Quantum Mechanics, and be able to apply the relevant methods to selected problems.
- become acquainted with special relativity, understand the meaning of its postulates as well as their deep consequences, and be able to solve simple problems.
- acquire the knowledge about some topics of electromagnetism (e.g. the potential equations, gauge transformations, retarded potentials), that are part of a solid background in classical physics and are not included in the introductory courses.
At the end of the course, the students will:
- know and understand the fundamentals of analytical mechanics, in the approaches by Lagrange, Hamilton and Hamilton-Jacobi, as well as their interconnections, and their analogies with the formalism used in Quantum Mechanics;
-know and understand the (main) reasons why Special Relativity was developed, its basic concepts, their profound meaning, and their difference with respect to those used in Newtonian mechanics.
- know the Lorentz transformations of coordinates and velocities, as well as the basics of the relativistic formulation of kinematics, dynamics and electromagnetism.
Moreover, the students are expected to become able to:
- derive the main equations of analytical mechanics, discussing the assumptions made and the relevant limits of validity
- derive the main equations of Special Relativity, properly define the relevant physical quantities, and show how they transform under changes of the reference frame
- solve simple problems of mechanics by using the various approaches described during the course and discuss the results in comparison with those provided by Newton’s approach.
- solve simple problems of Special relativity concerning: Lorentz transformations of coordinates and velocities; the Doppler effect; time dilation and length contraction; conservation of linear momentum and energy; transformation of the electromagnetic field, etc.
This knowledge and these abilities will contribute to the acquisition of the technical (calculation) skills of a Physical Engineer but also to the critical thinking and the ability to establish connections between different aspects of Physics usually treated in different courses.
- good knowledge of classical mechanics (from the Physics I course) and, in particular: kinematics and relative motion; dynamics of systems of particles and of rigid bodies.
- knowledge of basic electromagnetism (from the Physics II course) and in particular Maxwell's equations and wave propagation
- good knowledge of mathematical analysis (from Analysis I and Analysis II courses)
- good knowledge of classical mechanics (from the Physics I course) and, in particular: kinematics and relative motion; dynamics of systems of particles and of rigid bodies.
- knowledge of basic electromagnetism (from the Physics II course) and in particular Maxwell's equations and wave propagation
- good knowledge of mathematical analysis (from Analysis I and Analysis II courses) and linear algebra.
1) Analytical mechanics (35 h)
Survey of basic concepts of classical mechanics - Constraints and generalized coordinates - Principle of virtual work - D'Alembert principle - Generalized forces - Lagrangian function and Lagrange's equations - Generalized potentials - Examples of applications of the Lagrangian approach - Action and Hamilton's principle - Euler-Lagrange equations - Conservation theorems and symmetry properties - Cyclic coordinates and canonical momenta - Energy conservation - Legendre transformations and the canonical equations of Hamilton - Canonical transformations - Poisson's brackets - The Hamilton-Jacobi equations - Action variables.
2) Complements of electromagnetism (10 h)
Brief survey on Maxwell's equations - Maxwell stress tensor - Angular momentum of the EM field - Electromagnetic potentials - Gauge transformations - Maxwell's equations in terms of potentials - Solution of Maxwell's equations with sources - Kirchhoff integral theorem - Retarded potentials.
3) Special relativity (15 h)
Introduction: Galilean relativity and electromagnetism - The ether hypothesis and its crisis - Lorentz transformations - Length contraction and time dilation - Simultaneity and causality - Relativistic transformations of velocity - Proper time - Mass, impulse and energy - Doppler effect - Compton effect - Relativistic formulation of electromagnetism.
1) Analytical mechanics (25 h)
Survey of basic concepts of classical mechanics - Constraints and generalized coordinates - Principle of virtual work - D'Alembert principle - Generalized forces - Lagrangian function and Lagrange's equations - Generalized potentials - Examples of applications of the Lagrangian approach - Action and Hamilton's principle - Euler-Lagrange equations - Conservation theorems and symmetry properties - Cyclic coordinates and canonical momenta - Energy conservation - Legendre transformations and the canonical equations of Hamilton - Canonical transformations - Poisson's brackets - The Hamilton-Jacobi equation.
2) Complements of electromagnetism (10 h)
Brief survey on Maxwell's equations - Electromagnetic potentials - Gauge transformations - Maxwell's equations in terms of potentials - Solution of Maxwell's equations with sources - Kirchhoff integral theorem - Retarded potentials.
3) Special relativity (25 h)
Introduction: Galilean relativity and electromagnetism - The ether hypothesis and its crisis - Lorentz transformations - Length contraction and time dilation - Simultaneity and causality - Relativistic transformations of velocity - Doppler effect - Introduction to tensor calculus - Four-vectors - Spacetime graphs - The light cone - Relativistic dynamics - Mass, linear momentum and energy - Relativistic formulation of electromagnetism.
The course will consist of theoretical lectures, including the derivation of most important results and the description of some typical examples. The application of the theoretical concepts to the solution of relevant problems will be illustrated as well.
Exercises for self-training of the students will be proposed.
The course will mainly consist of theoretical lectures, including the step-by-step derivation of the important results (about 48 h).
The application of the theoretical concepts to the solution of relevant problems will be illustrated as well, and exercises for self-training of the students will be proposed. The time devoted to examples and exercises will be approximately equal to 12 h.
The structure of the course will not be affected by the teaching mode (online or onsite). In the case of "blended" teaching mode, lectures will be given in classroom but, at the same time, made available to students connected from remote through a suitable platform. The interaction with the teacher will be possible through the chat of the platform.
In any case, a weekly session of questions and answers (either in person or in remote) will be organized to improve the interaction between students and teacher.
H. Goldstein, J. L. Safko, C. P. Poole
Classical mechanics
III edition, Addison-Wesley 2002
M. Anselmino, S. Costa, E. Predazzi
Origine classica della fisica moderna
Levrotto&Bella, 1999 ISBN: 8882180352
1) The teacher's notes (in pdf) will be uploaded in the web page of this course, accessible through the Teaching Portal of Politecnico.
2) the recordings of the lectures of the previous academic year will be made available on the course webpage.
3) Examples of exercises and questions will be uploaded on the web page of the Course.
4) Suggested books (that can be found and bought online):
H. Goldstein, J. L. Safko, C. P. Poole
Classical mechanics
III edition, Addison-Wesley 2002
(for Analytical Mechanics)
M. Anselmino, S. Costa, E. Predazzi
Origine classica della fisica moderna
Levrotto&Bella, 1999 ISBN: 8882180352
(for Analytical Mechanics, Electromagnetism and Relativity)
R. Gautreau, W. Savin
Schaum's outline of Modern Physics -2nd edition
Mc Graw-Hill, 1999 ISBN: 0-07-024830-3
(for exercises of Relativity)
Modalità di esame: Prova orale obbligatoria;
Exam: Compulsory oral exam;
...
The exam is oral, and is aimed at testing:
- the knowledge of the topics listed in the program
- the ability of the student to connect different concepts and to apply the theoretical notions to the solution of selected problems.
Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.
Exam: Compulsory oral exam;
The exam is oral, and is aimed at testing:
- the knowledge of the topics listed in the program
- the ability of the student to connect different concepts and to apply the theoretical notions to the solution of selected problems.
The exam will thus consist in:
- one problem of analytical mechanics or relativity, to be solved completely (if particularly simple) or partially, by describing the main steps and the strategy of solution (max 10 points)
- two theoretical/conceptual questions about the topics of the course as listed in the detailed program that will be made available to students by the end of the lectures (max 10 points per each question).
The duration of the oral exam is approximately 45 minutes.
In general, the students will not be allowed to use books, notes or any other material during the exams. The teacher can however allow the access to the students' notes if particularly complicated formulas (difficult to remember) are necessary for the solution of the problems.
The minimum mark to pass the exam is 18/30.
The mark will be averaged with that obtained in the module "Quantum Physics".
In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.