Elementi di meccanica dei continui e meccanica quantistica

The “Quantum Mechanics” module of the course “Elements of continuum mechanics and quantum mechanics” has as its main objective an introductory theoretical treatment of Quantum Mechanics. In particular, both differential calculus and integral calculus are used appropriately in order to structurally connect quantum mechanics to classical mechanics in the framework of a non-local theory. The course program is divided into sections:

“Introductory elements for the birth of quantum mechanics”; “Theoretical formalization of quantum mechanics”. The two sections are closely interconnected and necessary for understanding the entire course both in the field of quantum mechanics and in the field of continuous system mechanics, connecting the basic concepts of quantum mechanics with those of classical mechanics and in particular of Analytical mechanics, both using the Lagrangian and Hamiltonian formalism.

With the "theoretical lessons" of this module the student will acquire the basic "theoretical knowledge" for:

 i) the study of quantum systems subject to forces of any nature.

ii) The mathematical formalization of Quantum Mechanics in both the Schrodinger Representation and the Heisenberg Representation.

iii) The study of conservation laws in physics and their connection with the symmetry properties of the physical system considered.

iv) the possibility of “finding ” and “solve” (also with “successive approximation” methods) the equations of motion, determining the evolutionary solutions for the physical system considered.

v) Some notes on the study of identical particles.

The objective of the course is to induce the student to "think", by relating and linking the various topics covered and acquiring new knowledge and skills.

To this end, according to with the teaching regulations of the CdS in Mathematics, it is expected that at the end of the course the student will have acquired:

- inductive and deductive reasoning ability;

- ability to schematize natural phenomena in terms of physical quantities, to set up a problem using suitable relationships (of an algebraic, integral or differential type) between physical quantities and to solve the problems with analytical and/or numerical methods;

- ability to understand simple experimental configurations in order to carry out measurements and analyze data.

The course will allow the student to acquire useful skills for various technical-professional opportunities, and in particular:

- For technological applications in the industrial and training sectors.

- For the acquisition and processing of data.

- To take care of modeling activities, analysis and related IT-physical implications.

Specifically, having to express the "expected learning outcomes", through the so-called "Dublin Descriptors", the Module "Quantum Mechanics" will therefore aim to achieve the following transversal skills:

Knowledge and understanding:

The Module "Quantum Mechanics" aims to provide mathematical tools (such as theorems, demonstrative procedures and algorithms) that allow students to tackle real applications: in applied mathematics, physics, computer science and many other fields. With these tools, the student will acquire "new abilities to understand and describe" the mathematical schemes hidden behind the physical processes studied during the course. This knowledge will also be very useful to understand new theoretical problems, which can be addressed both in subsequent studies and in the real world.

Ability to apply knowledge and understanding:

At the end of Module "Quantum Mechanics"  the student will be to acquire the "ability to apply knowledge and understanding" of the new mathematical techniques studied, both to concretely determine the "solutions of the equations of motion" associated with the physical problems studied during the course, and for the concrete resolution of possible new problems untreated during the course.

 Independent judgment:

The Module "Quantum Mechanics", based on a logical-deductive method, will give the student autonomous judgment skills to discern incorrect methods of demonstrations. Furthermore, the student, through logical reasoning, will have to face adequate problems of mechanics, and more generally of applied mathematics, seeking to solve them with the interactive help of the teacher.

Communication skills:

In the final exam, student must show that he has reached an adequate maturity in oral communication, both the various mathematical techniques learned and the physical problems described during the course.

Learning ability:

Students will be able to acquire the skills necessary to undertake subsequent studies (both in the framework of the master's degree and in the context of a possible doctorate) with a high degree of autonomy. In addition to proposing theoretical topics, the course deals with topics that may be useful in various fields and professional settings.

The part labeled as Introductory Module to "Quantum Mechanics" of the course of "Elements of continuum mechanics and quantum mechanics" will be carried out through theoretical lectures held by the teacher in the classroom. In these "theoretical lectures" the program will be divided into the sections reported in "Detailed Course Content" for the "Quantum Mechanics" Module. In each of the lectures the teacher will first address all the theoretical topics showing how these topics can be linked to possible applications and specific physical problems.

The part of the course of "Elements of continuum mechanics and quantum mechanics", consisting of the Module on "quantum mechanics", consists of a total of 3 CFU (corresponding to 7 hours each) for a total of 21 hours.

If the teaching is taught in mixed mode or remotely, the necessary changes may be introduced with respect to what was previously declared, in order to respect the expected program and reported in the syllabus.

Information for students with disabilities and/or DSA.

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CINAP (Center for Active and Participated Integration - Services for Disabilities and / or DSA) of our Department or the president of the course of study.

(Indispensable) Differential and integral calculus for functions of several variables.

(Indispensable) Basic physical concepts of classical mechanics, electromagnetism and Analytical Mechanics.

(Important) Symbology associated with physical quantities, dimensions and systems of units of measurement.

(Useful) Nomenclature and language properties for the elementary physical description of Analytical Mechanics and Mathematical Physics

Attendance is strongly recommended (consult the Academic Regulations of the Course of Studies)

Attendance signatures may be collected during the lessons.

Introductory elements for the formulation of Quantum Mechanics:

De Broglie wave theory. Fermat's principle and connections with the Maupertius Principle of Stationary Action. Formal analogies with the wave theory of the Electromagnetic Field. Particle aspects of electromagnetic radiation. Photoelectric effect. Compton effect. Recall of optics, diffraction and interference phenomena for an Electromagnetic Field. Diffraction and interference phenomena for particles with mass. Feynman's ideal experiment. Wave function and its interpretation in terms of probability amplitude.

Theoretical formalization of quantum mechanics (*):

Notation of Ket and Bra states in a Hilbert space. Scalar product and wave function. Observables, Operators and their properties. Matrix representation of operators. Eigenvalues and eigenvectors. Discrete spectrum and continuous spectrum of an operator. Projection operators. Commutator and anticommutator between two operators. Function of an operator. Unitary transformations and unitary operators. Infinitesimal and finite unitary transformations. Mean value of an observable. Dispersion around a central value. Position observable “r”. Symmetry transformations. Infinitesimal translations and generators, finite translations. Momentum observable “p”. Commutator between the components of the position and momentum operators, analogy with Poisson brackets. Wave function in the representation of coordinates and momentum. Uncertainty relation between the position operator and the momentum operator and its physical interpretation. Notes on the development of non-local theories, quantum statistics as consequences of the “position-momentum” uncertainty relation. Finite, infinitesimal rotations and generators. General properties of the angular momentum operator “J” and its eigenvalues. Orbital angular momentum and spin angular momentum. Matrix representation. Matrix representation of spin angular momentum, Pauli matrices. Calculus of commutators [p, f(r)], [r,f(p)], [J,f(r)], [J,f(p)]. Uncertainty relations for the components of the angular momentum operator. Time evolution of states. Time translation and time evolution operator. Infinitesimal and finite time translations. Hamiltonian operator. Schrodinger representation. Heisenberg representation. Time evolution equation for operators in the Heisenberg representation. Analogy with Classical Mechanics. Invariance of probability amplitudes, mean expectation values, Uncertainty Relations, in the two representations. Invariance of “H” and conservation laws. Schrodinger equation. Ehrenfest theorem and classical limit, applications. Notes on the theory of time-independent perturbations in the absence of degeneracy and in the presence of degeneracy. Applications. Notes on systems of identical and indistinguishable particles.

1) Teacher's Notes

2) S.A.Davidov "Meccanica Quantistica" , Edit. MIR 1981.

3) J. J. Sakurai "Meccanica Quantistica Moderna" Zanichelli 1990.

4) A. Messiah: "Quantum Mechanics", Vol. 1 e 2, ed. North-Holland.

5) R. P. Feynman, et al.: "The Feynman Lectures on Physics", Vol III, ed. Addison-Wesley.L.

6) L. I. Schiff, "Quantum Mechanics", McGraw-Hill Book Company 1949.

No ongoing tests will be carried out.

Verification of preparation is carried out only through oral exams, which take place during the periods provided in the academic calendars of the Department, on dates (exam sessions) published in the annual calendar of exam sessions (or Exam Calendar).

Verification of learning can also be carried out electronically, should the conditions require it.

In the criteria adopted for the evaluation of the oral exam, the following will be evaluated:

1) the relevance of the answers to the questions asked,

2) the level of detail of the contents displayed,

3) the ability to connect with other topics covered by the program and with topics already acquired in previous years courses, the ability to report examples,

4) the property of language and clarity of presentation.

The questions, reported below for the exam, do not constitute an exhaustive list but represent only some examples:

Connection between the Principle of Least Action and Fermat's Principle.

De Broglie wave theory.

Diffraction and interference phenomena for the Electromagnetic Field.

Diffraction and interference phenomena for particles with mass.

Observables, Operators and their properties. Matrix representation of operators.

Discrete spectrum and continuous spectrum of an operator.

Unitary transformations and unitary operators.

Symmetry transformations. Infinitesimal translations and generators.

Commutator between the components of the position and momentum operators, analogy with Poisson brackets.

Finite, infinitesimal rotations and generators.

General properties of the angular momentum operator "J" and its eigenvalues.

Uncertainty relations for the components of the angular momentum operator.

Schrodinger representation and Heisemberg representation.

Conservation Laws in Quantum Mechanics.

Schrodinger Equation.

Ehrenfest Theorem and Classical Limit.

Time-Independent Perturbation Theory.

Systems of Identical and Indistinguishable Particles.