00691 - Quantum Mechanics (A-L)

Learning outcomes

At the end of the course, the student has the basic knowledge of the foundations, the theory and the main applications of quantum
mechanics. In particular he/she is able to solve problems through the Schroedinger equation and its resolution methods, knows the
algebraic formalism and its main applications, the theory and the
applications of angular momentum and spin, can discuss simple
problems of perturbation theory.

Course contents

Module 1 - Part 1 (Prof. R. Zucchini)

1) From Classical Physics to Quantum Physics

Elements of wave theory of light, interference, and diffraction
Photoelectric effect and Compton effect
Particle theory of light
Matter waves, de Broglie theory and wave-particle duality
Davisson and Germer experiment
Taylor experiment
Atomic spectra
Franck and Hertz experiment
Bohr-Sommerfeld atomic model
Correspondence principle
Stern-Gerlach experiment
Angular momentum and spin in quantum physics and its quantization

2) The Schrödinger Equation

Wave equation and geometric optics
Hamilton-Jacobi equation and its relation to geometric optics
Semiclassical limit
Derivation of the Schrödinger equation
Wave function and its probabilistic interpretation
Time independent Schrödinger equation
Eigenfunctions and energy levels
Time evolution of the wave function
Schrödinger theory in momentum space
Uncertainty principle
Schrödinger equation for a particle with spin

3) Solving the Schrödinger Equation

Schrödinger equation in one dimension
Eigenfunctions and energy levels
Boxes and potential wells
One-dimensional harmonic oscillator
Schrödinger equation in three dimensions
Schrödinger equation for a central potential
Orbital angular momentum, parity, and spherical harmonics
Radial eigenfunctions
Spherical boxes and potential wells
The hydrogen atom
Other examples and applications

4) Collision Theory

Collisions in quantum physics
Scattering in one dimension
Reflection and transmission coefficients
Potential barriers


Module 1 - Part 2 (Prof. R. Zucchini)

5) Formalism of Quantum Mechanics

Bra, ket, and orthonormal bases
Self-adjoint operators, eigenkets, and eigenvalues of a self-adjoint operator
States and kets
Observables and self-adjoint operators
Measurement and state reduction
Probabilistic nature of quantum physics
Schrödinger, momentum, and Heisenberg representations
Quantum mechanics postulates
Quantization and canonical commutation rules
Ehrenfest theorem and semiclassical limit
Expectation values and uncertainty of an observable
Uncertainty principle
Compatible observables and simultaneous eigenstates

6) Elementary Applications

Schrödinger equation for a particle in an electromagnetic field
Two-state systems Harmonic oscillator in operator formalism
Other examples and applications

7) Theory of Angular Momentum

Commutation rules for angular momentum
Spectral theory of angular momentum
Sum of angular momenta and Clebsch-Gordan coefficients

8) Identical Particles

Identity and quantum indistinguishability
Spin and statistics, bosons and fermions
Pauli exclusion principle

9) Time-Independent Perturbation Theory

Perturbation and removal of degeneracy
Non-degenerate and degenerate perturbation theory
Perturbative expansion
Examples and applications

10) Time-Dependent Perturbation Theory

Schrödinger equation and evolution operator
Schrödinger, Heisenberg, and Dirac representations
Time-dependent perturbations
Examples and applications
There are no additional contents for non-attending students.


Module 2 - Exercises (Prof. I. Brivio)

Exercises on the following course topics:

One-dimensional potentials
Harmonic oscillator
Central potentials
Hydrogen-like atoms
Angular momentum and spin
Time-independent perturbation theory
Time-dependent perturbation theory

Readings/Bibliography

 

For the preparation of the exam, we recommend reading the course notes:

R. Zucchini
Quantum mechanics: Lecture Notes
Available on the Virtuale website

 

The following texts can be consulted for further information on the course contents.

 

P. A.M. Dirac
The Principles of Quantum Mechanics
Oxford University Press
ISBN-13: 978-0198520115
ISBN-10: 0198520115

C. Cohen-Tannoudji, B. Diu & F. Laloe
Quantum Mechanics I & II
Wiley-Interscience
ISBN 10: 047116433X
ISBN 13: 9780471164333

J. J. Sakurai & J. Napolitano
Modern Quantum Mechanics
Addison-Wesley
ISBN-13: 978-0805382914
ISBN-10: 0805382917

A. Galindo & P. Pascual
Quantum Mechanics I & II
Springer-Verlag
ISBN 978-3-642-83856-9
ISBN 978-3-642-84131-6

L. D. Landau, E. M. Lifshitz
Quantum Mechanics: Non-Relativistic Theory
Elsevier
ISBN: 9780080503486
ISBN: 9780750635394

Teaching methods

Classroom lectures on a blackboard or with the help of a projector

Classroom problem solving on a blackboard.

Assessment methods

Exam Structure

The exam covers the entire syllabus and consists of two parts:

1. Written exam: Theory questions and problems.
2. Oral exam: Discussion of the written exam results with additional theory questions and problems.

Both parts of the exam must be taken in the same session.

Prerequisites

There are no prerequisites for taking the exam. Attendance at lectures is not mandatory. There is no minimum score required in the written exam to access the oral exam. There is no separate exam for the exercise module.

Registration

You must register for the written and oral exams on AlmaEsami. Registrations are open from about one month before the exam until 2-3 days before. For organizational reasons, the oral exam is held in multiple sessions automatically distributed by the AlmaEsami system. Any changes in session or position within a session between two examinees must be communicated via email to the instructor before the start of the session.

Exam Sessions - Full Mode

The exam covers the entire syllabus, with sessions starting from the end of the course, normally distributed in the current academic year as follows:

• Early June
• Late June - early July
• Late July
• Early September
• First half of January of the following year
• Mid-February of the following year

An extraordinary session might be organized in November for students who need to graduate in December and are completing their thesis (with a written declaration from the supervisor), provided that this course is the only one yet to be recorded in the student's curriculum.

Written Exam

Duration: 180 minutes

Structure:

2 questions for each of the two parts of the course:

• One of type A (theory) - choose one of two proposals. Weight: 1/3.
• One of type B (problems) - choose one of two proposals. Weight: 2/3.

General Rules:

• The sheets must be numbered and include the student's name.
• The use of any documentary material or copying is prohibited, under penalty of cancellation of the exam.
• Submission of the paper requires valid identification.

Oral Exam

Held 4-5 days after the written exam. It consists of theory questions or exercises on topics chosen by the instructor. Duration: from a few minutes to about 30 minutes.

Partial Mode

Due to the complexity of the annual course, a two-part exam mode is also offered to facilitate the exam.

1. First part: Can be taken in the January or February sessions (only for part 1 of the course).
2. Second part: Can be taken in one of the sessions from June onwards.

The written exam for each part lasts 90 minutes and requires answering only the A and B questions concerning that part of the exam.

In the partial oral exam, questions will only cover the part (first or second) being taken.

Restrictions:

• Reserved only for third-year or out-of-course students attending.
• The first partial exam can only be taken once (January or February).
• In case of withdrawal, you automatically switch to the full mode.

Evaluation Criteria

Type A questions (theory): Maximum 15/90.

• Correctness and completeness of content.
• Relevance to the topic.
• Clarity and coherence of the exposition.
• Maximum length: three pages (penalty up to 5/90 if exceeded).

Type B questions (problems): Maximum 30/90.

• Correct approach to the solution.
• Conformity of the calculations.
• Correctness of the calculations.
• Mandatory explanatory comments (penalty up to 5/90 if missing).

Final Grade

The final grade, expressed in 30ths, is determined by the score of the written exam, possibly modified, for better or worse, by the oral exam performance. Honors are granted only in cases of exceptional mastery of the subject, clarity, and exposition virtuosity.

Retaking the Exam

The final grade can be refused only once. The grade obtained on the second attempt will be recorded without further option to refuse. You can accept a previously refused grade within the academic year in which the grade was obtained. Beyond this period, the grade is canceled, and the exam must be retaken.

Teaching tools

The following educational material is available on the Virtuale web site

1) Lecture notes in English

2) Texts of the problems proposed in the problem solving classes.

3) Texts of past written exams

Office hours

See the website of Roberto Zucchini

See the website of Ilaria Brivio