00691 - Quantum Mechanics (M-Z)

Learning outcomes

At the end of the course, the student possesses basic knowledge of the fundamentals, theory, and main applications of quantum mechanics. In particular, they are able to address problems using the Schrödinger equation and its solution methods, are familiar with algebraic formalism and its primary applications, understand the theory and applications of angular momentum and spin, and can discuss simple perturbation theory problems.

Course contents

Module 1 (Prof. F. Ravanini)

1) From Classical Physics to Quantum Physics

• Elements of wave theory of light, interference, and diffraction
• Black body radiation
• Photoelectric effect and Compton effect, particle theory of light
• Atomic spectra
• Bohr-Sommerfeld atomic model
• Correspondence principle

2) The Schrödinger Equation

• Matter waves, de Broglie theory and wave-particle duality
• Davisson and Germer experiment
• Double slit experiment
• Wave equation and geometric optics
• Wave function and its probabilistic interpretation
• Schrödinger equation
• Free particles, wave packets
• Wave mechanics principles
• Coordinate space and momentum space
• Averages, matrix elements of operators
• Time independent Schrödinger equation
• Eigenfunctions and energy levels
• Time evolution of the wave function

3) 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
  
• Quantization and canonical commutation rules
• Ehrenfest theorem and semiclassical limit
• Expectation values and uncertainty of an observable
• Uncertainty principle
• Compatible observables and simultaneous eigenstates

4) Solving the Schrödinger Equation in one dimension

• Schrödinger equation in one dimension
• Eigenfunctions and energy levels
• Boxes and potential wells
  
• One-dimensional harmonic oscillator

5) Theory of angular momentum

• Angular momentum in quantum mechanics and its quantization
  
• Spectral theory of angular momentum
• Orbital angular momentum, parity, and spherical harmonics
• Stern-Gerlach experiment, Spin
• Sum of angular momenta and Clebsch-Gordan coefficients

6) Schrödinger equation in three dimensions

• Central potentials
• Problems with spherical symmetry, radial wave functions
• Spherical boxes and potential wells
• The hydrogen atom
  

7) Collision Theory

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

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) Applications

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

11) 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. L. Pisani)

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

The following texts can be consulted for further insights into 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

Lectures at the blackboard or with the aid of a projector

Exercises at the 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 materials are available on the course's "Virtuale" webpage:

1. Lecture notes
2. Texts of the problems proposed in the exercises
3. Texts of past written exams

Office hours

See the website of Francesco Ravanini

See the website of Leonardo Pisani