00691 - Quantum Mechanics

Course Unit Page

Academic Year 2021/2022

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 Theory (Prof. Roberto Zucchini)


1) From classical physics to quantum physics

Undulatory theory of light, interference and diffraction
Photoelectric effect and Compton effect
Corpuscular theory of light
Material waves and de Broglie theory
Wave particle duality
Experience of Davisson and Germer
Atomic spectra
Experience of Franck and Hertz
Bohr-Sommerfeld atomic model
Correspondence principle
Experience of Stern and Gerlach
Angular momentum and spin in quantum physics
Spatial quantisation

2) The Schroedinger equation

The wave equation and geometric optics
Hamilton-Jacobi equation and its relation to geometric optics
Quasiclassical limit
Derivation of the Schroedinger equation
Wave function and its probabilistic interpretation
Energy eigenfunctions and levels
Time evolution of the wave function
Schroedinger equation for a particle with spin

3) Solution of the Schroedinger equation

Schroedinger equation in one dimension
Energy eigenfunctions and levels
Potential boxes and wells
The one-dimensional harmonic oscillator
Schroedinger equation in three dimensions
Schroedinger equation for a central potential
Orbital angular momentum, parity and spherical harmonics
Radial eigenfunctions
Spherical sotential boxes and wells
The hydrogen atom
Other examples and applications

4) Collision theory

Collision in quantum physics
Scattering in one dimension
Reflection and transmission coefficients
Potential barriers
Scattering in three dimensions
Differential and total scattering cross section
Scattering in a central potential
Born approximation
Partial waves expansion
Coulomb scattering
Examples and applications

5) Foundations of quantum physics

Basic quantum experiences
States, observables and measurement
Definition and eigenstates
Measurement and state reduction
Probabilistic nature of quantum physics
Spectrum of an observable
Superposition and completeness
Expectation values and uncertainty of an observable
Compatible observables and simultaneous eigenstates
Indetermination principle

6) Formalism of quantum mechanics

Bras, kets and orthonormal bases
Selfadjoint operators and eigenkets and eigenvalues of selfadjoint operators
States and kets
Observables and selfadjoint operators
Schroedinger, momentum and Heisenberg representations
Quantisation and canonical commutation relations
Ehrenfest theorem and quasiclassical limit

7) Elementary applications

Equazione di Schroedinger for a particle in an electromagnetic field
Two-state systems
The harmonic oscillator in the operator formalism
Other examples and applications

8) Angular momentum theory

Angular momentum commutation relations
Angular momentum spectral theory
Sum of angular momenta and Clebsh-Gordan coefficients
Wigner-Eckart theory
The hydrogen atom in the operator formalism
Pauli Theory of the spinning electron

9) Identical particles

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

10) Time independent perturbation theory

Perturbations and lift of degeneracy
Non degenerate and degenerate perturbation theory
Perturbative expansion
Examples and applications

11) Time dependent perturbation theory

Schroedinger equation and evolution operator
Time dependent perturbations
Schroedinger, Heisenberg and Dirac representation
Pulse perturbations
Periodic perturbations
Fermi golden rule
Adiabatic approximation
Examples and applications


No supplementary contents are envisaged for non-attending students.



Module 2 Problem solving  (Prof. Pierbiagio Pieri)


Problem solving in the following topics of the course


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




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

R. Zucchini
Quantum mechanics: Lecture Notes
Available on the Insegnamenti OnLine 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
ISBN 10: 047116433X
ISBN 13: 9780471164333

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

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

L. D. Landau, E. M. Lifshitz
Quantum Mechanics: Non-Relativistic Theory
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

The exam includes a written part with theory questions and problems about the course contents and a oral one consisting in a discussion of the results of the written part integrated with further theory questions and problems. The exam leads to the assignment of a comparative grade. The grade obtained by the student and determined by the score obtained in the written part possibly modified in the oral part. The written and oral part of the exam must mandatorily be taken during the same session.

There are no prerequisites for admission to the exam. In particular, classroom attendance is not necessary for taking it. There is no minimum score of the written part required for accessing the oral part. There is no a separate exam for the recitation module.

To take both the written and the oral part of the exam it is necessary to register for the relevant tests on AlmaEsami.
Students who fail to fulfill this requirement may be excluded from the exam.

The exam can be taken in two ways:

1) a partial examination on the first half of the program at mid-course followed by a partial examination on the second half of the program from the end of the course on;

2) an examination on the whole program from the end of the course on.

Only students enrolled in the third year in the current academic year may attend the partial examination. One can not take a partial exam on the first half of program after the end of the course.

As a rule, the student may repeat the examination at a second exam session if the grade obtained at the first one does not satisfy him/her within the same academic year. In that case, the grade obtained at the second attempt will be registered even if it is lower than that gotten at the first. The student can accept a grade he/she previously rejected within the academic year during which the grade was obtained. Over that term, the grade is canceled and the student must repeat the exam.

The student's grade takes into account not only his/her knowledge of the subject matter but also his/her ability of critical analysis and independent learning and the appropriateness of verbal expression.

The award of a cum laude grade is taken into account only for the student who has demonstrated an uncommon clarity of thought in the exposition of theory and virtuosity in the solution of problems in the written exam and after an additional oral exam on all the course contents confirming that the student possesses a degree of knowledge of the subject matter far above the average.

The total written exam comprises four themes, I - IV, divided into two parts: a, a theory question, and b, a problem.

The examining student is required to carry out one a part and one
b part chosen in the themes I, II and one a part and one b part
chosen in the themes III, IV. It is not necessary for a and b parts to belong to same theme.

The a and b parts can get a maximum score of 15/90 and 30/90, respectively. The maximum score the student can get in the written exam is therefore 90/90.

If more than one part a or b is carried out by the student in each of the two halves of the exam, only the one in which the student being examined has achieved the highest score is counted.

The total exam is three hours long.

The first partial written exam comprisess two themes, I, II, divided into two parts: a, a theory question, and b, a problem.

The examining student is required to carry out one a part and one b part chosen in the two themes. It is not necessary for the a and b parts to belong to same theme.

The a and b parts can get a maximum score of 15/90 and 30/90, respectively. The maximum score the student can get in the first partial written exam is therefore 45/90.

If more than one a or b part is performed by the examining student, only the one where the student has achieved the highest score will be counted.

The second partial written exam comprises two themes, III, IV, divided into two parts: a, a theory question, and b, a problem and has a rugulation similar to that of the first partial.

Each of the two partial exams is one hour and thirty minutes long.

The written exam sheets must be numbered sequentially and carry the name of the examining student written in bold and legible. Sheets that do not meet these requirements will not be evaluated.

The teacher does not answer any question about the content of the examination.

Use of any form of documental material is prohibited. It is forbidden to copy from the sheets of other students. The breaching student will be automatically excluded from the examination.

The examination will be considered as taken by the student only if he/she hands the exam sheets for the correction. Delivery is not mandatory.

Delivery of the exam sheets will be accepted only upon exhibition of a valid identification document by the examining student.

Teaching tools

The following educational material is available on the Insegnamenti OnLine web site

1) Lectute 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 Pierbiagio Pieri