00539 - Fundamentals of Theoretical Physics

Course Unit Page

Academic Year 2017/2018

Learning outcomes

At the end of the course, the student has the basic knowledge of the foundations, theory and applications of quantum mechanics, in particular the Schroedinger equation and its methods of resolution. 

Course contents

Module 1

 

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

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

Module 2

 

Problem solving in the following topics of the course

Hydrogenlike atoms
Harmonic oscillator
One-dimensional potentials
Spin
Perturbation theory

 

Readings/Bibliography

P. A.M. Dirac
The Principles of Quantum Mechanics
Oxford University Press

C. Cohen-Tannoudji, B. Diu & F. Laloe  
Quantum Mechanics  I & II
Wiley-Interscience

J. J. Sakurai & J. Napolitano
Modern Quantum Mechanics
Addison-Wesley

A. Galindo & P. Pascual
Quantum Mechanics  I & II
Springer-Verlag

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

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. 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 and 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.

 

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 two a parts and two b parts. chosen in the four themes. 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 two a or b parts are performed by the examining student, only the two where the student has achieved the highest score will be 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 arugulation 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's 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 AMS Campus web site

1) Lectute notes in English

2) Texts of the problems proposed in the problem solving classes with solutions in Italian

3) Texts of past written exams

 

Office hours

See the website of Roberto Zucchini

See the website of Francisco Manuel Soares Verissimo Gil Pedro