Learning outcomes

Introduction to the Mathematical Concepts of Quantum Mechanics through Postulates

Application of QM in atomic and molecular physics, nuclear physics and solid state physics (including quantum harmonic oscillator)

Kinetic moment and spin

Approximation methods for complex systems

 
 

Goals

The students will be familiar with the notions of kinetic moment and spin. The applications of QM in atomic and molecular physics, nuclear physics and solid state physics will be addressed : harmonic oscillator, symetries, Hydrogen atoms, approximation methods. These concepts will be preceded by an introduction to the mathematical concepts of quantum mechanics, through its postulates.

 
 

Content

The lecture propose an introduction at the use of kinetic moment and spin in quantum mechanics. It addresses also basics problem in physics : harmonic oscillator, symetries, hydrogen atom, approximation methods. These concepts will be preceded by an introduction to the mathematical concepts of quantum mechanics, through its postulates.

 
 

Table of contents

Part A
 
I. Review of Quantum Mechanics I
 
II. The Lebesgue Integral
 
III. Postulate of the State Vector
 
1. Banach Space
2. Hilbert Space and Separability
3. “Bra-ket” Formalism
 
IV. Postulate of Quantities and Observables
 
V. Postulates Related to the Measurement of Physical Quantities
 
VI. Schrödinger Equation and Path Integrals
 
Part B
 
VII. The Harmonic Oscillator in Quantum Physics
 
1. Harmonic Oscillator Hamiltonian
2. Quantization
3. Creation and Annihilation Operators
4. Expression of the Hamiltonian and Commutation Relations
5. Diagonalization of the Hamiltonian
6. Zero-Point Energy
7. Excitations and Particles
8. States of the Harmonic Oscillator in R Representation
 
VIII. Angular Momentum in Quantum Physics
 
1. Angular Momentum
2. The Angular Momentum Operator
3. Magnitude of Angular Momentum
4. Angular Momentum and Central Force
5. Simultaneous Diagonalization of  L^2  and  L_z 
6. Angular Momentum and R Representation
 
IX. Spin
 
1. Orbital Magnetic Moment and Angular Momentum
2. Half-Integer Spin Angular Momentum: Stern-Gerlach Experiment
3. Eigenstates and Spin Representation
4. Identical Particles: Bosons and Fermions
5. Spin Precession and Two-Level Systems
 
Part C
 
X. Composition of Angular Momenta
 
XI. Density Operator
 
XII. Multi-Dimensional Systems
 
1. Separable Hamiltonian
2. Hamiltonian and Central Potential
3. Hydrogen Atoms
4. Hybrid Orbitals
 
XIII. Stationary Approximation Methods
 
1. Stationary Perturbation for Non-Degenerate States
2. Perturbation of the Harmonic Oscillator
3. Stationary Perturbation for Degenerate States
4. Fine Structure of the Hydrogen Atom
 
XIV. Time-Dependent Approximation Methods
 
1. Introduction
2. Sinusoidal Perturbation
3. Fermi’s Golden Rule
 
 

Teaching methods

Use of the blackboard, projections and time for the resoluation of problems (individually and by groups) are alternate

 
 

Assessment method

Oral exams with preparation (50 %) during the exam session for theory

Written exams (50%) during the exam session for exercises

If one of the two grades is inferior to 8, the global exam is automatically considered failed (independently of the grade average).

A student that during the first session obtained a mark of a least 10/20 either for the exercises or for the entire theory part benefits from a partial exemption of either the exercises or the entire theory for the second exam session.

During the oral exam, the student will draw two questions, each covering a different part of the course. An insufficient grade on one of the two questions may result in a failure of the entire theory exam.

 
 

Sources, references and any support material

C. Cohen-Tannoudji, B. Diu et F. Laloë, Mécanique quantique I (Editions Hermann, Collection : Enseignement des sciences, 1997)

C. Cohen-Tannoudji, B. Diu et F. Laloë, Mécanique quantique II (Editions Hermann, Collection : Enseignement des sciences, 1997)

J.-M. Lévy-Leblond, F. Balibar, Quantique : Rudiments (Dunod, Collection : Les cours de reference, 2007)

C. Ngô, H. Ngô ,Physique quantique : Introduction - Cours et exercices corrigés (Dunod, Collection : Sciences sup physique, 2005)

B.H. Bransden, C.J. Joachain. Quantum Mechanics. Pearson Education (2000)

Mécanique Quantique. C. Aslungul. De Boeck - Larcier (2007)

Quantique. Fondements et applications. J.-P. Pérez, R. Charles, O. Pujol. De Boeck (2013)

 
 

Language of instruction

French
Training Study programme Block Credits Mandatory
Bachelor in Physics Standard 0 6
Bachelor in Physics Standard 3 6