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. Wave-particle duality and quantum measurement

II. Mathematical formulation of the state vector postulate


  • Infinite-dimensional topological vector spaces ; Banach spaces
  • Hilbert spaces and orthonormal basis of observable states

III. Schrödinger's wave mechanics and Heisenberg's matrix mechanics


  • Parseval theorem and separable Hilbert spaces ; isomorphism between H and l^2
  • Non-locality : Lebesgue intégral and Lebesgue spaces

IV. Dirac's BRA-KET formalism


  • Topological dual and Riesz's representation theorem
  • Generalized functions and distribution ; Dirac's delta revisited
  • Rigged Hilbert spaces (Gel'Fand triplets)


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. Multi-Dimensional Systems

 

1. Separable Hamiltonian

2. Hamiltonian and Central Potential

3. Hydrogen Atoms

4. Hybrid Orbitals

 

XII. Stationary Approximation Methods

 

1. Stationary Perturbation for Non-Degenerate States

2. Stationary Perturbation for Degenerate States

3. Fine Structure of the Hydrogen Atom

 

XIII. Time-Dependent Approximation Methods

 

1. Introduction

2. Sinusoidal Perturbation

3. Fermi’s Golden Rule

 

XIV. Introduction to numerical methods

1. Born-Oppenheimer approximation

2. Independent model

3. Slater determinant

4. One nucleus and several electrons systems

5. Several nuclei and several electrons systems

Teaching methods

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

 
 

Assessment method

For the theory course: oral exam with written preparation (50%).


For the exercises: oral exam with written preparation (50%).


The exam will automatically be considered failed if the student obtains a grade lower than 8/20 either for the exam relating to the “Theory” part or for the exam relating to the “Exercises” part (regardless of the overall average of the two grades).


A student who, in the first session, obtained a grade of at least 10/20, either for the tutorials or for the entire theory, is partially exempt from the activity they passed for the second session.


During the oral exam for the theory course, the student will draw two questions, each of which will cover a different part of the course (the theory course is subdivided into three parts, corresponding to the subjects taught by the three teachers). An insufficient grade for one of the two questions may result in failure for the entire theory exam.


A student who fails the theory exam in the first session but has obtained at least a grade of 14/20 for the other part of the theory on which they were tested may be exempted from this third of the theory subject for the second session exam, if they wish.

Sources, references and any support material

E. Prugovecki, "Quantum Mechanics in Hilbert space", Dover, 2006.

L. Debnath, P. Mikusinski, "Introduction to Hilbert spaces with applications", Eslevier Academic press, 2005.

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