# M1: Advanced Quantum Mechanics

2022-2023

**Lectures** 09:00-11:30, Wednesdays. Location: Université Paris Cité (14/09, 21/09, 28/09, 05/10, 12/10, 19/10, 26/10, 09/11, 16/11, 23/11, 30/11, 07/12). Room #279F (except on 07/12 in room 579F), Halle aux farines bldg.

**Tutorials** by Prof. Michael Joyce, Thursdays. Location: Sorbonne Université (15/09, 22/09, 29/09, 06/10, 13/10, 20/10, 27/10, 17/11, 24/11, 01/12, 08/12, 15/12).

Documents for the tutorials can be retrieved on Moodle.

Mid-term and final for 2021-2022.

**Mid-term: **On Thursday Nov. 10th, 2022 (time slot of the tutorials, location tba). Questions and solutions of the 2021 midterm.

**Final: **No documents, no electronic devices allowed. Week of January 2nd, 2023.

**Fall break:** no lectures/tutorials on Nov 2nd and 3rd, 2022.

**Useful References**

[JJS] J. J. Sakurai, Modern Quantum Mechanics, Addison Wesley

[RS] R. Shankar, Principles of Quantum Mechanics, Kluwer academic

[MLB] M. Le Bellac, Quantum Physics, Cambridge UP

[CCT] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, vol 1 and 2, Wiley

**Chapter 1 States, measurements, information**

**Lecture 1 - 14/09/2022**

A review of the founding postulates underpinning quantum mechanics and a discussion of some of their implications. Evolution operator, Heisenberg inequalities. Hamiltonian, Particle in a potential and solving an eigenvalue problem.

The material covered can found in [JJS], chapter 1, and in [MLB], chapter 4.

**Lecture 2 - 21/09/2022**

Basics of information theory. Density matrix. Entanglement, tensor spaces.

The material covered can found in [MLB], chapter 6. It is interesting to read the original papers of EPR and of Aspect et al. (here and there).

**Lecture 3 - 2****8****/09**

Entanglement and how to quantify it. Reduced density matrix obtained by partially tracing the density matrix. Schmidt decomposition. The EPR paradox et Bell inequality.

Some (handwritten) notes for those curious about singular value/Schmidt decomposition**.**

**Lecture 4 - 05/10**

EPR Paradaox and Bell inequality.

**Chapter 2 Path integral formulation of quantum mechanics**

We realized that an alternative formulation of quantum mechanical expectation values was possible. It is based on a new mathematical object, called path-integrals. The interesting features of this alternative approach are its ability to easily connect to classical physics, and to ease approximation methods (for the latter, you have to believe me).

**Lecture ****5**** - ****12****/10**

**Chapter 3 Symmetries in quantum mechanics **

Passive and active transformations. Review of classical physics. Wigner's theorem.

**Chapter 4 Angular momentum and spin**

What is a rotation in real space and how to characterize it fully.

**Lecture ****6**** - 1****9****/10**

We studied how spatial rotations in real space are represented in the Hilbert space of states by some unitary operators. We realized that the algebra of the infinitesimal spatial rotations is the same as that of the quantum operator **J** used to represent them. But we derived the spectrum of **J**. This chapter 4 is covered by [RS], chapter 12, or by [JJS], chapter 3, or by [MLB], chapter 10.

**Lecture 7 - 26/10**

We will show how to add two angular momentum operators, resulting in yet another angular momentum operator with *j *ranging from |*j*1-*j*2| to *j*1+*j*2. We will focus on the case where the Hilbert space is spanned by the position |x> kets only, where **J** is often denoted by **L** and is called an orbital momentum. The important added property with respect to a generic **J** (where 2*j *is an integer) is that the quantum number *l i*tself is an integer.

**Lecture ****8**** - ****09****/1****1**

We have realized that some experiments could not be explaing without introducing an extra bit to the angular momentum, which we have called spin. A few models of condensed matter physics are introduced.

**Chapter 5 Particle in a central force field**

We investigate the motion of a particle in a central force field, classically, then frrom the quantum mechanical point of view. The bottom of the spectrum is easy to obtain. Spherical harmonics and the radial component of the wave function are introduced. The short-distance behavior is identified.

**Lecture ****9**** - ****16****/11**

We will derive the energy levels of the hydrogen atom. This is nicely done in [RS] chapter 13, or in [MLB] chapter 14 for more advanced material on how the spin-orbit coupling lifts the degeneracy of the energy levels.

**Chapter 6 Approximation methods**

**Lecture 10 - 23/11 **

We begin with the variational method for finding the ground state energy and the ground state wave function that is based on an inequality that is at the basis of the approximation. We show how it can be adapted to excited states. A very physical presentation if that of [RS], chapter 16.

We work out the example of a particle in a quartic potential with the variational method. Then we introduce the WKB method, which captures quantum fluctuations in the semi-classical limit. This can be read in [RS] chapter 16, but I also advise reading the article by Holstein and Switf, Path integrals and the WKB approximation.

**Lecture 1****1**** - ****30****/1****1**** **

We will cover the last approximation method based on the separation of the Hamiltonian into two parts, one for which we know everything, and one, hopefully small, that we shall consider as a perturbation. A good reference is [JJS] chapter 5.

**Chapter 7 Systems of identical particles**

We explain why one more postulate is necessary to deal with systems of identical particules. We state the spin-statistics theorem. A basis of states for fermions is worked out.

**Lecture 1****2**** - 0****7****/12 **

Bosons. Applications to the Helium atom spectrum.