# M1: Advanced Quantum Mechanics

2021-2022

**Lectures** 09:00-11:30, Wednesdays. Location: Université de Paris,

**Tutorials** by Prof. Michael Joyce, 13:30-16:00, Thursdays. Location: Sorbonne Université. Uopdated text available on Moodle, but here is the Nov 2021 version.

**Mid-term: **On Wed. Nov. 10th, same room and same time slot as the lectures. Questions and solutions.

**Final: **No documents, no electronic devices allowed.

**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

**Lecture 1 - 15/09**

**Chapter 1 States, measurements, information**

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 - 22/09**

**Chapter 1 cont'd. ** Basics of information theory. Density matrix. Entanglement and how to quantify 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 - 29/09**

**Chapter 1 cont'd. **The EPR paradox et Bell inequality.

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

**Chapter 2 Symmetries in quantum mechanics**

A reminder of what symmetries tell us in classical mechanics (Noether's theorem, momentum, angular momentum, energy). From the quantum mechanics points of view, symmetries should be expressed in a way compatible with classical mechanics. The material coveree in this chapter can be found in chapter 8 of [MLB] or in chapters 4 then 3 of [JJS], or in chapter 11 of [RS].

**Lecture 4 - 06/10**

**Chapter 2 cont'd. **Symmetry groups and their representation. Wigner's theorem (the proof will not be given but it can be found in the appendix of the first chapter of the first volume of Weinberg's Quantum Field Theory textbook. Infinitesimal generators will be briefly discussed (and they will be seen in greater detail, for rotations, in our **chapter 5** titled **Angular momentum and spin**).

**Chapter 3 Path integral formulation of quantum mechanics**

Propagator of a free particle, and the idea of slicing a time interval in very small subunits. A very good introduction can be found in [RS], chapter 8. An interesting read would be the connection from the path integral formulation back to the Schrödinger equation. Shankar also discussed the interest of path integrals for perturbation theory. Else, there is the classic text by Feynman and Hibbs. It is easy to spend too much time on this topic: as I said during the lectures, this is an important topic for your culture, and should you work in a field where path integrals are needed, you'll learn more about them when necessary.

**Lecture ****5**** - ****13****/10**

**Chapter 3 cont'd. **We showed how a matrix element of the evolution operator could be expressed as a sum over all possible classical paths weighted by a phase factor whose argument is basically the classical action of these paths. This allows an unexpected connection with classical trajectories, yet extended in time. This also bridges quantum mechanics to classical mechanics.

**Chapter ****4**** ****Angular momentum and spin**

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 stopped short of deriving the spectrum of **J**. This chapter 4 is covered by [RS], chapter 12, or by [JJS], chapter 3, or by [MLB], chapter 10.

Here is a handwritten note justifying how a spatial rotation by an arbitrary angle around an arbitray axis can be written in some intrinsic form. This is the formula (8.23) in chapter 8 of [MLB] that I have been using repeatedly. Make sure you can rederive it smoothly.

**Lecture ****6**** - ****20****/10**

**Chapter ****4**** cont'd. **We have derived the spectrum of the angular momentum operator. We have also shown 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 have then focused 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. Same bibliographical references as in the previous lecture.

**Lecture 7 - 27/10 **

**Chapter 4 cont'd. **We will talk about the component of the angular momentum acting in the subspace of vectors not spanned by |x> (which is the spin). I will show that without the existence of extra degrees of freedom (beyond position) some experiments could not have a rational interpretation.

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

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. We stopped short of solving the ODE for the radial part of the wavefunction of the electron, but we discussed an argument giving an approximate form of the low lying energy levels at fixed but arbitrary eigenvalue of the orbital momentum **L**^2.

**Fall break - 03/11**

**Lectur****e time slot**** - ****10/11 : Mid-term**

**Lecture 8 - 17/11 **

**Chapter 5 cont'd. **We'll follow up with the full spectrum and we will discuss several effects that lift the degeneracy of the states.

**Chapter 6 Approximation methods**

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.

**Lecture 9 - 24/11 **

**Chapter 6 cont'd **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 10 - 01/12 **

**Chapter 6 cont'd. **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.

**Lecture 11 - 08/12 **

**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 12 - 15/12 **

**Chapter 7 cont'd. **Bosons. Applications. Towards "second quantization".

Hints for the solution of the midterm.

Review of the important results of the lectures.