Introductory Quantum Mechanics

Introduction

For an incorrect statement of what quantum mechanics is, see the Wikipedia article which starts: "Quantum mechanics is the study of mechanical systems whose dimensions are close to or below the atomic scale, such as molecules, atoms, electrons, protons and other subatomic particles. Quantum mechanics is a fundamental branch of physics with wide applications. Quantum theory generalizes classical mechanics and provides accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits. The effects of quantum mechanics are typically not observable on macroscopic scales, but become evident at the atomic and subatomic level. There are however exceptions to this rule such as superfluidity."

Before the course begins, think about the Wikipedia definition, and in what ways it is wrong.

Pre-requisites

If you want to take this course then I will assume that you have some knowledge of the standard methods of classical mechanics: namely, configuration and phase space, Lagragian and Hamiltonian methods. If you want to refresh your memory of these things keep a book on classical mechanics handy.

It would help if you brush up on your knowledge of vectors and matrices. If you don't know how to find the eigenvalues and eigenvectors of a matrix, and various properties related to these notions, then you need to look up any standard book on mathematical techniques.

Drop Test

If you think you know the material that will be covered in this course, then you are strongly urged to take a drop test. Send me mail at the email address "sgupta+qm2008" (at mailhost) asking to register for the test. A drop test will be administered on August 23 to those who register by August 15. Make sure that you don't miss the deadline, because this will be your last chance to gain credits by not taking the course.

Course contents

Quantum states and their time evolution: We deal with the question of how to specify the states of a physical system, and how these states change with time. Such questions go by the name of kinematics. We discuss the simplest applications of these concepts. These lead us to the notion of uncertainty and wave packets. We also examine collections (ensembles) of systems and the description of such collections. [Lecture 1 (Aug 4)] [Lecture 2 (Aug 6)]
Mathematical techniques: We revisit the kinematics of quantum mechanics and discover that there are simple mathematical structures underlying them. These are the notions of vectors and matrices (linear vector spaces and operators on them, if you want to use fancy language). These notions are used for a detailed study of various two-level systems. [Lecture 3 (Aug 8)] [Lecture 4 (Aug 11)] [Lecture 5 (Aug 13)]
Square well and cousins in one dimension: We discuss the dynamics of particles in one space dimension when the potential is piecewise constant. We discuss the stationary states of the system with positive and negative energies, and how they enter the dynamics. We study tunnelling, which is an interesting physical effect absent in classical systems, scattering, and symmetry breaking. We examine universal properties of quantum systems which follow essentially from the uncertainty principle. [Lecture 6 (Aug 18)] [Lecture 7 (Aug 20)] [Lecture 8 (Aug 22)]
Other one dimensional problems: We solve the harmonic oscillator problem using raising and lowering operators to reduce the second order differential equation (Schroedinger's equation) to a first order differential equation. We construct the wavefunctions using Hermite polynomials. We factorize other Hamiltonians and show that such factorizable Hamiltonians often have larger symmetries than the apparent ones. [Lecture 9 (Aug 25)] [Lecture 10 (Aug 27)] [Lecture 11 (Aug 29)] [Lecture 12 (Sep 22)] [Lecture 20 (Oct 20)] [Lecture 21 (Oct 24)]
Three dimensional problems: We examine the motion of a single particle in three spatial dimensions. The first class of motions we examine have full rotational symmetry; these include the symmetric harmonic oscillator and motion in a Coulomb potential. In order to solve these problems we use rotational symmetry and reduce the problem to a one-dimensional problem. Next we examine some problems with lesser symmetries (for example, the hydrogen molecular ion $H$ ₂⁺). [Lecture 13 (Sep 24)] [Lecture 14 (Sep 26)] [Lecture 15 (Sep 29)] [Lecture 16 (Oct 01)] [Lecture 17 (Oct 06)] [Lecture 19 (Oct 17)]
Multiparticle states: We examine the wavefunctions of two-particle states and the symmetries that they can have under interchange of particles. We write down general multi-particle states built out of single particle states. We touch upon the method of second quantization, and show how it simplifies book-keeping. [Lecture 18 (Oct 03)]
The path integral: An alternative formulation of quantum mechanics uses the notion of sums over paths: a generalization of the basic physics of the two-slit experiment. We study this formalism and check that it is equivalent to the usual formulation in terms of states and operators. [Lecture 22 (Oct 29)] [Lecture 23 (Oct 31)]

References

Classical Mechanics H. Goldstein 2001 Narosa: This is the standard reference for the parts of classical mechanics which you might need to know for this course, namely, the notions of configuration and phase space, and Lagrange and Hamilton formulations of classical dynamics.

Mathematical Methods for Physicists G. Arfken 200? Academic Press

This is a standard reference for most of the mathematical methods that are needed for this course: matrices and vectors, complex variables, differential equations, special functions and elementary group theory.

The Principles of Quantum Mechanics P. A. M. Dirac 1982 Oxford University Press

This is the standard reference for quantum mechanics, by one of the founders of the subject. If you've taken the drop test, and don't have to worry about submitting assignments, you might want to read through this book. It is very well written, and you will never regret the time spent on it.

Quantum Mechanics (Non-relativistic Theory) L. D. Landau and E. M. Lifschitz 2005 Elsevier

A comprehensive, and, currently definitive, textbook on the subject. This course will roughly cover the first five chapters of this book. If you want an overview of theoretical physics, you will want to start with this series of books.

The Feynman Lectures on Physics (Vol 3) R. P. Feynman, R. B. Leighton and M. Sands 1997 Narosa

If you've never come across quantum mechanics before, you might want to read through this marvellously insightful book. The first lecture, and much else, is based on this.