Lectures on Quantum Mechanics

The following sections are included:

• Contents

• Preface

In this lecture, let us review some of the essential features of classical mechanics which we will use in the study of quantum mechanics.

In the following few lectures, we will recapitulate some of the mathematical concepts that we will need for a detailed understanding of quantum mechanics.

In the next few lectures, we will introduce the basic concepts of quantum mechanics. However, let us first discuss the reasons for going beyond the classical description of physical systems, which we have discussed in the first chapter.

In these lectures, we will apply the principles of quantum mechanics described in the last chapter to study some simple, one dimensional quantum mechanical systems. This will bring out the essential features of quantum mechanical systems and will also illustrate the differences of such systems from their corresponding classical counterparts.

The following sections are included:

• Harmonic oscillator in one dimension

• Matrix formulation

• Solution of the Schrödinger equation

• Hermite polynomials

• Discussion of the results

• Density matrix

• Planck’s law

• Oscillator in higher dimensions

• Selected problems

Symmetries play an important role in the study of physical systems – both in classical mechanics as well as in quantum mechanics. In these lectures we will start with a review of classical symmetry transformations before going into a discussion of symmetries in quantum mechanics.

As we have seen, angular momentum operator is the generator of rotations. In the next few lectures we will study the algebra of rotations in three dimensions by studying the algebraic properties of the angular momentum operators.

In earlier lectures, we solved some quantum mechanical systems in one dimension. In the next few lectures, we will discuss the solutions of some three dimensional systems with spherically symmetric potentials.

The hydrogen atom is an important physical system that can be exactly solved. In the next few lectures, we will study this system in detail to bring out various quantum mechanical features. We note that, unlike the systems we have studied so far, the hydrogen atom is a two particle system. It consists of a negatively charged electron interacting electromagnetically with a positively charged proton. Thus, let us first set up the formalism to study such systems.

So far, we have studied quantum mechanical systems that can be exactly solved. Most often, however, we are confronted with systems which are very difficult (if not impossible) to solve exactly. That is, the Hamiltonians for such systems cannot be diagonalized exactly in a simple way. In such a case, we look for approximate methods for finding the eigenvalues and eigenstates of the Hamiltonian. And sometimes, these approximate methods give results which are amazingly close to the true experimental values. There are various approximate methods which one can apply to different physical problems. We will discuss all these methods systematically starting with the variational method.

In the following lectures, we will study another very powerful approximation method, known as the WKB approximation, that is used in studying quantum mechanical systems subjected to complicated potentials.

Perturbation characterizes a series of iterative methods by which one obtains an approximate solution to a difficult problem. The idea is to introduce a small parameter ϵ into the problem and obtain a solution as a series in powers of ϵ. This method is very powerful and is used in all branches of physics.

Let us consider a system whose Hamiltonian H₀ is time independent and let us assume that we know how to solve for its eigenvalues and eigenfunctions exactly. We know that the eigenstates, in this case, will be stationary states.

From our study of the angular momentum in chapter 7, we have learned so far the following things…

Scattering is a valuable probe to study the structure of particles when we cannot directly see them. For example, it is through scattering experiments that we have learned that the hydrogen atom consists of a nucleus and an electron going around it. We know that electrons are point particles also from the results of scattering experiments. Furthermore, the scattering experiments have told us that protons consist of yet other constituents – the quarks. Therefore, we see that scattering is an essential tool in our understanding of the quantum nature of particles. We will, therefore, spend some time studying this topic. However, let me begin by recapitulating some of the features of scattering theory in classical physics.

We have so far studied only non-relativistic quantum mechanical systems. In the following lectures, we would like to extend the discussion to a relativistic quantum mechanical system describing a single particle.

Let us recapitulate very briefly what we have learnt so far in quantum mechanics. We have seen that, given a set of dynamical variables in classical mechanics, we go over to quantum mechanics by promoting these variables to operators. In particular the Hamiltonian or the total energy of a classical system also becomes an operator. These operators operate on an infinite dimensional Hilbert space labeled by time.

The following sections are included:

• Index