We have laid down the basic techniques for solving second-order homogeneous differential equations. In this appendix, we apply these techniques to some differential equations of special interest, which appeared at various parts of the text. We perform series solutions to identify the functions that emerge as solutions of these equations. Note that for some of these differential equations, the point x = 0 is an ordinary point so that two independent series solutions are possible. For some others, the point x = 0 is a regular singular point. We do not point out which differential equation belongs to which category, because it is easy to make that decision from the look of any equation.

In this chapter, we describe the simplest possible system, viz., a free particle. To keep the notation simple, we will often pretend in this chapter that the particle moves only in one spatial dimension, so that there is only one coordinate x and its corresponding momentum p. However, the notation can be trivially modified to make the discussion applicable to motion of particles in 3D space.

Solution of free Schrödinger equation

We have introduced the Schrödinger equation in Chapter 3, which says that the time evolution of the state vector is governed by the Hamiltonian of the system:

If the Hamiltonian does not have any explicit time dependence, the solution of this equation is simply

as given in Eq. (4.15, p. 85). This solution is valid irrespective of whether the particle is free. For a free particle, the extra simplification comes from the fact that the Hamiltonian involves only the momentum operator and not the position operator. Therefore, H commutes with all components of the momentum, and so the momentum eigenstates are also energy eigenstates. This means that we can use the eigenstates of the momentum operator as the energy eigenstates and write

where Ep is the energy eigenvalue that depends on the momentum eigenvalue p. This feature can also be understood as a consequence of translational invariance.

In this chapter, we shall treat quantum mechanical problems where the system Hamiltonian either has explicit time dependence or has some parameter that is time-dependent. In either case, the Schrödinger equation is time-dependent and we are going to discuss techniques, either perturbative or exact, to solve such equations. In §12.1, we shall discuss the general formalism for treating such problems. Next, in §§ 12.2 and 12.3, we provide an analysis of the timedependent Hamiltonian for a two-level system. This will be followed by §12.4, where perturbative solutions to time-dependent Schrödinger equation shall be outlined. Finally, we analyze some aspects of Hamiltonians with periodic time dependence in §12.5.

Since a great many of the examples of quantum systems described in this book involve solutions of second-order differential equations, in this appendix we lay down the basic techniques for finding these solutions. We do not care to be completely general. We only discuss the kind of equations that we encounter for the purpose of this book.

Symmetries play a very important role in quantum mechanics, much more so than in classical mechanics. The reason for this is the vector space structure of the state vectors of a quantum system, as will be clear in the discussion of this chapter.

Symmetry and conservation

Our intuitive notion of symmetry is through geometrical objects. A square is more symmetric than a rectangle, whereas a circle is more symmetric than both. We can make the notion quantitative if we consider what are the operations on these geometrical objects that produce a result that is indistinguishable from the original one. For a rectangle, rotations in its plane by 180 or its multiples produce an identical figure. For a square, rotation by 90 and its multiples do the job, and this is why a square is more symmetric than a rectangle. For a circle, any rotation in its plane leaves the shape unchanged, and that is why the circle is more symmetric than either the square or the rectangle.

We can extend this notion to mathematical expressions as well. For example, suppose we are considering a particle in a 1D space, with a potential that obeys the condition

In this case, we can say that the potential is symmetric under the transformation that changes the sign of x, something that we will denote by x→ −x. From the Heisenberg equation of the system, Eq. (4.4, p. 83), we see that this transformation on x would also imply that p changes sign: p → −p. Since the kinetic energy term in the Hamiltonian is quadratic in p, it does not change under this transformation. Thus, the entire Hamiltonian is invariant under the transformation if the potential obeys Eq. (5.1).

Let us now discuss what happens if we apply a transformation to all vectors in a vector space and find that the physical consequences of the transformed vectors are indistinguishable from those of the original ones. In that case, we would call the transformation a symmetry transformation.

Quantum mechanics is an essential part of any physics undergraduate or graduate curriculum for its wide range of applicability to different branches of physics. It is taught at different levels, starting from second or third year undergraduate to final year graduate programmes. Through these courses students learn to appreciate the details of the subject at various levels. It is therefore difficult to have a single book which caters to students at all levels. This gap is precisely what we aim to fill here.

The book bore out of several courses on quantum mechanics taught by both the authors at several institutions such as the Saha Institute of Nuclear Physics, the University of Calcutta, and the Indian Association for the Cultivation of Science. These included both undergraduate and graduate level courses. These courses made us realize the necessity of a book which not only provides a clear introduction to the basic tenets of the subject but also presents a thorough discussion of several advanced topics. The latter seemed particularly difficult to find in a single book, which served as a motivation for writing this book.

We have not included a discussion of the old quantum theory in our book. We assumed that the reader would be familiar with that development, which was initiated by Planck and Einstein. That theory deals with particle-like properties of energy. The subject matter of our book is the other side of the wave–particle duality that is the hallmark of modern physics, i.e., the theory which treats matter as waves. The title of this book clearly shows that this theory, `quantum mechanics', will be the sole concern of this book.

Any book on quantum mechanics requires introduction to certain mathematical techniques such as group theory, linear algebra, and differential equations. We provide a somewhat detailed introduction to the first two topics since it is our understanding that these topics may not be taught in detail in other courses that physics students usually encounter before taking their first course on quantum mechanics. However, the last topic is usually well discussed in a generic physics curriculum; we have therefore relegated its discussion to the appendices, making every effort to keep it self-contained.

In this chapter, we shall discuss a few problems that are not usually addressed for a first course in quantum mechanics. However, these topics are becoming increasingly important for modern-day research in several areas of theoretical physics. We have tried to make the range of topics discussed as broad as possible; however, they are by no means exhaustive.

Coherent states

In this section, we shall discuss coherent states. Such states can be constructed for a wide variety of quantum mechanical systems. We shall, however, restrict our discussion to coherent states of harmonic oscillators, because the concept and analytic treatment of coherent states are particularly simple for harmonic oscillators.

Simple harmonic oscillators were discussed in Chapter 7. The concept of coherent states of such oscillators comes from noting that the annihilation operator a annihilates the ground state: a|0〉 = 0. It can be said that the ground state of the oscillator is an eigenstate of the annihilation operator with eigenvalue equal to zero.

As stated in the concluding sections of Chapter 1, in the light of the interference experiments, we would like to treat the states of a system as vectors. In this chapter, we discuss the general theory of vectors, and how the state vectors fit into that theoretical structure.

Introduction

During the first encounter with vectors, in high school, a student learns that a vector is something that has both magnitude and direction. This definition is unsatisfactory for reasons that we will discuss now. In order to apply this definition, one needs to first define the magnitude of a vector, which is not a vector. The direction is represented by the angles subtended by the vector with some fixed vectors. So we need to develop the notion of the angle between two vectors, and the angle is again not a vector. Alternatively, a vector is described in terms of its components, which are projections of the vector on some fixed axes, or basis vectors. This just means that the definition is incomplete, because now we need to define the basis vectors, i.e., the fixed axes. Rather, a vector should be defined by the fundamental properties of the set of vectors itself.

Let us then look for operations on vectors that produce vectors. We can take some hint from the set of numbers. Given two numbers, we can add them and get a number. We can also multiply them and get a number. There are certain properties that addition and multiplication of two numbers possess, for example, both should be commutative and both should be associative.

In elementary texts on vectors, we certainly learn how to add them, so addition is a property defined on vectors as well. However, multiplication cannot be defined. Although in high school or early college we learn about dot products and cross products of vectors, they do not qualify for our purpose. Dot product does not qualify because the result of dot product of two vectors is not a vector: it is a scalar. The reason cross product does not qualify is that it can be defined only in three dimensional (3D) space, where there can be a unique direction that is perpendicular to two given vectors.

In this chapter, we will formulate the basic equations of quantum mechanics that govern the time evolution of a quantum system.

Quantum bracket

In Chapter 1, we discussed several formulations for classical mechanics. We now see that none of them is suitable for quantum mechanics because they involve position, velocity or momentum, none of which can be defined for a particle because of the uncertainty principle.

What is worse, the discussion of Chapter 2 seems to indicate that dynamical variables should be represented by operators on a vector space. The Lagrangian and the Hamiltonian formalisms require taking derivatives with respect to dynamical variables. One cannot take derivatives with respect to an operator.

Faced with these problems, we may wonder if there is a formulation of classical mechanics where one does not have to take derivatives with respect to position and other dynamical variables. Indeed, such a formalism exists, and it involves constructs called Poisson brackets.

In classical mechanics, any dynamical variable F, i.e., any function F of positions x and momenta p, evolves according to the rule

where H is the Hamiltonian of the system (which, by the way, is a dynamical variable itself), and the Poisson bracket for any two dynamical variables is

defined to be

where xa indicate generalized coordinates, pa the corresponding momenta, and the sum is over all such generalized coordinates.

Obviously, this definition is meaningless in the realm of quantum mechanics, where xa and pa are operators in a Hilbert space. To obtain a suitable modification of the definition that can be used in quantum mechanics, let us first note that the Poisson bracket, defined above, has the following properties:

And, in addition, if there is a constant K, which can be seen as a special case of a dynamical variable, we should have

for any dynamical variable A. These are the properties that can be used to find all Poisson brackets, starting from the basic brackets

The important point is that as long as one needs to take Poisson brackets between dynamical variables which involve only positive integral powers of position and momenta, it is not necessary to invoke the definition of Eq. (3.2).

It is often said in science fiction as well as in books for science popularization that quantum mechanics makes everything probabilistic: nothing is certain. This is not quite true. The state vector of a quantum system evolves in time according to very precise rules, as shown in §3.3. Alternatively, we can think of operators to evolve in time according to well-determined rules, as described in §3.2. In this chapter, we show various ways of discussing the time evolution of quantum systems, often giving examples with a system whose state vectors belong to a 2D vector space.

Heisenberg equations and classical mechanics

We already commented that the Heisenberg equation looks very similar to the classical equation of motion written in terms of Poisson brackets. As we saw, it can be obtained from the classical equation by making the replacement

for any dynamical variable A, which is represented by an operator  in quantum mechanics. Once the brackets are evaluated, the Heisenberg equations are the same as the classical Hamilton's equations, although they do not mean exactly the same thing.

Take, for example, a free particle in 1D. The Hamiltonian is

Hamilton's equations of motion for a classical free particle are:

We assume that the electron carries quantised angular momentum as it moves around the positively charged proton in the hydrogen atom. This leads to a quantisation of the total energy of the electron. The Bohr model is a semiclassical model, combining classical orbits with the idea of quantising a physical quantity. This model explains some of the features of the real hydrogen atom (e.g. energy quantisation), but fails to explain other properties, such as the shape and the related binding properties.