Quantum mechanics originally developed for describing non-relativistic systems. It was a natural question to ask how the formalism can be extended to relativistic systems. Besides, it turned out that some of the properties of nonrelativistic systems can be understood naturally in the light of the relativistic theory. One example is the spin of the electron, which has to be introduced in an ad-hoc manner in non-relativistic theory, but appears naturally in the relativistic theory, as we will explain in this chapter. The aim of this chapter is to indicate, rather than elaborate, the kind of questions that can be asked and answered using hints of relativity. A full-edged relativistic theory takes us beyond quantum mechanics, as we argue in §15.1. We also clarify that when we talk of relativity, we only mean the special theory. The general theory of relativity is not discussed at all.

Conict between relativity and quantum mechanics

The special theory of relativity can be interpreted as a theory of the geometry of the 4D space-time, composed of the three spatial dimensions and time. Thus, time and space are treated on equal footing in special relativity.

This is what causes a conict with quantum mechanics. We have used time as a parameter with respect to which we study the evolution of systems. On the other hand, the spatial coordinates indicating the position are treated as operators, acting on the vector space of state vectors. There is no way that one can make a truly relativistic quantum theory if one maintains this status for time and space.

There are two ways out of the impasse. First, we might contemplate making time an operator as well, just as the position coordinates are. In §3.8, we have argued that this cannot be done. The second alternative is to make the spatial coordinates parameters, just like time is. In this case, one needs to study the evolution of objects in space and time, and the objects to be studied should therefore be some kind of functions of position and time. In physics, a function of space and time is called a field. Examples are electromagnetic and gravitation fields, which can depend on both position and time. In order to do relativistic quantum mechanics, i.e., do quantum mechanics in a way that is compatible with the special theory of relativity, one must therefore do some kind of field theory.

Classical mechanics is governed by Newton's laws of motion. It has been very successful, for over three centuries, in explaining motions of objects that we see around us. However, around the beginning of the twentieth century, when it came to understanding the properties of small systems like an atom, classical mechanics seemed inadequate. In this chapter, we will review the basic formulas of classical mechanics and indicate why it could not describe small systems.

Classical mechanics

Classical mechanics seeks a description of the motion of a particle, by specifying the path of motion of the particle, i.e., the position and velocity of the particle at any given instant. In the Newtonian formulation, this is done by invoking the idea of forces, and using Newton's second law of motion, which says that the rate of change of momentum of a particle is equal to the force that acts on the particle:

If we know the forces as a function of time, we can in principle solve this equation. It is a second-order differential equation in time, so we will need two initial conditions to solve the position vector r as a function of time. In particular, if we know the position r and velocity c = dr/dt at an initial instant, we can solve for the position and velocity of the particle at any instant, given the knowledge of the force F.

There are alternative ways of formulating classical mechanics. One of these is the Lagrangian formulation. In this formulation, one defines a function L of coordinates and velocities of all particles in the system, called the Lagrangian. The equation of motion is then given by

where the xa's denote different independent coordinates and ẋa's their time derivatives, i.e., the corresponding velocities.

Delta function

In earlier chapters, we have analyzed the quantum behavior of particles in many different kinds of Hamiltonian. Some of these Hamiltonians are quite contrived and used for exemplary purposes only. Some are realistic cases, like the case of the hydrogen atom in Chapter 9. In this chapter, we continue discussing realistic interactions, viz., interactions of particles with magnetic fields.

The Hamiltonian

Classical electrodynamics is described by Maxwell equations, which connect the derivatives of the electric field E and magnetic field B to the sources of the electromagnetic field, the charge density ρ and the current density J. All the quantities mentioned are in general functions of position and time. The behavior of a particle of charge q in such a field is described by the Lorentz force law. In the SI units, this force is given by

If we want to solve the classical problem of the path of a particle in an electromagnetic field, we can just set up Newton's equation of motion with the force shown above.

Clearly, this method does not help us solve the corresponding quantum mechanical problem, because the concept of force is not used in the formulation of quantum mechanics. We need to set up the Hamiltonian of the particle in an electromagnetic field. This cannot be done by using the field strength vectors E and B. Instead, we need to use the potentials A and ϕ, which are related to the fields by the relations

With these potentials, there is a very simple prescription for finding the Hamiltonian of a charged particle in an electromagnetic field.

We mentioned in Chapter 5 that the wavefunction of a system consisting of several identical particles has some special symmetries, related to the permutation of the particles. In this chapter, we discuss the issue in detail, throwing light on some consequences of the symmetry.

Many problems that we come across in quantum systems cannot be solved exactly. In fact, exactly solvable problems form only a small fraction of quantum problems that one encounters in realistic physical systems. In the absence of such exact solutions of the Schrödinger equation, one needs to resort to several methods providing approximate solutions. In this chapter we shall list some of these methods for time-independent Hamiltonians.

Variational method

The variational method is often used for a quick estimation of the ground state energy of a Hamiltonian whose exact eigenvalue and eigenstates are unknown. We first describe this method in general terms and then follow the discussion up with examples.

The method

To elucidate the technique, we first note that a trial wavefunction ψ trial for any quantum Hamiltonian H always satisfies the inequality

where E0 is the exact ground state energy of H. To see why this holds, first let us expand ψ triali in the eigenbasis of H:

where |n〉 is nth eigenstate of H with energy En. Since the eigenstates of H form a complete basis, this can always be done. The eigenstates |n〉, as well as the coeficients cn that appear in Eq. (11.2), can be determined only if we can solve the Hamiltonian problem exactly.

In this chapter, we shall discuss the formulation of scattering of a quantum particle from a potential. Throughout most of this chapter, we shall be interested in the state of the quantum particle for r ≫ a, where a is the range of the potential. Our main focus shall be, following directions from classical scattering theory, on computing the differential scattering cross-section for such a particle. The formalism necessary for this purpose is discussed in §13.1 and §13.2. This is followed by a discussion of the perturbative schemes in §13.3 and partial wave analysis in §13.4. Next, we consider the scattering cross-section due to the long-range Coulomb potential in §13.5.

Quantum mechanics urges us to look at anything in two complementary ways: as particle or as wave. The two descriptions are very different, and therefore intuitively some of the basic questions of quantum mechanics seem diffcult to reconcile. In this chapter, we will outline some of the problems with this reconciliation.

Waves vs particles

Let us start with something as simple as a double-slit experiment, described in §1.5. In Figure 1.2 (p. 13), we have given a schematic representation of the apparatus, and also the resulting interference pattern obtained on the screen. For the convenience of description, let us say that we are talking of interference of light.

It is easy to understand the experiment if we think in terms of waves. Light waves are coming from the source. Some part of the wavefront escapes through one hole and some escapes through the other. On the other side, waves arriving through the two slits superpose and produce the interference pattern.

What if we now want to describe the same phenomenon using light particles, or photons? Should we now say, in order to stay as close to the wave description as possible, that the photons have gone through both slits?

It might naively seem that there is nothing wrong with that. After all, there are many photons in a beam of light. So, it is certainly conceivable that some of them have gone through one slit and some through the other.

But if that happened, each photon would have ended up in a region directly behind one slit or the other, with maybe a little scatter because of the finite widths of the slits and also the finite size of the source. The pattern on the screen would have looked something like what we see in Figure 16.1a. But in reality, we observe the interference pattern which, on the plane of the screen, looks schematically like what has been shown in Figure 16.1b.

To escape the embarrassment, one will then have to say that a photon also must have gone through both slits, just like a wave can go through both. There is no way of saying which slit the photon has passed through, so we have to admit that the photon could have passed through both slits.

We have introduced many functions in Appendix C. Here, we discuss some of their properties. Defined as a solution of a certain differential equation, special functions are defined only up to an arbitrary multiplicative constant. However, in this appendix, we assume some particular normalization for each of these functions. Some of these preferred normalization conditions have already been mentioned in Appendix C. Others will be mentioned as and when necessary.

Generating functions

Different polynomials introduced earlier in the appendix can be seen as coeffcients in the power series expansion of some algebraic functions. To illustrate the point, we show here how to derive the generating functions of some sets of orthogonal polynomials.

In Chapter 6, we have discussed free particles. We carried out the discussion mostly in 1D space, although we mentioned that generalization to arbitrary dimensions is obvious and straightforward. We now start discussing particles in a potential. Problems of this kind and their solutions are quite sensitive to the number of spatial dimensions. In this chapter, we discuss particles subjected to a potential in a 1D space. Problems with higher-dimensional space will be discussed in later chapters of the book.

Stationary states

The Schrödinger equation, introduced in Chapter 3, admits particularly simple analytical solution in 1D for a variety of potentials V (x). In this chapter, we are going to look at some of them to gain an insight into the methods of analytical solution of the Schrödinger equation in 1D.

From this chapter onward, we will work almost exclusively in the coordinate representation.

The saddle-point method is widely used in many physics problems for approximately evaluating integrals in which the integrand has one or a few sharp peaks. The basic idea is that the region around the sharp peak gives the dominant contribution to the integral. So, if we can find the contribution around the peak, we have an estimate of the integral. The reason for the name of the method will be explained below.

If f(z) has a sharp maxima, the main contribution to the integral would come from there, as we said in the prelude to this appendix. There are now several issues to worry about. First, the original contour may not pass through the stationary points. This is not a big concern because we are considering integrals for which the integrand is analytic everywhere. Therefore, the result should be independent of the path, because the integral of a differentiable complex function around any closed path is zero. We need to deform the contour C to another contour C′ that passes through the stationary points.

The second issue is that for differentiable complex functions, there is really no maxima. The Cauchy-Riemann conditions imply that both real and imaginary parts of a complex analytic function satisfy the 2D Laplace's equation. Therefore, if the second derivative is positive along one direction, it must be negative along the perpendicular direction. Thus, all stationary points are saddle points, and the method gets the name saddle-point method because of the association with these points.

So, passing the contour through the saddle point is not enough to ensure that we will obtain the dominant contribution to the integral. The deformed contour C′ must go through the direction along which the function really attains a maxima and falls sharply at points away from the maxima. Our estimate of the integral will be best if we find the path where the maxima is the steepest, and for this reason the method is also called the method of steepest descent.

One of the main differences between 1D problems discussed in Chapter 7 and their higher-dimensional counterparts is the presence of rotational degrees of freedom. For reasons elaborated in Chapter 5, the description of such rotational motion necessitates the introduction of the concept of angular momentum, which will be the main focus of this chapter.

General considerations

It was mentioned in §5.7.2 that the rotation algebra admits both difierential and matrix representations. In order to discuss some general properties of both kinds of representations, it is customary to denote the angular momentum generators by J. The letter L would then apply only for the difierential generators or orbital angular momenta, and the letter S will be used for spin if necessary.

To reiterate, the angular momentum algebra is the set of the following commutator relations:

In a compact notation, we can write all three relations as

where the indices take the ‘values’ x, y, z, and εαβγ is the completely antisymmetric Levi-Civita symbol defined earlier in 5.7.

In order to discuss the angular momentum states, we have to first set up a basis on the vector space. Earlier in Chapter 2, we said that the eigenstates of a Hermitian operator can be used as a basis. We can choose, for this purpose, the eigenvectors of one of the three components of the angular momentum. These states will not be eigenstates of the other two components, since the operators for difierent components of J do not commute, as seen in Eq. (8.2). Conventionally, one chooses the eigenstates of Jz. This is, of course, no special assumption: we can always consider the basis as the eigenvectors of angular momentum component in one direction and call this direction to be the z direction.

But there is a problem. There is a lot of degeneracy among the eigenstates of Jz. Whenever there is degeneracy, the eigenstates cannot be uniquely defined, because any combination of states in a degenerate subspace of eigenvectors also qualifies as an eigenvector. For a basis, we need well-defined vectors, with no ambiguity in the defintion.

We have discussed 1D problems in Chapter 7. In this chapter, we discuss mainly 3D problems, although some of the generalities might apply to spaces whose dimensions are difierent.

Generalities

The stationary state form of the Schrödinger equation for a single particle system was given in Eq. (4.24, p. 86). For non-relativistic systems with velocity-independent potential energies, the general form of the Hamiltonian would be

for a particle of mass M. Accordingly, the Schrӧdinger equation for stationary states has the form

The Laplacian operator, ▿2, involves second-order derivatives with respect to the coordinates. This is the difierential equation we want to solve in this chapter for various choices of the potential.

Particle in a 3D box

The problem of a particle in an infinite rectangular-shaped 3D potential well is just an extension of the same problem in 1D. So, the solution can be borrowed from what we have already done in §7.2, but there will be some new features of the solutions which need some attention.

The potential is infinite except within the box defined by

Inside the box, the potential vanishes. The wavefunctions for the stationary states can be written in the form

and the difierential equation corresponding to each factor will be the same as the corresponding 1D problem.