Quantum Mechanics

The following sections are included:

• Contents

• Preface

Quantum mechanics is a general theory. It is presumed to apply to everything, from subatomic particles to galaxies. But interest is naturally focussed on those phenomena that are most distinctive of quantum mechanics, some of which led to its discovery. Rather than retelling the historical development of quantum theory, which can be found in many books,* I shall illustrate quantum phenomena under three headings: discreteness, diffraction, and coherence. It is interesting to contrast the original experiments, which led to the new discoveries, with the accomplishments of modern technology…

Certain mathematical topics are essential for quantum mechanics, not only as computational tools, but because they form the most effective language in terms of which the theory can be formulated. These topics include the theory of linear vector spaces and linear operators, and the theory of probability. The connection between quantum mechanics and linear algebra originated as an apparent by-product of the linear nature of Schrödinger’s wave equation. But the theory was soon generalized beyond its simple beginnings, to include abstract “wave functions” in the 3N-dimensional configuration space of N paricles, and then to include discrete internal degrees of freedom such as spin, which have nothing to do with wave motion. The structure common to all of those diverse cases is that of linear operators on a vector space. A unified theory based on that mathematical structure was first formulated by P. A. M. Dirac, and the formulation used in this book is really a modernized version of Dirac’s formalism…

The following sections are included:

• Basic Theoretical Concepts
  • Mechanical aspect
  • Statistical aspect

• Conditions on Operators

• General States and Pure States
  • Pure states

• Probability Distributions
  • Further reading for Chapter 2
  • Problems

The results of Ch. 2 constitute what is sometimes called “the formal structure of quantum mechanics”. Although much has been written about its interpretation, derivation from more elementary axioms, and possible generalization, it has by itself very little physical content. It is not possible to solve a single physical problem with that formalism until one obtains correspondence rules that identify particular dynamical variables with particular operators. This will be done in the present chapter.

The fundamental physical variables, such as linear and angular momentum, are closely related to space–time symmetry transformations. The study of these transformations serves a dual purpose: a fundamental one by identifying the operators for important dynamical variables, and a practical one by introducing the concepts and techniques of symmetry transformations.

The following sections are included:

• Coordinate Representation

• The Wave Equation and Its Interpretation

• Galilei Transformation of Schrödinger’s Equation

• Probability Flux

• Conditions on Wave Functions

• Energy Eigenfunctions for Free Particles

• Tunneling

• Path Integrals
  • Classical limit of the path integral
  • Imaginary time and statistical mechanics
  • Discussion of the path integral method
  • Further reading for Chapter 4
  • Problems

The following sections are included:

• Momentum Representation

• Momentum Distribution in an Atom

• Bloch’s Theorem

• Diffraction Scattering: Theory
  • Diffraction by a periodic array
  • Double slit diffraction

• Diffraction Scattering: Experiment

• Motion in a Uniform Force Field

• Further reading for Chapter 5

• Problems

A harmonic oscillator is an object that is subject to a quadratic potential energy, which produces a restoring force against any displacement from equilibrium that is proportional to the displacement. The Hamiltonian for such an object whose motion is confined to one dimension is

where P is the momentum, Q is the position, and M is the mass. It is easily shown, by solving the classical equations of motion, that ω is the frequency of oscillation (in radians per unit time). The harmonic oscillator is important because it provides a model for many kinds of vibrating systems, including the electromagnetic field (see Ch. 19). Its solution also illustrates important techniques that are useful in other applications.

In Ch. 3 we showed that the generators of space-time symmetry operations are related to fundamental dynamical variables. In particular, the generators of rotations are the angular momentum operators (in units of ħ). Rotational symmetry plays a very important role in many physical problems, especially in atomic and nuclear physics. Many useful conclusions can be deduced from the transformation properties of various observables under rotations. Therefore it is useful to develop the theory of angular momentum and rotations in considerable detail.

In this chapter we return to the fundamental development of quantum theory. The formal structure of the theory, set forth abstractly in Ch. 2, involves two basic concepts, dynamical variables and states, and their mathematical representations. In Ch. 3 we determined the particular operators that correspond to particular dynamical variables. It would have been logical to proceed next to the discussion of the mathematical representation of particular physical states. That discussion has been delayed until now because the intervening four chapters have made it possible to treat some specific cases, instead of merely discussing state preparation and state determination in general terms…

A typical experimental run consists of state preparation followed by measurement. The former of these operations was analyzed in the preceding chapter, and the latter will now be treated. An analysis of measurement is required for completeness, but even more important, it turns out to be very useful in elucidating the correct interpretation of the concept of a state in quantum mechanics.

One of the distinctive characteristics of quantum mechanics, in contrast to classical mechanics, is the existence of bound states corresponding to discrete energy levels. Some of the conditions under which this happens will be discussed in this chapter.

The theory of the motion of a charged particle in a magnetic field presents several difficult and unintuitive features. The derivation of the quantum theory does not require the classical theory; nevertheless it is useful to first review the classical theory in order to show that some of these unintuitive features are not peculiar to the quantum theory, but rather that they are characteristic of motion in a magnetic field.

Because of their obvious importance, stationary states (energy eigenstates) have played a prominent role in most of the cases treated so far. But there are many phenomena in which the time dependence is the most interesting feature.

In the simplest case of a pure state and a time-independent Hamiltonian, it is possible to formally express the time dependence in terms of energy levels and the corresponding stationary states. The equation of motion for this case is . If the eigenvalues and eigenvectors of H are known, , and if we can expand the initial state vector as a series of these eigenvectors, , then the time-dependent state vector is simply given by . This method has important uses, but it is not adequate in all cases, and it is necessary to devise methods that treat time-dependent states in their own right, without attempting to reduce them to stationary states.

When symmetry transformations were first considered in Sec. 3.1, it was pointed out that a theorem due to Wigner proves that such a transformation may, in principle, be implemented by a unitary (linear) operator or by an antiunitary (antilinear) operator. An operator U is said to be unitary or antiunitary if the mapping is one-to-one and A linear operator L, by definition, satisfies the relation

Classical mechanics has been verified in a very wide domain of experience, so if quantum mechanics is correct it must agree with classical mechanics in the appropriate limit. Ideally we would like to exhibit quantum mechanics as a broader theory, encompassing classical mechanics as a special limiting case. Loosely stated, the limit must be one in which ħ is negligibly small compared with the relevant dynamical parameters. However, the matter is quite subtle. One cannot merely define ħ → 0 to be the classical limit. That limit is not well defined mathematically unless one specifies what quantities are to be held constant during the limiting process. Moreover, there are conceptual problems that are at least as important as the mathematical problem…

The following sections are included:

• Why Phase Space Distributions?

• The Wigner Representation
  • Time dependence of the Wigner function

• The Husimi Distribution
  • Indeterminacy relation for the Husimi distribution

• Further reading for Chapter 15

• Problems

The phenomenon of scattering was first mentioned in Sec. 2.1 of this book as an illustration of the fact that quantum mechanics does not predict the outcome of an individual measurement, but rather the statistical distribution or probabilities of all possible outcomes. Scattering is even more important than that illustration would indicate, much of our information about the interaction between particles being derived from scattering experiments. Entire books have been written on the subject of scattering theory, and this chapter will cover only the basic topics.

In this chapter we discuss the properties of systems of identical particles, following principles that were first expounded by Messiah and Greenberg (1964). Three successively stronger expressions of particle identity can be distinguished: (1) permutation symmetry of the Hamiltonian, which leads to degeneracies and selection rules, as does any symmetry; (2) permutation symmetry of all observables, which leads to a superselection rule; and (3) the symmetrization postulate, which restricts the states for a species of particle to be of a single symmetry type (either symmetric or antisymmetric). The stronger principles in this sequence cannot be deduced from the weaker principles, and some misleading arguments in the literature will be corrected.

Most of the problems and applications of quantum mechanics that we considered prior to Ch. 17 effectively involved only single particle states. This was due in part to a deliberate selection of simple problems. In other cases the degrees of freedom could be decoupled, resulting in an effective one-particle problem. The separation of the two-particle scattering problem into CM and relative coordinates is a good example of this decoupling. In this chapter we shall consider some many-body problems that cannot be reduced to effective one-body problems. The new features that emerge are of both practical and fundamental significance.

The development of electromagnetic theory from Coulomb’s law to Maxwell’s equations is accompanied by a change in the concept of the field from a merely passive agent that mediates the interactions between particles to a dynamical system in its own right. One may expect the dynamics of the EM field to be governed by quantum mechanics, as is the dynamics of particles. This expectation, however, cannot be justified a priori, and we shall need to seek experiments which can distinguish between classical and quantum electrodynamics…

In this chapter we shall show that some simple results about correlations turn out to have very profound and puzzling consequences about the nature of the world, as it is described by quantum mechanics. Certain ideas that seem natural, and indeed almost inevitable, from the point of view of special relativity have consequences that are contradicted by quantum mechanics. The starting point of the investigation was an argument by Einstein, Podolsky, and Rosen in 1935, but it was not pursued until 30 years later, by J. S. Bell, who obtained much more significant results that had not been anticipated in any previous work. Bell’s work has led to an interesting series of experiments, which have confirmed the numerical correctness of the predictions of QM, but have not made the implications of the results seem any less strange.

The following sections are included:

• Appendix A Schur’s Lemma

• Appendix B Irreducibility of Q and P

• Appendix C Proof of Wick’s Theorem

• Appendix D Solutions to Selected Problems

• Bibliography

• Index