Quantum Mechanics A Modern Depeloment

Although there are many textbooks that deal with the formal apparatus of
quantum mechanics and its application to standard problems, before the first
edition of this book (Prentice–Hall, 1990) none took into account the developments in the foundations of the subject which have taken place in the last
few decades. There are specialized treatises on various aspects of the foundations of quantum mechanics, but they do not integrate those topics into the
standard pedagogical material. I hope to remove that unfortunate dichotomy,
which has divorced the practical aspects of the subject from the interpretation and broader implications of the theory. This book is intended primarily
as a graduate level textbook, but it will also be of interest to physicists and
philosophers who study the foundations of quantum mechanics. Parts of the
book could be used by senior undergraduates.
The first edition introduced several major topics that had previously been
found in few, if any, textbooks. They included:
– A review of probability theory and its relation to the quantum theory.
– Discussions about state preparation and state determination.
– The Aharonov–Bohm effect.
– Some firmly established results in the theory of measurement, which are
useful in clarifying the interpretation of quantum mechanics.
– A more complete account of the classical limit.
– Introduction of rigged Hilbert space as a generalization of the more familiar
Hilbert space. It allows vectors of infinite norm to be accommodated
within the formalism, and eliminates the vagueness that often surrounds
the question whether the operators that represent observables possess a
complete set of eigenvectors.
– The space–time symmetries of displacement, rotation, and Galilei transformations are exploited to derive the fundamental operators for momentum,
angular momentum, and the Hamiltonian.
– A charged particle in a magnetic field (Landau levels).
– Basic concepts of quantum optics.
– Discussion of modern experiments that test or illustrate the fundamental
aspects of quantum mechanics, such as: the direct measurement of the
momentum distribution in the hydrogen atom; experiments using the single crystal neutron interferometer; quantum beats; photon bunching and
antibunching.
– Bell’s theorem and its implications.
This edition contains a considerable amount of new material. Some of the
newly added topics are:
– An introduction describing the range of phenomena that quantum theory
seeks to explain.
– Feynman’s path integrals.
– The adiabatic approximation and Berry’s phase.
– Expanded treatment of state preparation and determination, including the
no-cloning theorem and entangled states.
– A new treatment of the energy–time uncertainty relations.
– A discussion about the influence of a measurement apparatus on the environment, and vice versa.
– A section on the quantum mechanics of rigid bodies.
– A revised and expanded chapter on the classical limit.
– The phase space formulation of quantum mechanics.
– Expanded treatment of the many new interference experiments that are
being performed.
– Optical homodyne tomography as a method of measuring the quantum
state of a field mode.
– Bell’s theorem without inequalities and probability.
The material in this book is suitable for a two-semester course. Chapter 1
consists of mathematical topics (vector spaces, operators, and probability),
which may be skimmed by mathematically sophisticated readers. These topics
have been placed at the beginning, rather than in an appendix, because one
needs not only the results but also a coherent overview of their theory, since
they form the mathematical language in which quantum theory is expressed.
The amount of time that a student or a class spends on this chapter may vary
widely, depending upon the degree of mathematical preparation. A mathematically sophisticated reader could proceed directly from the Introduction to
Chapter 2, although such a strategy is not recommended.
The space–time symmetries of displacement, rotation, and Galilei transformations are exploited in Chapter 3 in order to derive the fundamental
operators for momentum, angular momentum, and the Hamiltonian. This
approach replaces the heuristic but inconclusive arguments based upon
analogy and wave–particle duality, which so frustrate the serious student. It
also introduces symmetry concepts and techniques at an early stage, so that
they are immediately available for practical applications. This is done without
requiring any prior knowledge of group theory. Indeed, a hypothetical reader
who does not know the technical meaning of the word “group”, and who
interprets the references to “groups” of transformations and operators as
meaning sets of related transformations and operators, will lose none of the
essential meaning.
A purely pedagogical change in this edition is the dissolution of the old
chapter on approximation methods. Instead, stationary state perturbation
theory and the variational method are included in Chapter 10 (“Formation of
Bound States”), while time-dependent perturbation theory and its applications
are part of Chapter 12 (“Time-Dependent Phenomena”). I have found this to
be a more natural order in my teaching. Finally, this new edition contains
some additional problems, and an updated bibliography.
Solutions to some problems are given in Appendix D. The solved problems
are those that are particularly novel, and those for which the answer or the
method of solution is important for its own sake (rather than merely being
an exercise).
At various places throughout the book I have segregated in double
brackets, [[ ··· ]], comments of a historical comparative, or critical nature.
Those remarks would not be needed by a hypothetical reader with no
previous exposure to quantum mechanics. They are used to relate my
approach, by way of comparison or contrast, to that of earlier writers, and
sometimes to show, by means of criticism, the reason for my departure from
the older approaches