The Physics of Quantum Mechanics

This book grew out of classes given for many years to the second-year un
dergraduates of Merton College, Oxford. The University lectures that the
students were attending in parallel were restricted to the wave-mechanical
methods introduced by Schr¨odinger, with a very strong emphasis on the
time-independent Schr¨odinger equation. The classes had two main aims: to
introduce more wide-ranging concepts associated especially with Dirac and
Feynman, and to give the students a better understanding of the physical
implications of quantum mechanics as a description of how systems great
and small evolve in time.
While it is important to stress the revolutionary aspects of quantum
mechanics, it is no less important to understand that classical mechanics is
just an approximation to quantum mechanics. Traditional introductions to
quantum mechanics tend to neglect this task and leave students with two
independent worlds, classical and quantum. At every stage we try to explain
how classical physics emerges from quantum results. This exercise helps
students to extend to the quantum regime the intuitive understanding they
have developed in the classical world. This extension both takes much of the
mystery from quantum results, and enables students to check their results
for common sense and consistency with what they already know.
A key to understanding the quantum–classical connection is the study
of the evolution in time of quantum systems. Traditional texts stress instead
the recovery of stationary states, which do not evolve. We want students to
understand that the world is full of change– that dynamics exists– precisely
because the energies of real systems are always uncertain, so a real system is
never in a stationary state; stationary states are useful mathematical abstrac
tions but are not physically realisable. We try to avoid confusion between
the real physical novelty in quantum mechanics and the particular way in
which it is convenient to solve its governing equation, the time-dependent
Schr¨odinger equation.
Quantum mechanics emerged from efforts to understand atoms, so it
is natural that atomic physics looms large in traditional courses. However,
atoms are complex systems in which tens of particles interact strongly with
each other at relativistic speeds. We believe it is a mistake to plunge too soon
into this complex field. We cover atoms only in so far as we can proceed with
a reasonable degree of rigour. This includes hydrogen and helium in some
detail (including a proper treatment of Thomas precession), and a qualitative
sketch of the periodic table. But is excludes traditional topics such as spin
orbit coupling schemes and the physical interpretation of atomic spectra.
We devote a chapter to the adiabatic principle, which opens up a won
derfully rich range of phenomena to quantitative investigation. We also de
vote a chapter to scattering theory, which is both an important practical
application of quantum mechanics, and a field that raises some interesting
conceptual issues and makes one think carefully about how we compute re
sults in quantum mechanics.
When one sits down to solve a problem in physics, it’s vital to identify
the optimum coordinate system for the job– a problem that is intractable
in the coordinate system that first comes to mind, may be trivial in another
system. Dirac’s notation makes it possible to think about physical problems
in a coordinate-free way, and makes it straightforward to move to the chosen
coordinate system once that has been identified. Moreover, Dirac’s notation
brings into sharp focus the still mysterious concept of a probability ampli
tude. Hence, it is important to introduce Dirac’s notation from the outset,
and to use it for an extensive discussion of probability amplitudes and why
they lead to qualitatively new phenomena.
In the winter of 2008/9 the book was used as the basis for the second
year introductory quantum-mechanics course in Oxford Physics. At the out
set there was a whiff of panic in the air, emanating from tutors as well as
students. Gradually more and more participants grasped what was going
on and appreciated the intellectual excitement of the subject. Although the
f
inal feedback covered the full gamut of opinion from “incomprehensible” to
“the best course ever” there were clear indications that many students and
some tutors had risen to the challenge and gained a deeper understanding of
this difficult subject than was previously usual. Several changes to the text
of this revised edition were made in response to feedback from students and
tutors. It was clear that students needed to be given more time to come to
terms with quantum amplitudes and Dirac notation. To this end some work
on spin-half systems and polarised light has been introduced to Chapter 1.
The students found orbital angular momentum hard, and the way this is
handled in what is now Chapter 7 has been changed.
The major change from the first edition is Chapter 6, a new chapter
on composite systems. It starts with material transferred from the end of
Chapter 2 of the first edition, and then discusses entanglement, the Einstein
Podolski–Rosen experiment and Bell inequalities. Sections on quantum com
puting, density operators, thermodynamics and the measurement problem
follow. It is most unusual for the sixth chapter of a second-year textbook
to be able to take students to the frontier of human understanding, as this
chapter does. More minor changes include the addition of a section on the
Heisenberg picture to Chapter 4, the correction of a widespread misunder
standing about the singlet-triplet splitting in helium in Chapter 10, and the
addition of thermodynamics to the applications of the adiabatic principle
discussed in Chapter 11.
Problem solving is the key to learning physics and most chapters are
followed by a long list of problems. These lists have been extensively revised
since the first edition and printed solutions prepared. The solutions to starred
problems, which are mostly more-challenging problems, are now available
online1 and solutions to other problems are available to colleagues who are
teaching a course from the book. In nearly every problem a student will either
prove a useful result or deepen his/her understanding of quantum mechanics
and what it says about the material world. Even after successfully solving a
problem we suspect students will find it instructive and thought-provoking
to study the solution posted on the web.
We are grateful to several colleagues for comments on the first edition,
particularly Justin Wark for alerting us to the problem with the singlet
triplet splitting. Fabian Essler and John March-Russell made several con
structive suggestions. We thank our fellow Mertonian Artur Ekert for stim
ulating discussions of material covered in Chapter 6 and for reading that
chapter in draft form