From Classical to
Quantum Mechanics

This book provides a pedagogical introduction to the formalism, foundations and applications of quantum mechanics. Part I covers the basic material that is necessary to an
understanding of the transition from classical to wave mechanics. Topics include classical
dynamics, with emphasis on canonical transformations and the Hamilton–Jacobi equation;
the Cauchy problem for the wave equation, the Helmholtz equation and eikonal approximation; and introductions to spin, perturbation theory and scattering theory. The Weyl
quantization is presented in Part II, along with the postulates of quantum mechanics. The
Weyl programme provides a geometric framework for a rigorous formulation of canonical
quantization, as well as powerful tools for the analysis of problems of current interest in
quantum physics. In the chapters devoted to harmonic oscillators and angular momentum
operators, the emphasis is on algebraic and group-theoretical methods. Quantum entanglement, hidden-variable theories and the Bell inequalities are also discussed. Part III is
devoted to topics such as statistical mechanics and black-body radiation, Lagrangian and
phase-space formulations of quantum mechanics, and the Dirac equation.
This book is intended for use as a textbook for beginning graduate and advanced
undergraduate courses. It is self-contained and includes problems to advance the reader’s
understanding.

Giampiero Esposito received his PhD from the University of Cambridge in
1991 and has been INFN Research Fellow at Naples University since November 1993. His
research is devoted to gravitational physics and quantum theory. His main contributions
are to the boundary conditions in quantum field theory and quantum gravity via functional integrals.
Giuseppe Marmo has been Professor of Theoretical Physics at Naples University
since 1986, where he is teaching the first undergraduate course in quantum mechanics.
His research interests are in the geometry of classical and quantum dynamical systems,
deformation quantization, algebraic structures in physics, and constrained and integrable
systems.
George Sudarshan has been Professor of Physics at the Department of Physics
of the University of Texas at Austin since 1969. His research has revolutionized the
understanding of classical and quantum dynamics. He has been nominated for the Nobel

Prize six times and has received many awards, including the Bose Medal in 1977.

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FROM CLASSICAL TO
QUANTUM MECHANICS
An Introduction to the Formalism, Foundations
and Applications

Giampiero Esposito, Giuseppe Marmo
INFN, Sezione di Napoli and
Dipartimento di Scienze Fisiche,
Universit`
a Federico II di Napoli

George Sudarshan
Department of Physics,
University of Texas, Austin

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  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521833240
© G. Esposito, G. Marmo and E. C. G. Sudarshan 2004
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2004
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For Michela, Patrizia, Bhamathi, and Margherita, Giuseppina, Nidia

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Contents

Preface
Acknowledgments
Part I From classical to wave mechanics

page xiii
xvi
1

1
1.1
1.2
1.3
1.4

1.5
1.6
1.7
1.8
1.9
1.10
1.11

Experimental foundations of quantum theory
The need for a quantum theory
Our path towards quantum theory
Photoelectric effect
Compton effect
Interference experiments
Atomic spectra and the Bohr hypotheses
The experiment of Franck and Hertz
Wave-like behaviour and the Bragg experiment
The experiment of Davisson and Germer
Position and velocity of an electron
Problems
Appendix 1.A The phase 1-form

3
3
6
7
11
17
22
26

27
33
37
41
41

2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

Classical dynamics
Poisson brackets
Symplectic geometry
Generating functions of canonical transformations
Hamilton and Hamilton–Jacobi equations
The Hamilton principal function
The characteristic function
Hamilton equations associated with metric tensors
Introduction to geometrical optics
Problems
Appendix 2.A Vector fields

43

44
45
49
59
61
64
66
68
73
74

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viii

Contents
Appendix
Appendix
Appendix
Appendix

2.B Lie algebras and basic group theory
2.C Some basic geometrical operations
2.D Space–time
2.E From Newton to Euler–Lagrange

76

80
83
83

3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11

Wave equations
The wave equation
Cauchy problem for the wave equation
Fundamental solutions
Symmetries of wave equations
Wave packets
Fourier analysis and dispersion relations
Geometrical optics from the wave equation
Phase and group velocity
The Helmholtz equation
Eikonal approximation for the scalar wave equation
Problems


86
86
88
90
91
92
92
99
100
104
105
114

4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10

Wave mechanics
From classical to wave mechanics
Uncertainty relations for position and momentum
Transformation properties of wave functions
Green kernel of the Schră

odinger equation
Example of isometric non-unitary operator
Boundary conditions
Harmonic oscillator
JWKB solutions of the Schră
odinger equation
From wave mechanics to Bohr–Sommerfeld
Problems
Appendix 4.A Glossary of functional analysis
Appendix 4.B JWKB approximation
Appendix 4.C Asymptotic expansions

115
115
128
131
136
142
144
151
155
162
167
167
172
174

5
5.1
5.2

5.3
5.4
5.5
5.6
5.7
5.8
5.9

Applications of wave mechanics
Reflection and transmission
Step-like potential; tunnelling eect
Linear potential
The Schră
odinger equation in a central potential
Hydrogen atom
Introduction to angular momentum
Homomorphism between SU(2) and SO(3)
Energy bands with periodic potentials
Problems

176
176
180
186
191
196
201
211
217
220


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ix

Appendix 5.A Stationary phase method
Appendix 5.B Bessel functions

221
223

6
6.1
6.2
6.3
6.4
6.5
6.6

Introduction to spin
Stern–Gerlach experiment and electron spin
Wave functions with spin
The Pauli equation
Solutions of the Pauli equation
Landau levels
Problems
Appendix 6.A Lagrangian of a charged particle

Appendix 6.B Charged particle in a monopole field

226
226
230
233
235
239
241
242
242

7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12

Perturbation theory
Approximate methods for stationary states
Very close levels
Anharmonic oscillator

Occurrence of degeneracy
Stark effect
Zeeman effect
Variational method
Time-dependent formalism
Limiting cases of time-dependent theory
The nature of perturbative series
More about singular perturbations
Problems
Appendix 7.A Convergence in the strong resolvent sense

244
244
250
252
255
259
263
266
269
274
280
284
293
295

8
8.1
8.2
8.3

8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13

Scattering theory
Aims and problems of scattering theory
Integral equation for scattering problems
The Born series and potentials of the Rollnik class
Partial wave expansion
The Levinson theorem
Scattering from singular potentials
Resonances
Separable potential model
Bound states in the completeness relationship
Excitable potential model
Unitarity of the Mă
oller operator
Quantum decay and survival amplitude
Problems

297
297
302

305
307
310
314
317
320
323
324
327
328
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Contents
Part II

Weyl quantization and algebraic methods

337

9
9.1
9.2
9.3
9.4
9.5

9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13

Weyl quantization
The commutator in wave mechanics
Abstract version of the commutator
Canonical operators and the Wintner theorem
Canonical quantization of commutation relations
Weyl quantization and Weyl systems
The Schră
odinger picture
From Weyl systems to commutation relations
Heisenberg representation for temporal evolution
Generalized uncertainty relations
Unitary operators and symplectic linear maps
On the meaning of Weyl quantization
The basic postulates of quantum theory
Problems

339
339
340
341
343

345
347
348
350
351
357
363
365
372

10
10.1
10.2
10.3
10.4
10.5
10.6

Harmonic oscillators and quantum optics
Algebraic formalism for harmonic oscillators
A thorough understanding of Landau levels
Coherent states
Weyl systems for coherent states
Two-photon coherent states
Problems

375
375
383
386

390
393
395

11
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8

Angular momentum operators
Angular momentum: general formalism
Two-dimensional harmonic oscillator
Rotations of angular momentum operators
Clebsch–Gordan coefficients and the Regge map
Postulates of quantum mechanics with spin
Spin and Weyl systems
Monopole harmonics
Problems

398
398
406
409
412
416

419
420
426

12
12.1
12.2
12.3
12.4
12.5
12.6

Algebraic methods for eigenvalue problems
Quasi-exactly solvable operators
Transformation operators for the hydrogen atom
Darboux maps: general framework
SU (1, 1) structures in a central potential
The Runge–Lenz vector
Problems

429
429
432
435
438
441
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Contents
13
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10

xi

From density matrix to geometrical phases
The density matrix
Applications of the density matrix
Quantum entanglement
Hidden variables and the Bell inequalities
Entangled pairs of photons
Production of statistical mixtures
Pancharatnam and Berry phases
The Wigner theorem and symmetries
A modern perspective on the Wigner theorem
Problems

445
446

450
453
455
459
461
464
468
472
476

Part III

477

Selected topics

14
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
14.10
14.11
14.12


From classical to quantum statistical mechanics
Aims and main assumptions
Canonical ensemble
Microcanonical ensemble
Partition function
Equipartition of energy
Specific heats of gases and solids
Black-body radiation
Quantum models of specific heats
Identical particles in quantum mechanics
Bose–Einstein and Fermi–Dirac gases
Statistical derivation of the Planck formula
Problems
Appendix 14.A Towards the Planck formula

479
480
481
482
483
485
486
487
502
504
516
519
522
522


15
15.1
15.2
15.3
15.4
15.5
15.6

Lagrangian and phase-space formulations
The Schwinger formulation of quantum dynamics
Propagator and probability amplitude
Lagrangian formulation of quantum mechanics
Green kernel for quadratic Lagrangians
Quantum mechanics in phase space
Problems
Appendix 15.A The Trotter product formula

526
526
529
533
536
541
548
548

16
16.1
16.2
16.3

16.4
16.5

Dirac equation and no-interaction theorem
The Dirac equation
Particles in mutual interaction
Relativistic interacting particles. Manifest covariance
The no-interaction theorem in classical mechanics
Relativistic quantum particles

550
550
554
555
556
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16.6
16.7

Contents
From particles to fields
The Kirchhoff principle, antiparticles and QFT

564
565


References
Index

571
588

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Preface

The present manuscript represents an attempt to write a modern monograph on quantum mechanics that can be useful both to expert readers,
i.e. graduate students, lecturers, research workers, and to educated readers who need to be introduced to quantum theory and its foundations. For
this purpose, part I covers the basic material which is necessary to understand the transition from classical to wave mechanics: the key experiments
in the development of wave mechanics; classical dynamics with emphasis on canonical transformations and the Hamilton–Jacobi equation; the
Cauchy problem for the wave equation, the Helmholtz equation and the
eikonal approximation; physical arguments leading to the Schră
odinger
equation and the basic properties of the wave function; quantum dynamics in one-dimensional problems and the Schră
odinger equation in a central
potential; introduction to spin and perturbation theory; and scattering
theory. We have tried to describe in detail how one arrives at some ideas
or some mathematical results, and what has been gained by introducing
a certain concept.
Indeed, the choice of a first chapter devoted to the experimental foundations of quantum theory, despite being physics-oriented, selects a set
of readers who already know the basic properties of classical mechanics and classical electrodynamics. Thus, undergraduate students should
study chapter 1 more than once. Moreover, the choice of topics in chapter 1 serves as a motivation, in our opinion, for studying the material
described in chapters 2 and 3, so that the transition to wave mechanics is
as smooth and ‘natural’ as possible. A broad range of topics are presented

in chapter 7, devoted to perturbation theory. Within this framework, after
some elementary examples, we have described the nature of perturbative
series, with a brief outline of the various cases of physical interest: regular perturbation theory, asymptotic perturbation theory and summability methods, spectral concentration and singular perturbations. Chapter
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xiv

Preface

8 starts along the advanced lines of the end of chapter 7, and describes a
lot of important material concerning scattering from potentials.
Advanced readers can begin from chapter 9, but we still recommend
that they first study part I, which contains material useful in later investigations. The Weyl quantization is presented in chapter 9, jointly with
the postulates of the currently accepted form of quantum mechanics. The
Weyl programme provides not only a geometric framework for a rigorous formulation of canonical quantization, but also powerful tools for the
analysis of problems of current interest in quantum mechanics. We have
therefore tried to present such a topic, which is still omitted in many
textbooks, in a self-contained form. In the chapters devoted to harmonic
oscillators and angular momentum operators the emphasis is on algebraic
and group-theoretical methods. The same can be said about chapter 12,
devoted to algebraic methods for the analysis of Schră
odinger operators.
The formalism of the density matrix is developed in detail in chapter 13,
which also studies some very important topics such as quantum entanglement, hidden-variable theories and Bell inequalities; how to transfer the
polarization state of a photon to another photon thanks to the projection
postulate, the production of statistical mixtures and phase in quantum
mechanics.

Part III is devoted to a number of selected topics that reflect the authors’ taste and are aimed at advanced research workers: statistical mechanics and black-body radiation; Lagrangian and phase-space formulations of quantum mechanics; the no-interaction theorem and the need for
a quantum theory of fields.
The chapters are completed by a number of useful problems, although
the main purpose of the book remains the presentation of a conceptual
framework for a better understanding of quantum mechanics. Other important topics have not been included and might, by themselves, be the
object of a separate monograph, e.g. supersymmetric quantum mechanics, quaternionic quantum mechanics and deformation quantization. But
we are aware that the present version already covers much more material
than the one that can be presented in a two-semester course. The material in chapters 9–16 can be used by students reading for a master or
Ph.D. degree.
Our monograph contains much material which, although not new by itself, is presented in a way that makes the presentation rather original with
respect to currently available textbooks, e.g. part I is devoted to and built
around wave mechanics only; Hamiltonian methods and the Hamilton–
Jacobi equation in chapter 2; introduction of the symbol of differential operators and eikonal approximation for the scalar wave equation in chapter
3; a systematic use of the symbol in the presentation of the Schrăodinger
equation in chapter 4; the Pauli equation with time-dependent magnetic
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Preface

xv

fields in chapter 6; the richness of examples in chapters 7 and 8; Weyl
quantization in chapter 9; algebraic methods for eigenvalue problems in
chapter 12; the Wigner theorem and geometrical phases in chapter 13;
and a geometrical proof of the no-interaction theorem in chapter 16.
So far we have defended, concisely, our reasons for writing yet another
book on quantum mechanics. The last word is now with the readers.

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Acknowledgments

Our dear friend Eugene Saletan has kindly agreed to act as merciless
reviewer of a first draft. His dedicated efforts to assess our work have
led to several improvements, for which we are much indebted to him.
Comments by Giuseppe Bimonte, Volodya Man‘ko, Giuseppe Morandi,
Saverio Pascazio, Flora Pempinelli and Patrizia Vitale have also been
helpful.
Rosario Peluso has produced a substantial effort to realize the figures
we needed. The result shows all his skills with computer graphics and
his deep love for fundamental physics. Charo Ivan Del Genio, Gabriele
Gionti, Pietro Santorelli and Annamaria Canciello have drawn the last set
of figures, with patience and dedication. Several students, in particular
Alessandro Zampini and Dario Corsi, have discussed with us so many
parts of the manuscript that its present version would have been unlikely
without their constant feedback, while Andrea Rubano wrote notes which
proved very useful in revising the last version.
Our Italian sources have not been cited locally, to avoid making unhelpful suggestions for readers who cannot understand textbooks written
in Italian. Here, however, we can say that we relied in part on the work
in Caldirola et al. (1982), Dell’Antonio (1996), Onofri and Destri (1996),
Sartori (1998), Picasso (2000) and Stroffolini (2001).
We are also grateful to the many other students of the University of
Naples who, in attending our lectures and asking many questions, made
us feel it was appropriate to collect our lecture notes and rewrite them
in the form of the present monograph.
Our Editor Tamsin van Essen at Cambridge University Press has provided invaluable scientific advice, while Suresh Kumar has been assisting
us with TeX well beyond the call of duty.


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Part I
From classical to wave mechanics

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1
Experimental foundations
of quantum theory

This chapter begins with a brief outline of some of the key motivations
for considering a quantum theory: the early attempts to determine the
spectral distribution of energy density of black bodies; stability of atoms
and molecules; specific heats of solids; interference and diffraction of
light beams; polarization of photons. The experimental foundations of
wave mechanics are then presented in detail, but in a logical order quite
different from its historical development: photo-emission of electrons by
metallic surfaces, X- and γ-ray scattering from gases, liquids and solids,
interference experiments, atomic spectra and the Bohr hypotheses, the
experiment of Franck and Hertz, the Bragg experiment, diffraction of
electrons by a crystal of nickel (Davisson and Germer), and measurements of position and velocity of an electron.
1.1 The need for a quantum theory

In the second half of the nineteenth century it seemed that the laws
of classical mechanics, developed by the genius of Newton, Lagrange,
Hamilton, Jacobi and Poincar´e, the Maxwell theory of electromagnetic
phenomena and the laws of classical statistical mechanics could account
for all known physical phenomena. Still, it became gradually clear, after
several decades of experimental and theoretical work, that one has to formulate a new kind of mechanics, which reduces to classical mechanics in a
suitable limit, and makes it possible to obtain a consistent description of
phenomena that cannot be understood within the classical framework. It
is now appropriate to present a brief outline of this new class of phenomena, the systematic investigation of which is the object of the following
sections and of chapters 4 and 14.
(i) In his attempt to derive the law for the spectral distribution of energy
density of a body which is able to absorb all the radiant energy falling
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4

Experimental foundations of quantum theory

upon it, Planck was led to assume that the walls of such a body consist
of harmonic oscillators, which exchange energy with the electromagnetic
field inside the body only via integer multiples of a fundamental quantity ε0 . At this stage, to be consistent with another law that had been
derived in a thermodynamical way and was hence of universal validity,
the quantity ε0 turned out to be proportional to the frequency of the
radiation field, ε0 = hν, and a new constant of nature, h, with dimension
[energy] [time] and since then called the Planck constant, was introduced
for the first time. These problems are part of the general theory of heat
radiation (Planck 1991), and we have chosen to present them in some

detail in chapter 14, which is devoted to the transition from classical to
quantum statistical mechanics.
(ii) The crisis of classical physics, however, became even more evident
when attempts were made to account for the stability of atoms and
molecules. For example, if an atomic system, initially in an equilibrium
state, is perturbed for a short time, it begins oscillating, and such oscillations are eventually transmitted to the electromagnetic field in its
neighbourhood, so that the frequencies of the composite system can be
observed by means of a spectrograph. In classical physics, independent
of the precise form of the forces ruling the equilibrium stage, one would
expect to be able to include the various frequencies in a scheme where
some fundamental frequencies occur jointly with their harmonics. In contrast, the Ritz combination principle (see section 1.6) is found to hold,
according to which all frequencies can be expressed as differences between
some spectroscopic terms, the number of which is much smaller than the
number of observed frequencies (Duck and Sudarshan 2000).
(iii) If one tries to overcome the above difficulties by postulating that the
observed frequencies correspond to internal degrees of freedom of atomic
systems, whereas the unknown laws of atomic forces forbid the occurrence
of higher order harmonics (Dirac 1958), it becomes impossible to account
for the experimental values of specific heats of solids at low temperatures
(cf. section 14.8).
(iv) Interference and diffraction patterns of light can only be accounted for
using a wave-like theory. This property is ‘dual’ to a particle-like picture,
which is instead essential to understanding the emission of electrons by
metallic surfaces that are hit by electromagnetic radiation (section 1.3)
and the scattering of light by free electrons (section 1.4).
(v) It had already been a non-trivial achievement of Einstein to show that
the energy of the electromagnetic field consists of elementary quantities
W = hν, and it was as if these quanta of energy were localized in space
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1.1 The need for a quantum theory

5

(Einstein 1905). In a subsequent paper, Einstein analysed a gas composed of several molecules that was able to emit or absorb radiation, and
proved that, in such processes, linear momentum should be exchanged
among the molecules, to avoid affecting the Maxwell distribution of velocities (Einstein 1917). This ensures, in turn, that statistical equilibrium
is reached. Remarkably, the exchange of linear momentum cannot be obtained, unless one postulates that, if spontaneous emission occurs, this
happens along a well-defined direction with corresponding vector u, so
that the linear momentum reads as
p=

W
hν
h
u=
u = u.
c
c
λ

(1.1.1)

In contrast, if a molecule were able to emit radiation along all possible
directions, as predicted by classical electromagnetic theory, the Maxwell
distribution of velocities would be violated. There was, therefore, strong
evidence that spontaneous emission is directional. Under certain circumstances, electromagnetic radiation behaves as if it were made of elementary quantities of energy W = hν, with speed c and linear momentum
p as in Eq. (1.1.1). One then deals with the concept of energy quanta of
the electromagnetic field, later called photons (Lewis 1926).

(vi) It is instructive, following Dirac (1958), to anticipate the description
of polarized photons in the quantum theory we are going to develop. It
is well known from experiments that the polarization of light is deeply
intertwined with its corpuscular properties, and one comes to the conclusion that photons are, themselves, polarized. For example, a light beam
with linear polarization should be viewed as consisting of photons each
of which is linearly polarized in the same direction. Similarly, a light
beam with circular polarization consists of photons that are all circularly
polarized. One is thus led to say that each photon is in a given polarization state. The problem arises of how to apply this new concept to
the spectral resolution of light into its polarized components, and to the
recombination of such components. For this purpose, let us consider a
light beam that passes through a tourmaline crystal, assuming that only
linearly polarized light, perpendicular to the optical axis of the crystal,
is found to emerge. According to classical electrodynamics, if the beam
is polarized perpendicularly to the optical axis O, it will pass through
the crystal while remaining unaffected; if its polarization is parallel to
O, the light beam is instead unable to pass through the crystal; lastly, if
the polarization direction of the beam forms an angle α with O, only a
fraction sin2 α passes through the crystal.
Let us assume, for simplicity, that the incoming beam consists of one
photon only, and that one can detect what comes out on the other side
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6

Experimental foundations of quantum theory

of the crystal. We will learn that, according to quantum mechanics, in a
number of experiments the whole photon is detected on the other side of
the crystal, with energy equal to that of the incoming photon, whereas,

in other circumstances, no photon is eventually detected. When a photon
is detected, its polarization turns out to be perpendicular to the optical
axis, but under no circumstances whatsoever shall we find, on the other
side of the crystal, only a fraction of the incoming photon. However, on
repeating the experiment a sufficiently large number of times, a photon
will eventually be detected for a number of times equal to a fraction
sin2 α of the total number of experiments. In other words, the photon is
found to have a probability sin2 α of passing through the tourmaline, and
a probability cos2 α of being, instead, absorbed by the tourmaline. A deep
property, which will be the object of several sections from now on, is then
found to emerge: when a series of experiments are performed, one can only
predict a set of possible results with the corresponding probabilities.
As we will see in the rest of the chapter, the interpretation provided
by quantum mechanics requires that a photon with oblique polarization
can be viewed as being in part in a polarization state parallel to O, and
in part in a polarization state perpendicular to O. In other words, a state
of oblique polarization results from a ‘superposition’ of polarizations that
are perpendicular and parallel to O. It is hence possible to decompose
any polarization state into two mutually orthogonal polarization states,
i.e. to express it as a superposition of such states.
Moreover, when we perform an observation, we can tell whether the
photon is polarized in a direction parallel or perpendicular to O, because
the measurement process makes the photon be in one of these two polarization states. Such a theoretical description requires a sudden change
from a linear superposition of polarization states (prior to measurement)
to a state where the polarization of the photon is either parallel or perpendicular to O (after the measurement).
Our brief outline has described many new problems that the general
reader is not expected to know already. Now that his intellectual curiosity
has been stimulated, we can begin a thorough investigation of all such
topics. The journey is not an easy one, but the effort to understand what
leads to a quantum theory will hopefully engender a better understanding

of the physical world.
1.2 Our path towards quantum theory
Unlike the historical development outlined in the previous section, our
path towards quantum theory, with emphasis on wave mechanics, will
rely on the following properties.
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1.3 Photoelectric effect

7

(i) The photoelectric effect, Compton effect and interference phenomena provide clear experimental evidence for the existence of photons.
‘Corpuscular’ and ‘wave’ behaviour require that we use both ‘attributes’,
therefore we need a relation between wave concepts and corpuscular concepts. This is provided for photons by the Einstein identification (see
appendix 1.A)
k · dx − ω dt =

1
p · dx − p0 dx0 .
¯h

(1.2.1a)

More precisely, light has a corpuscular nature that becomes evident thanks to the photoelectric and Compton effects, but also a wave-like nature
as is shown by interference experiments. Although photons are massless,
one can associate to them a linear momentum p = h
¯ k, and their energy
equals ¯hω = hν.
(ii) The form of the emission and absorption spectra, and the Bohr hypotheses (section 1.6). Experimental evidence of the existence of energy

levels (section 1.7).
(iii) The wave-like behaviour of massive particles postulated by de Broglie
(1923) and found in the experiment of Davisson and Germer (1927, diffraction of electrons by a crystal of nickel). For such particles one can perform
the de Broglie identification
p · dx − p0 dx0 = h
¯ k · dx − ω dt .

(1.2.1b)

It is then possible to estimate when the corpuscular or wave-like aspects
of particles are relevant in some physical processes.
1.3 Photoelectric effect
In the analysis of black-body radiation one met, for the first time, the
hypothesis of quanta: whenever matter emits or absorbs radiation, it does
so in a sequence of elementary acts, in each of which an amount of energy
ε is emitted or absorbed proportional to the frequency ν of the radiation:
ε = hν, where h is the universal constant known as Planck’s constant. We
are now going to see how the ideas developed along similar lines make
it possible to obtain a satisfactory understanding of the photoelectric
effect.
The photoelectric effect was discovered by Hertz and Hallwachs in 1887.
The effect consists of the emission of electrons from the surface of a solid
when electromagnetic radiation is incident upon it (Hughes and DuBridge
1932, DuBridge 1933, Holton 2000). The three empirical laws of such
an effect are as follows (see figures 1.1 and 1.2; the Millikan experiment
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Experimental foundations of quantum theory

C

A

I

P
R

Fig. 1.1. The circuit used in the Millikan experiment. The energy with which
the electron leaves the surface is measured by the product of its charge with
the potential difference against which it is just able to drive itself before being
brought to rest. Millikan was careful enough to use only light for which the illuminated electrode was photoelectrically sensitive, but for which the surrounding
walls were not photosensitive.

I

C
B
A

V0

V

Fig. 1.2. Variation of the photoelectric current with voltage, for given values of
the intensity.


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