esposito g., marmo g., sudarshan g. from classical to quantum mechanics.. formalism, foundations, applications

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 eikon

From Classical to Quantum Mechanics

This book provides a pedagogical introduction to the formalism, foundations and cations 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 approxi- mation; 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 entan- glement, 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.

appli-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 func- tional integrals.

Giuseppe Marmohas 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 Sudarshanhas 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.

i

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

iii

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press

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© G Esposito, G Marmo and E C G Sudarshan 2004

2004

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Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York www.cambridge.org

hardback

eBook (NetLibrary) eBook (NetLibrary) hardback

For Michela, Patrizia, Bhamathi, and Margherita, Giuseppina, Nidia

v

1.8 Wave-like behaviour and the Bragg experiment 27

2.3 Generating functions of canonical transformations 49

2.7 Hamilton equations associated with metric tensors 66

vii

Appendix 2.B Lie algebras and basic group theory 76

Appendix 2.C Some basic geometrical operations 80

Appendix 2.E From Newton to Euler–Lagrange 83

3.10 Eikonal approximation for the scalar wave equation 105

4.2 Uncertainty relations for position and momentum 128 4.3 Transformation properties of wave functions 131 4.4 Green kernel of the Schr¨ odinger equation 136 4.5 Example of isometric non-unitary operator 142

4.8 JWKB solutions of the Schr¨ odinger equation 155

Appendix 4.A Glossary of functional analysis 167

5.4 The Schr¨ odinger equation in a central potential 191

Contents ix

6.1 Stern–Gerlach experiment and electron spin 226

Appendix 6.A Lagrangian of a charged particle 242

Appendix 6.B Charged particle in a monopole field 242

7.1 Approximate methods for stationary states 244

7.9 Limiting cases of time-dependent theory 274

Appendix 7.A Convergence in the strong resolvent sense 295

8.2 Integral equation for scattering problems 302 8.3 The Born series and potentials of the Rollnik class 305

8.9 Bound states in the completeness relationship 323

Part II Weyl quantization and algebraic methods 337

9.3 Canonical operators and the Wintner theorem 341 9.4 Canonical quantization of commutation relations 343

9.7 From Weyl systems to commutation relations 348 9.8 Heisenberg representation for temporal evolution 350

9.10 Unitary operators and symplectic linear maps 357

9.12 The basic postulates of quantum theory 365

10.1 Algebraic formalism for harmonic oscillators 375 10.2 A thorough understanding of Landau levels 383

11.3 Rotations of angular momentum operators 409 11.4 Clebsch–Gordan coefficients and the Regge map 412 11.5 Postulates of quantum mechanics with spin 416

12.2 Transformation operators for the hydrogen atom 432

12.4 SU (1, 1) structures in a central potential 438

Contents xi

13.4 Hidden variables and the Bell inequalities 455

13.9 A modern perspective on the Wigner theorem 472

14.9 Identical particles in quantum mechanics 504

14.11 Statistical derivation of the Planck formula 519

15.1 The Schwinger formulation of quantum dynamics 526

15.3 Lagrangian formulation of quantum mechanics 533 15.4 Green kernel for quadratic Lagrangians 536

Appendix 15.A The Trotter product formula 548

16.3 Relativistic interacting particles Manifest covariance 555 16.4 The no-interaction theorem in classical mechanics 556

16.6 From particles to fields 564 16.7 The Kirchhoff principle, antiparticles and QFT 565

The present manuscript represents an attempt to write a modern graph on quantum mechanics that can be useful both to expert readers,i.e graduate students, lecturers, research workers, and to educated read-ers who need to be introduced to quantum theory and its foundations Forthis purpose, part I covers the basic material which is necessary to under-stand the transition from classical to wave mechanics: the key experiments

mono-in the development of wave mechanics; classical dynamics with sis on canonical transformations and the Hamilton–Jacobi equation; theCauchy problem for the wave equation, the Helmholtz equation and theeikonal approximation; physical arguments leading to the Schr¨odingerequation and the basic properties of the wave function; quantum dynam-ics in one-dimensional problems and the Schr¨odinger equation in a centralpotential; introduction to spin and perturbation theory; and scatteringtheory We have tried to describe in detail how one arrives at some ideas

empha-or some mathematical results, and what has been gained by introducing

mechan-as smooth and ‘natural’ mechan-as possible A broad range of topics are presented

in chapter 7, devoted to perturbation theory Within this framework, aftersome elementary examples, we have described the nature of perturbativeseries, with a brief outline of the various cases of physical interest: regu-lar perturbation theory, asymptotic perturbation theory and summabil-ity methods, spectral concentration and singular perturbations Chapter

xiii

8 starts along the advanced lines of the end of chapter 7, and describes alot of important material concerning scattering from potentials.

Advanced readers can begin from chapter 9, but we still recommendthat they first study part I, which contains material useful in later inves-tigations The Weyl quantization is presented in chapter 9, jointly withthe postulates of the currently accepted form of quantum mechanics TheWeyl programme provides not only a geometric framework for a rigor-ous formulation of canonical quantization, but also powerful tools for theanalysis of problems of current interest in quantum mechanics We havetherefore tried to present such a topic, which is still omitted in manytextbooks, in a self-contained form In the chapters devoted to harmonicoscillators and angular momentum operators the emphasis is on algebraicand group-theoretical methods The same can be said about chapter 12,devoted to algebraic methods for the analysis of Schr¨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 entangle-ment, hidden-variable theories and Bell inequalities; how to transfer thepolarization state of a photon to another photon thanks to the projectionpostulate, the production of statistical mixtures and phase in quantummechanics

Part III is devoted to a number of selected topics that reflect the thors’ taste and are aimed at advanced research workers: statistical me-chanics and black-body radiation; Lagrangian and phase-space formula-tions of quantum mechanics; the no-interaction theorem and the need for

au-a quau-antum theory of fields

The chapters are completed by a number of useful problems, althoughthe main purpose of the book remains the presentation of a conceptualframework for a better understanding of quantum mechanics Other im-portant topics have not been included and might, by themselves, be theobject of a separate monograph, e.g supersymmetric quantum mechan-ics, quaternionic quantum mechanics and deformation quantization But

we are aware that the present version already covers much more materialthan the one that can be presented in a two-semester course The ma-terial in chapters 9–16 can be used by students reading for a master orPh.D degree

Our monograph contains much material which, although not new by self, is presented in a way that makes the presentation rather original withrespect to currently available textbooks, e.g part I is devoted to and builtaround wave mechanics only; Hamiltonian methods and the Hamilton–Jacobi equation in chapter 2; introduction of the symbol of differential op-erators and eikonal approximation for the scalar wave equation in chapter3; a systematic use of the symbol in the presentation of the Schr¨odingerequation in chapter 4; the Pauli equation with time-dependent magnetic

it-Preface xvfields in chapter 6; the richness of examples in chapters 7 and 8; Weylquantization in chapter 9; algebraic methods for eigenvalue problems inchapter 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 anotherbook on quantum mechanics The last word is now with the readers

Our dear friend Eugene Saletan has kindly agreed to act as mercilessreviewer of a first draft His dedicated efforts to assess our work haveled 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 beenhelpful.

Rosario Peluso has produced a substantial effort to realize the figures

we needed The result shows all his skills with computer graphics andhis deep love for fundamental physics Charo Ivan Del Genio, GabrieleGionti, Pietro Santorelli and Annamaria Canciello have drawn the last set

of figures, with patience and dedication Several students, in particularAlessandro Zampini and Dario Corsi, have discussed with us so manyparts of the manuscript that its present version would have been unlikelywithout their constant feedback, while Andrea Rubano wrote notes whichproved very useful in revising the last version

Our Italian sources have not been cited locally, to avoid making helpful suggestions for readers who cannot understand textbooks written

un-in Italian Here, however, we can say that we relied un-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 ofNaples 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 vided invaluable scientific advice, while Suresh Kumar has been assisting

pro-us with TeX well beyond the call of duty

xvi

Part I

From classical to wave mechanics

metallic surfaces, X- and γ-ray scattering from gases, liquids and solids,

interference experiments, atomic spectra and the Bohr hypotheses, theexperiment of Franck and Hertz, the Bragg experiment, diffraction ofelectrons by a crystal of nickel (Davisson and Germer), and measure-ments 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 electromagneticphenomena and the laws of classical statistical mechanics could accountfor all known physical phenomena Still, it became gradually clear, afterseveral decades of experimental and theoretical work, that one has to for-mulate a new kind of mechanics, which reduces to classical mechanics in asuitable limit, and makes it possible to obtain a consistent description ofphenomena that cannot be understood within the classical framework It

is now appropriate to present a brief outline of this new class of ena, the systematic investigation of which is the object of the followingsections and of chapters 4 and 14

phenom-(i) In his attempt to derive the law for the spectral distribution of energydensity of a body which is able to absorb all the radiant energy falling

3

upon it, Planck was led to assume that the walls of such a body consist

of harmonic oscillators, which exchange energy with the electromagneticfield inside the body only via integer multiples of a fundamental quan-

tity ε0 At this stage, to be consistent with another law that had beenderived 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 introducedfor the first time These problems are part of the general theory of heatradiation (Planck 1991), and we have chosen to present them in somedetail in chapter 14, which is devoted to the transition from classical toquantum statistical mechanics

(ii) The crisis of classical physics, however, became even more evidentwhen attempts were made to account for the stability of atoms andmolecules For example, if an atomic system, initially in an equilibriumstate, is perturbed for a short time, it begins oscillating, and such os-cillations are eventually transmitted to the electromagnetic field in itsneighbourhood, so that the frequencies of the composite system can beobserved by means of a spectrograph In classical physics, independent

of the precise form of the forces ruling the equilibrium stage, one wouldexpect to be able to include the various frequencies in a scheme wheresome fundamental frequencies occur jointly with their harmonics In con-trast, the Ritz combination principle (see section 1.6) is found to hold,according to which all frequencies can be expressed as differences betweensome spectroscopic terms, the number of which is much smaller than thenumber of observed frequencies (Duck and Sudarshan 2000)

(iii) If one tries to overcome the above difficulties by postulating that theobserved frequencies correspond to internal degrees of freedom of atomicsystems, whereas the unknown laws of atomic forces forbid the occurrence

of higher order harmonics (Dirac 1958), it becomes impossible to accountfor 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 forusing a wave-like theory This property is ‘dual’ to a particle-like picture,which is instead essential to understanding the emission of electrons bymetallic 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 thatthe energy of the electromagnetic field consists of elementary quantities

W = hν, and it was as if these quanta of energy were localized in space

1.1 The need for a quantum theory 5(Einstein 1905) In a subsequent paper, Einstein analysed a gas com-posed of several molecules that was able to emit or absorb radiation, andproved that, in such processes, linear momentum should be exchangedamong the molecules, to avoid affecting the Maxwell distribution of ve-locities (Einstein 1917) This ensures, in turn, that statistical equilibrium

is reached Remarkably, the exchange of linear momentum cannot be tained, unless one postulates that, if spontaneous emission occurs, this

ob-happens along a well-defined direction with corresponding vector  u, so

that the linear momentum reads as

evidence that spontaneous emission is directional Under certain

circum-stances, electromagnetic radiation behaves as if it were made of

elemen-tary 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 deeplyintertwined with its corpuscular properties, and one comes to the conclu-sion that photons are, themselves, polarized For example, a light beamwith linear polarization should be viewed as consisting of photons each

of which is linearly polarized in the same direction Similarly, a lightbeam with circular polarization consists of photons that are all circularly

polarized One is thus led to say that each photon is in a given

polar-ization state The problem arises of how to apply this new concept to

the spectral resolution of light into its polarized components, and to therecombination of such components For this purpose, let us consider alight beam that passes through a tourmaline crystal, assuming that onlylinearly 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 sin2α passes through the crystal.

Let us assume, for simplicity, that the incoming beam consists of onephoton only, and that one can detect what comes out on the other side

of the crystal We will learn that, according to quantum mechanics, in anumber of experiments the whole photon is detected on the other side ofthe 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 opticalaxis, 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 fractionsin2α of the total number of experiments In other words, the photon is

found to have a probability sin2α of passing through the tourmaline, and

a probability cos2α of being, instead, absorbed by the tourmaline A deep

property, which will be the object of several sections from now on, is thenfound to emerge: when a series of experiments are performed, one can onlypredict 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 polarizationcan be viewed as being in part in a polarization state parallel to O, and

in part in a polarization state perpendicular toO In other words, a state

of oblique polarization results from a ‘superposition’ of polarizations thatare 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 thephoton is polarized in a direction parallel or perpendicular toO, because

the measurement process makes the photon be in one of these two larization states Such a theoretical description requires a sudden changefrom a linear superposition of polarization states (prior to measurement)

po-to a state where the polarization of the phopo-ton is either parallel or

per-pendicular to O (after the measurement).

Our brief outline has described many new problems that the generalreader is not expected to know already Now that his intellectual curiosityhas been stimulated, we can begin a thorough investigation of all suchtopics The journey is not an easy one, but the effort to understand whatleads 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, ourpath towards quantum theory, with emphasis on wave mechanics, willrely on the following properties

1.3 Photoelectric effect 7(i) The photoelectric effect, Compton effect and interference phenom-ena 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 con-cepts This is provided for photons by the Einstein identification (seeappendix 1.A)

tha-as is shown by interference experiments Although photons are mtha-assless,

one can associate to them a linear momentum  p = ¯ hk, and their energy

equals ¯hω = hν.

(ii) The form of the emission and absorption spectra, and the Bohr potheses (section 1.6) Experimental evidence of the existence of energylevels (section 1.7)

hy-(iii) The wave-like behaviour of massive particles postulated by de Broglie(1923) and found in the experiment of Davisson and Germer (1927, diffrac-tion of electrons by a crystal of nickel) For such particles one can performthe de Broglie identification

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, thehypothesis 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 photoelectriceffect

The photoelectric effect was discovered by Hertz and Hallwachs in 1887.The effect consists of the emission of electrons from the surface of a solidwhen 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

C I

R P

A

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 illu- minated electrode was photoelectrically sensitive, but for which the surrounding walls were not photosensitive.

V

B A

1.3 Photoelectric effect 9quoted therein should not be confused with the measuremt of the electroncharge, also due to Millikan).

(i) The electrons are emitted only if the frequency of the incoming

radia-tion is greater than a certain value ν0, which is a peculiar property of the

metal used in the experiment, and is called the photoelectric threshold.

(ii) The velocities of the electrons emitted by the surface range from 0

to a maximum value of vmax The kinetic energy corresponding to vmax

depends linearly on the frequency ν: Tmax= k(ν − ν0), k > 0 Tmax doesnot depend on the intensity of the incoming radiation

(iii) For a given value of the frequency ν of the incoming radiation, the

number of electrons emitted per cm2 per second is proportional to theintensity

These properties cannot be understood if one assumes that classicalelectromagnetic theory rules the phenomenon In particular, if one as-sumes that the energy is uniformly distributed over the metallic surface,

it is unclear how the emission of electrons can occur when the intensity

of the radiation is extremely low (which would require a long time beforethe electron would receive enough energy to escape from the metal) Theexperiments of Lawrence and Beans showed that the time lag between theincidence of radiation on a surface and the appearance of (photo)electrons

is less than 10−9 s

However, the peculiar emission of electrons is naturally accounted for,

if Planck’s hypothesis is accepted More precisely, one has to assume thatthe energy of radiation is quantized not only when emission or absorptionoccur, but can also travel in space in the form of elementary quanta

of radiation with energy hν Correspondingly, the photoelectric effect

should be thought of as a collision process between the incoming quanta ofradiation and the electrons belonging to the atoms of the metallic surface.According to this quantum scheme, the atom upon which the photon falls

receives, all at once, the energy hν As a result of this process, an electron can be emitted only if the energy hν is greater than the work function

If the inequality (1.3.1) is satisfied, the electron can leave the metallic

plate with an energy which, at the very best, is W = hν − W0, which

This agrees completely with the second law, (ii) Lastly, upon varying theintensity of the incoming radiation, the number of quanta falling upon thesurface in a given time interval changes, but from the above formulae it

is clear that the energy of the quanta, and hence of the electrons emitted,

is not affected by the intensity

In the experimental apparatus (see figure 1.1), ultraviolet or X-rays fallupon a clean metal cathode, and an electrode collects the electrons that

are emitted with kinetic energy T = hν − W0 If V0 is the potential forwhich the current vanishes, one has (see figure 1.3)

ex-Society) A linear relation is found between V0and ν, and the slope of the

corre-sponding line is numerically equal toh

e The intercept of such a line on the ν axis

is the lowest frequency at which the metal in question can be photoelectrically active.

1.4 Compton effect 11

The plot of V0(ν) is a straight line that intersects the ν-axis when ν = ν0.The slope of the experimental curve makes it possible to measure Planck’sconstant (for this purpose, Millikan used monochromatic light) The value

of the ratio h e is 4.14 × 10 −15 V s, with h = 6.6 × 10 −27 erg s.

Einstein made a highly non-trivial step, by postulating the existence

of elementary quanta of radiation which travel in space This was farmore than what Planck had originally demanded in his attempt to un-derstand black-body radiation Note also that, strictly, Einstein was notaiming to ‘explain’ the photoelectric effect When he wrote his funda-mental papers (Einstein 1905, 1917), the task of theoretical physicists wasnot quite that of having to understand a well-established phenomenology,since the Millikan measurements were made 10 years after the first Ein-stein paper Rather, Einstein developed some far-reaching ideas which,

in particular, can be applied to account for all known aspects of thephotoelectric effect Indeed, in Einstein (1905), the author writes asfollows

The wave theory of light, which operates with continuous spatial functions, has worked well in the representation of purely optical phe- nomena and will probably never be replaced by another theory It should

be kept in mind, however, that the optical observations refer to time erages rather than instantaneous values In spite of the complete exper- imental confirmation of the theory as applied to diffraction, reflection, refraction, dispersion, etc., it is still conceivable that the theory of light which operates with continuous spatial functions may lead to contradic- tions with experience when it is applied to the phenomena of emission and transformation of light.

av-It seems to me that the observations associated with blackbody radiation, fluorescence, the production of cathode rays by ultraviolet light, and other related phenomena connected with the emission or transformation of light are more readily understood if one assumes that the energy of light is discontinuously distributed in space In accordance with the assumption

to be considered here, the energy of a light ray spreading out from a point source is not continuously distributed over an increasing space but consists

of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed

as complete units.

1.4 Compton effect

Classically, a monochromatic plane wave of electromagnetic nature

car-ries momentum according to the relation p = E c Since E is quantized, one

Crystalslits

Lead collimating

ScattererIncident beam

Source

X-ray

θ

Fig 1.4 Experimental setup for the Compton experiment.

is naturally led to ask whether the momentum is carried in the form ofquanta with absolute value hν c The Compton effect (Compton 1923a,b)provides clear experimental evidence in favour of this conclusion, andsupports the existence of photons For this purpose, the scattering of

monochromatic X- and γ-rays from gases, liquids and solids is

stud-ied in the laboratory (see figure 1.4) Under normal circumstances, theX-rays pass through a material of low atomic weight (e.g coal) A spec-trograph made out of crystal collects and analyses the rays scattered in

a given direction One then finds, jointly with the radiation scattered

by means of the process we are going to describe, yet another ation which is scattered without any change of its wavelength There

radi-exist two nearby lines: one of them has the same wavelength λ as the incoming radiation, whereas the other line has a wavelength λ
> λ.

The line for which the wavelength remains unaffected can be accountedfor by thinking that the incoming photon also meets the ‘deeper un-derlying’ electrons of the scattering material For such processes, themass of the whole atom is involved, which reduces the value of the shift

λ
−λ significantly, so that it becomes virtually unobservable We are now

going to consider the scattering process involving the external electron

only.

Let us assume that the incoming radiation consists of photons having

frequency ν Let me be the rest mass of the electron, v its velocity after collision with the photon and let ν
be the frequency of the scatteredphoton The conservation laws that make it possible to obtain a theoret-ical description of the phenomenon are the conservation of energy andmomentum, and the description has to be considered within a relativistic

setting We denote by l the unit vector along the direction of the ing photon, and by  u the unit vector along the direction of emission of

incom-the scattered photon (see figure 1.5)

Fig 1.5. A photon with linear momentum  p0 collides with an electron at rest

and is scattered with momentum  p1 , while the electron recoils with momentum

which, jointly with Eq (1.4.1), are three equations from which one may

evaluate φ, the frequency ν
of the scattered X-ray, and the momentum

p of the electron as functions of the scattering angle θ Here attention

is focused on the formula for wavelength shift First, setting β ≡ v

m2 e

obtained from Eq (1.4.7) and the well-known relation between frequency

and wavelength: ν/c = 1/λ, ν
/c = 1/λ
Hence one finds

Interestingly, the wavelength shift is maximal when cos θ = −1, and it

vanishes when cos θ = 1 In the actual experiments, the scattered

pho-tons are detected if in turn, they meet an atom that is able to absorbthem (provided that such an atom can emit, by means of the photoelec-tric effect, an electron, the passage of which is visible on a photographicplate)

1.4 Compton effect 15

We can thus conclude that photons behave exactly as if they were

particles with energy hν and momentum hν c According to relativity ory, developed by Einstein and Poincar´e, the equation p = E c is a peculiarproperty of massless particles Thus, we can say that photons behave likemassless particles

the-The frequency shift is a peculiar property of a quantum theory whichrelies on the existence of photons, because in the classical electromagnetictheory no frequency shift would occur To appreciate this, let us considerthe classical description of the phenomenon On denoting the position

vector in R3by r, with Cartesian coordinates (x, y, z), and by k the wave vector with corresponding components (k x , k y , k z), the electric field of

the incoming plane wave of frequency ν = 2π ω may be written in the form



E =  E0cos

k · r − ωt, (1.4.10) where the vector  E0 has components

E0x , E0y , E0z

independent of

(x, y, z, t) Strictly, one has then to build a wave packet from these

el-ementary solutions of the Maxwell equations, but Eq (1.4.10) is all weneed to obtain the classical result The electric field which varies in spaceand time according to Eq (1.4.10) generates a magnetic field that alsovaries in space and time in a similar way This is clearly seen from one ofthe vacuum Maxwell equations, i.e (we do not present the check of the

Maxwell equations for the divergences of  E and  B, but the reader can

easily perform it)

k · r − ωt (1.4.15)

The coefficients of the sin function on the right-hand sides of Eqs.(1.4.13)–(1.4.15) are easily seen to be minus the components along the

X, Y, Z axes, respectively, of the vector product k ∧  E0, and hence onefinds

its oscillatory motion, the electron begins to radiate a field which, at a

distance R, has components with magnitude (Jackson 1975)

|  E
| = |  B
| = e

c2R¨r sin φ, (1.4.24)

1.5 Interference experiments 17

where c is the velocity of light and φ is the angle between the scattered

beam and the line along which the electron oscillates Substituting forthe acceleration, one finds

turn, radiate electromagnetic waves of the same frequency, leading to the

so-called Thomson scattering This is a non-relativistic scattering process,

which describes X-ray scattering from electrons and γ-ray scattering from protons For a particle of charge q and mass m, the total Thompson

scattering cross-section (recall that the cross-section describes basicallythe probability of the scattering process) reads as (Jackson 1975)

σT= 8π3

of photons? There are, indeed, various devices that can produce

interfer-ence fringes For example, a source S of monochromatic light illuminates

an opaque screen where two thin slits, close to each other, have been

produced In passing through the slits, light is diffracted On a plate L

located a distance from the slits, interference fringes are observed in thearea where overlapping occurs of the diffraction patterns produced from

the slits A and B, i.e where light is simultaneously received from A and

B (see figure 1.6).

Another device is the Fresnel biprism (Born and Wolf 1959): the

mono-chromatic light emitted from S is incident on two coupled prisms P1

B S

A

L

fringes

Fig 1.6 Diffraction pattern from a double slit.

and P2; light rays are deviated from P1 and P2 as if they were emitted

from two (virtual) coherent sources S
and S

As in the previous device,

interference fringes are observed where light emitted both from P1 and

P2 is collected (see figure 1.7)

elec-Interestingly, the Fresnel biprism makes it possible to produce

inter-ference fringes with electrons The source S is replaced by an electron

gun and the biprism is replaced by a metallic panel where a slit has beenproduced At the centre of the slit, a wire of silver-plated quartz is main-tained at a potential slightly greater than the potential of the screen Theelectrons are deviated by the electric field of the slit, and they reach thescreen as if they were coming from two different sources (see figure 1.8).For simplicity, we can consider the Fresnel biprism and talk about pho-tons, but of course this discussion can be repeated in precisely the sameway for electrons

How can one interpret the interference experiment in terms of photons?

It is clear that bright fringes result from the arrival of several photons,whereas no photons arrive where dark fringes are observed It thereforeseems that the various photons interact with each other so as to give

rise, on plate L, to an irregular distribution of photons, and hence bright

as well as dark fringes are observed If this is the case, what is going to

happen if we reduce the intensity of the light emitted by S until only one photon at a time travels from the source S to the plate L? The answer is that we have then to increase the exposure time of the plate L, but even-

tually we will find the same interference fringes as discussed previously.Thus, the interpretation based upon the interaction among photons isincorrect: photons do not interfere with each other, but the only possibleconclusion is that the interference involves the single photon, just as inthe case of the superposition for polarization However, according to a

particle picture, a photon (or an electron) starting from S and arriving

at L, either passes through A or passes through B We shall say that, if it

passes through A, it is in the state ψ A (the concept of state will be fully

defined in chapter 9), whereas if it passes through B it is in the state ψ B.But if this were the correct description of the possible options, we would

be unable to account for the interference fringes Indeed, if the photon is

in the state ψ A this means, according to what we said above, that slit B

can be neglected (it is as if it had been closed down) Under such tions, it should be possible for the photon to arrive at all points on plate

condi-L of the diffraction pattern produced from A, and hence also at those

points where dark fringes occur The same holds, with A replaced by B,

if we say that the photon is in the state ψ B This means that a third tion should be admissible, inconceivable from the classical viewpoint, and

op-different from ψ A and ψ B We shall then say that photons are in a state

ψ C , different from both ψ A and ψ B , but ψ Cshould ‘be related’, somehow,

with both ψ A and ψ B In other words, it is incorrect to say that photons

pass through A or through B, but it is as if each of them were passing, at the same time, through both A and B This conclusion is suggested by

the wave-like interpretation of the interference phenomenon: if only slit

A is open, there exists a wave A(x, y, z, t) in between the screen and L,

whereas, if only slit B is open, there exists a wave B(x, y, z, t) in between the screen and L If now both slits are opened up, the wave involved is neither A(r, t) nor B(r, t), but C(r, t) = A(r, t) + B(r, t).

But then, if the photon is passing ‘partly through A and partly through

B’, what should we expect if we place two photomultipliers F1 and F2in

front of A and B, respectively, and a photon is emitted from S (see figure 1.9)? Should we expect that F1 and F2 register, at the sametime, the passage of the photon? If this were the case, we would haveachieved, in the laboratory, the ‘division’ of a photon! What happens,however, is that only one of the two photomultipliers registers the pas-sage of the photon, and upon repeating the experiment several times

one finds that, on average, half of the events can be ascribed to F1 and

half of the events can be ascribed to F2 Does this mean that the

exis-tence of the state ψ C is an incorrect assumption? Note, however, thatthe presence of photomultipliers has made it impossible to observe theinterference fringes, since the photons are completely absorbed by suchdevices

At this stage, one might think that, with the help of a more ticated experiment, one could still detect which path has been followed

sophis-by photons, while maintaining the ability to observe interference fringes

For this purpose, one might think of placing a mirror S1 behind slit A, and another mirror S2 behind slit B (see figure 1.10) Such mirrors can

be freely moved by hypothesis, so that, by observing their recoil, one

1.5 Interference experiments 21

B A

B S

could (in principle) understand whether the photon passed through A or, instead, through slit B Still, once again, the result of the experiment is

negative: if one manages to observe the recoil of a mirror, no interferencefringes are detected The wave-like interpretation of the failure is as fol-lows: the recoil of the mirror affects the optical path of one of the rays,

to the extent that interference fringes are destroyed In summary, we canmake some key statements

(i) Interference fringes are also observed by sending only one photon at atime Thus, the single photon is found ‘to interfere with itself’

(ii) It is incorrect to say that the single photon passes through slit A or through slit B There exists instead a third option, represented by a state

ψ C , and deeply intertwined with both ψ A and ψ B

(iii) A measurement which shows whether the photon passed through A

or through B perturbs the state of the photon to such an extent that

no interference fringes are detected Thus, either we know which slit the

photon passed through, or we observe interference fringes We cannot

achieve both goals: the two possibilities are incompatible

1.6 Atomic spectra and the Bohr hypotheses

The frequencies that can be emitted by material bodies form their sion spectrum, whereas the frequencies that can be absorbed form theirabsorption spectrum For simplicity, we consider gases and vapours

emis-A device to obtain the emission and absorption spectra works as follows(see figure 1.11) Some white light falls upon a balloon containing gas or

a vapour; a spectrograph, e.g a prism P1, splits the light transmittedfrom the gas into monochromatic components, which are collected on a

plate L1 On L1 one can see a continuous spectrum of light transmittedfrom the gas, interrupted by dark lines corresponding to the absorptionfrequencies of the gas These dark lines form the absorption spectrum Toinstead obtain the emission spectrum, one has to transmit some energy

to the gas, which will eventually emit such energy in the form of magnetic radiation This can be achieved in various ways: by heating thematerial, by an electrical discharge, or by sending light into the material

electro-as we outlined previously By referring to this latter celectro-ase for simplicity, if

we want to analyse the emitted light, we shall perform our observations

in a direction orthogonal to that of the incoming light (to avoid being

disturbed by such light) A second prism P2 is inserted to decompose

the radiation emitted from the gas, and this is collected on plate L2 On

L2 one can see, on a dark background, some bright lines corresponding

1.6 Atomic spectra and the Bohr hypotheses 23

frequencies ν greater than a certain value ν1, it is also possible to observe,

in the emission spectrum, lines corresponding to frequencies smaller than

ν1 To account for the emission and absorption spectra, Bohr made someassumptions (Bohr 1913) that, as in the case of Einstein’s hypothesis,disagree with classical physics, which was indeed unable to account forthe properties of the spectra The basic idea was that privileged orbitsfor atoms exist that are stable If the electrons in the atom lie on one ofthese orbits, they do not radiate Such orbits are discrete, and hence the