An Introduction to AdvancedQuantum Physics

Hans P. PaarUniversity of California San Diego, USA

A John Wiley and Sons, Ltd., Publication

ayyappan9780470665091.jpg

An Introduction to AdvancedQuantum Physics

An Introduction to AdvancedQuantum Physics

Hans P. PaarUniversity of California San Diego, USA

A John Wiley and Sons, Ltd., Publication

This edition first published 2010 2010 John Wiley and Sons, Ltd.

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Library of Congress Cataloging-in-Publication DataPaar, Hans P.

An introduction to advanced quantum physics / Hans P. Paar.p. cm.

Includes bibliographical references and index.ISBN 978-0-470-68676-8 (cloth) – ISBN 978-0-470-68675-1 (pbk.)1. Quantum theory – Textbooks. I. Title.QC174.12P33 2010530.12 – dc22

2009054392

A catalogue record for this book is available from the British Library.

ISBN: H/bk 978-0-470-68676-8 P/bk 978-0-470-68675-1

Typeset in 10/12 Sabon by Laserwords Private Limited, Chennai, IndiaPrinted and bound in Great Britain by TJ International, Padstow, Cornwall

www.wiley.com

Contents

Preface ix

PART 1 Relativistic Quantum Physics 1

1 Electromagnetic Radiation and Matter 31.1 Hamiltonian and Vector Potential 31.2 Second Quantization 10

1.2.1 Commutation Relations 101.2.2 Energy 121.2.3 Momentum 171.2.4 Polarization and Spin 191.2.5 Hamiltonian 23

1.3 Time-Dependent Perturbation Theory 241.4 Spontaneous Emission 28

1.4.1 First Order Result 281.4.2 Dipole Transition 301.4.3 Higher Multipole Transition 32

1.5 Blackbody Radiation 361.6 Selection Rules 39

Problems 44

2 Scattering 492.1 Scattering Amplitude and Cross Section 492.2 Born Approximation 52

2.2.1 Schrödinger Equation 522.2.2 Green’s Function Formalism 522.2.3 Solution of the Schrödinger Equation 55

vi CONTENTS

2.2.4 Born Approximation 582.2.5 Electron-Atom Scattering 59

2.3 Photo-Electric Effect 632.4 Photon Scattering 67

2.4.1 Amplitudes 672.4.2 Cross Section 722.4.3 Rayleigh Scattering 732.4.4 Thomson Scattering 75Problems 78

3 Symmetries and Conservation Laws 813.1 Symmetries and Conservation Laws 81

3.1.1 Symmetries 813.1.2 Conservation Laws 82

3.2 Continuous Symmetry Operators 843.2.1 Translations 843.2.2 Rotations 86

3.3 Discrete Symmetry Operators 873.4 Degeneracy 89

3.4.1 Example 893.4.2 Isospin 90Problems 91

4 Relativistic Quantum Physics 934.1 Klein-Gordon Equation 934.2 Dirac Equation 95

4.2.1 Derivation of the Dirac Equation 954.2.2 Probability Density and Current 101

4.3 Solutions of the Dirac Equation, Anti-Particles 1044.3.1 Solutions of the Dirac Equation 1044.3.2 Anti-Particles 108

4.4 Spin, Non-Relativistic Limit and Magnetic Moment 1114.4.1 Orbital Angular Momentum 1114.4.2 Spin and Total Angular Momentum 1124.4.3 Helicity 1144.4.4 Non-Relativistic Limit 116

4.5 The Hydrogen Atom Re-Revisited 120Problems 124

5 Special Topics 1275.1 Introduction 1275.2 Measurements in Quantum Physics 1275.3 Einstein-Podolsky-Rosen Paradox 1295.4 Schrödinger’s Cat 133

CONTENTS vii

5.5 The Watched Pot 1355.6 Hidden Variables and Bell’s Theorem 137

Problems 140

PART 2 Introduction to Quantum Field Theory 143

6 Second Quantization of Spin 1/2 and Spin 1 Fields 1456.1 Second Quantization of Spin 12 Fields 145

6.1.1 Plane Wave Solutions 1456.1.2 Normalization of Spinors 1466.1.3 Energy 1486.1.4 Momentum 1516.1.5 Creation and Annihilation Operators 151

6.2 Second Quantization of Spin 1 Fields 155Problems 159

7 Covariant Perturbation Theory and Applications 1617.1 Covariant Perturbation Theory 161

7.1.1 Hamiltonian Density 1617.1.2 Interaction Representation 1657.1.3 Covariant Perturbation Theory 168

7.2 W and Z Boson Decays 1717.2.1 Amplitude 1717.2.2 Decay Rate 1737.2.3 Summation over Spin 1747.2.4 Integration over Phase Space 1797.2.5 Interpretation 181

7.3 Feynman Graphs 1837.4 Second Order Processes and Propagators 185

7.4.1 Annihilation and Scattering 1857.4.2 Time-Ordered Product 187Problems 193

8 Quantum Electrodynamics 1958.1 Electron-Positron Annihilation 1958.2 Electron-Muon Scattering 201

Problems 204

Index 207

Preface

Over the years, material that used to be taught in graduate school made itsway into the undergraduate curriculum to create room in graduate coursesfor new material that must be included because of new developments invarious fields of physics. Advanced Quantum Physics and its relativisticextension to Quantum Field Theory are a case in point. This book isintended to facilitate this process. It is written to support a second course inQuantum Physics and attempts to present the material in such a way that itis accessible to advanced undergraduates and starting graduate students inPhysics or Electrical Engineering.

This book consists of two parts.Part 1, comprising Chapters 1 through 5, contains the material for a

second course in Quantum Physics. This is where concepts from classicalmechanics, electricity and magnetism, statistical physics, and quantumphysics are pulled together in a discussion of the interaction of radiationand matter, selection rules, symmetries and conservation laws, scattering,relativistic quantum physics, questions related to the validity of quantumphysics, and more. This is material that is suitable to be taught as partof an undergraduate quantum physics course for physics and electricalengineering majors. Surprisingly, there is no undergraduate textbook thattreats this material at the undergraduate level, although it is (or ought to be)taught at many institutions.

In Part 2, comprising Chapters 6 through 8, we present elementary Quan-tum Field Theory. That this material should be studied by undergraduatesis controversial but I expect it will become accepted practice in the future.This material is intended for undergraduates that are interested in the topicsdiscussed and need it, for example, in a course on elementary particle physicsor condensed matter. Traditionally such a course is taught in the beginningof graduate school. When teaching particle physics to advanced undergrad-uate students I felt that the time was ripe for an elementary introductionto quantum field theory, concentrating on only those topics that have an

x PREFACE

application in particle physics at that level. I have also taught the materialin Chapters 6 through 8 to undergraduates and have found that they hadno problem in understanding the material and doing the homework.

It is hoped that the presentation of the material is such that any goodundergraduate student in physics or electrical engineering can follow it, andthat such a student will be motivated to continue the study of quantum fieldtheory beyond its present scope. Additionally, beginning graduate studentsmay also find it of use.

Please communicate suggestions, criticisms and errors to the author athpaar@ucsd.edu.

Hans P. PaarJanuary 2010

PREFACE xi

UNITS AND METRIC

It is customary in advanced quantum physics to use natural units. These arecgs units with Planck’s constant divided by 2π and the velocity of light setto unity. Thus we set � = c = 1.

In natural units we have for example that the Bohr radius of the hydrogenatom is 1/αm with α = e2/(4π ) the fine-structure constant and m the massof the electron. Likewise we have that the energy of the hydrogen atom isαm/(2n2) and the classical radius of the electron is α/m.

When results of calculations have to be compared with experiment,we must introduce powers of � and c. This can be done easily withdimensional analysis using, for example, that the product �c = 0.197 GeVfmand (�c)2 = 0.389 GeV2mb.

In summations I use the Einstein convention that requires one to sumover repeated indices, 1 to 3 for Latin and 1 to 4 for Greek letter indices.

Three-vectors are written bold faced such as x for the coordinate vector.We use the ‘East-coast metric’, introduced by Minkovski and made popularby Pauli in special relativity. Its name is obviously US-centric. In it, four-vectors have an imaginary fourth component. The alternative is to use ametric tensor as in General Relativity. This is called the ‘West-coast metric’,made popular by the textbook on Field Theory by Bjorken and Drell, butis overkill for our purposes. Thus xµ = (x, it), pµ = (p, iE), and kµ = (k, iω)with k the wave vector and ω the angular frequency. A traveling wave canbe written as exp ikx = exp i(kxx + kyy + kzz − ωt). Squaring four-vectors,we get for example p2 = pµpµ = p2 − E2 which is −m2 (unfortunatelynegative) and k2 = 0 for electromagnetic radiation. When integrating infour-dimensional space we use d4x which one might think is equal to d3xidtbut this is not so. By d4x we mean d3xdt (the West-coast metric does nothave this inconsistency, sorry). With this convention the gamma matrices inthe Dirac equation are all Hermitian.

We also find that we can write the commutation relations of p and x onthe one hand and E and t on the other in relativistically covariant formas [pµ, xν] = δµν/i where δµν is the Kronecker delta. Thus it is seen thatthe (at first sight odd) difference in sign of the two original commutationrelations is required by relativistic invariance.

The partial derivatives ∂/∂xµ will often be abbreviated to ∂µ. For example,the Lorentz condition ∇A + ∂φ/∂t = 0 can be written as ∂µAµ = 0, showingthat the Lorentz condition is relativistically covariant. Furthermore we havethat ∂2φ = ∂µ∂µφ = (∇2 − ∂2/∂t2)φ, an expression that is useful in writingdown a Lorentz invariant wave equation for the function φ.

xii PREFACE

In some equations the notation h.c. appears. This differs from the usualh.c. in that h.c. preserves the order of operators to which it is applied. Sothe h.c. of AB is B†A† while the h.c. of AB is A†B†.

We do not follow the convention of some textbooks in which e stands forthe absolute value of the electron charge; we use e = −1.602 × 10−19 C.

Part IRelativistic

Quantum Physics

1Electromagnetic Radiationand Matter

1.1 HAMILTONIAN AND VECTOR POTENTIAL

The classical Hamiltonian describing the interaction of a particle with massm and charge e with an electromagnetic field with vector potential A andscalar potential φ is

H = 12m

(p − eA)2 + eφ (1.1)

where p is the momentum of the particle. For example, A could be thevector potential of an external magnetic field while φ could be the Coulombpotential due to the presence of another charge. We do not follow theconvention where e stands for the absolute value of the electron charge; inour case e = −1.602 × 10−19 C for an electron. This Hamiltonian is derivedby casting the Lorentz force in the Hamiltonian formalism. It can also beobtained from the Hamiltonian of a free particle

H = p2

2m(1.2)

with the substitutions

p → p − eA H → H − eφ (1.3)

This is called the ‘Minimal Substitution’. The substitutions (1.3) can bewritten in covariant form as

pµ → pµ − eAµ (1.4)

An Introduction to Advanced Quantum Physics Hans P. Paar 2010 John Wiley & Sons, Ltd

4 ELECTROMAGNETIC RADIATION AND MATTER

with pµ = (p, iE) and Aµ = (A, iφ), c = 1. The word ‘minimal’ indicatesthat no additional terms that would in principle be allowed are included(experiment is the arbiter). An example would be a term that accounts forthe intrinsic magnetic moment of the particle. As is known from the study ofan atom in an external static magnetic field, the Hamiltonian Equation (1.1)already accounts for a magnetic moment associated with orbital angularmomentum. To obtain the Hamiltonian of the quantized system we use asalways the replacements

p → 1i∇ H (or E) → −1

i∂

∂t(1.5)

or in covariant formpµ → 1i

∂

∂xµ= 1

i∂µ (1.6)

The requirement that the substitutions in Equation (1.5) be covariant, thatis, that they can be written in the form of Equation (1.6), explains theminus sign in Equation (1.5). The substitutions are a manifestation of ‘FirstQuantization’ in which momenta and energies, and functions dependentupon these, become operators. They lead to the commutation relations

[pµ, xν] = 1i δµν (1.7)

where δµν is the Kronecker delta.The question arises as to how to quantize the electromagnetic field.

We know from the hypothesis of Planck and its extension by Einsteinin the treatment of black body radiation and the photo-electric effectrespectively that electromagnetic energy is quantized with quanta equal �ω.The replacements Equation (1.6) are of no use for an explanation. We willaddress this issue fully in the next subsection when we introduce ‘SecondQuantization’ in which the vector and scalar potentials and thus the electricand magnetic fields become operators. This is a prototype of relativisticquantum field theory.

In preparation for quantization of the electromagnetic field, we will brieflyreview the arguments that lead to the wave equation for A with φ = 0 (theCoulomb Gauge). Recall from classical electromagnetism that when thevector and scalar potentials are transformed into new ones by the Gaugetransformation

A → A′ = A + ∇χ φ → φ′ = φ − ∂χ∂t

(1.8)

HAMILTONIAN AND VECTOR POTENTIAL 5

that the values of electric and magnetic fields E and B do not change. Thisis so because under the Gauge transformation in Equation (1.8) we have

E = −∇φ − ∂A∂t

→ E′ = −∇φ′ − ∂A′

∂t= E (1.9)

andB = ∇ × A → B′ = ∇ × A′ = A (1.10)

where we used in Equation (1.9) that the order of ∇ and ∂/∂t can beexchanged while we used in Equation (1.10) that ∇ × ∇χ = 0 for all χ . Thefunction χ is arbitrary (unconstrained). The minus sign in Equation (1.8) isnecessary for E to remain unchanged under the Gauge transformation. Theminus sign also follows if we require that the relations Equation (1.8) canbe written in a covariant form

Aµ → A′µ = Aµ + ∂µχ (1.11)

Writing out the fourth component of Aµ = (A, iφ) in Equation (1.11),one finds the second relation of Equation (1.8). One can make a Gaugetransformation to find new A′ and φ′ such that they satisfy the Lorentzcondition

∇ · A′ + ∂φ′

∂t= ∂µA′µ = 0 (1.12)

as follows. If ∇ · A + ∂φ/∂t = f (x, t) �= 0 then the new A′ and φ′ will satisfythe Lorentz condition in Equation (1.12) if χ is required to satisfy theinhomogeneous wave equation

∇2χ − ∂2χ

∂t2= −f (x, t) (1.13)

The Lorentz condition is seen to be covariant as well. The Lorentz conditionsimplifies the differential equations for the vector and scalar potentials to

∇2A − ∂2A

∂t2= −4π j ∇2φ − ∂

2φ

∂t2= −4πρ (1.14)

or∂µ∂µAα = ∂2Aα = 0 (1.15)

showing that the wave equations for the potentials are covariant as theyshould be. This is the Lorentz Gauge.

6 ELECTROMAGNETIC RADIATION AND MATTER

There is more freedom left in the choice of A and φ in that a furtherGauge transformation as in Equation (1.11) can be made that results in thescalar potential φ being zero if we require that the new χ satisfies

φ = ∂χ∂t

and ∇2χ − ∂2χ

∂t2= 0 (1.16)

compared with Equation (1.13). This is the Coulomb or Radiation Gauge.The electric and magnetic fields are now given by

E = −∂A∂t

B = ∇ × A (1.17)

compared with the relations for E and B used in Equation (1.9) andEquation (1.10). The Lorentz condition in Equation (1.12) in the CoulombGauge is

∇ · A = 0 (1.18)We introduce the wave vector k and the corresponding angular frequencyω with k2 = ω2. It follows from Equation (1.18) for a traveling wave of theform A(x, t) = A0exp(ik · x − ωt) that

k · A = 0 (1.19)

so k and A are perpendicular to each other. From the first relation inEquation (1.17) we find that E = iωA, so E and A are parallel and thusk and E are also perpendicular. The second relation in Equation (1.17)gives B = ik × A = k × E/ω. This relation shows that B is in phase withE and is perpendicular to both k and E, so all three vectors E, B, k aremutually perpendicular. They form a right-handed triplet in that orderbecause E × B = E × (k × E)/ω = E2k/ω where we used that E · k = 0. Wedefine the polarization of the electromagnetic field as the direction of theelectric field. This is so because the effects of the electric field dominatethose of the magnetic field, for example in the exposure of photographicfilm. Because the electric field is perpendicular to its momentum we say thatthe electromagnetic field is transversely polarized. The cross product E × Bequals the Poynting vector S, which is in the direction of k as it should be.Because Gauge transformations do not change the physical properties ofthe electromagnetic field, the last conclusion about the orientation of E, Band k holds as well in the Lorentz Gauge and indeed in general. Note thatby transforming away φ we have not removed a Coulomb potential thatmight be present, we only removed the scalar potential associated with theelectromagnetic field described by the coupled E and B.

The reader is urged to review this material from the text used in clas-sical electromagnetism. A problem about Gauge transformations and theCoulomb Gauge is provided at the end of this chapter.

HAMILTONIAN AND VECTOR POTENTIAL 7

We stress that the sequence of two Lorentz transformations has left theelectric and magnetic fields unchanged, so the physical properties of thesystem have not been affected. This is in analogy with elementary classicalmechanics where it is shown that the potential energy U(r) is defined upto a constant because adding a constant to U does not change the forceF = −∇U and does not change the physical properties of the system.

The total Hamiltonian of the system consisting of a charged particle inan electromagnetic field with no other charges and currents but the onesassociated with the particle in Equation (1.1) consists of the part given inEquation (1.1) and the energy of the electromagnetic field

Eem = 18π∫

d3x(|E|2 + |B|2) (1.20)

Because E or B may be complex we use absolute values in the integrand.The homogeneous wave Equation (1.14) for A becomes

∇2A − ∂2A

∂t2= 0 (1.21)

This equation describes ‘free’ electromagnetic fields, that is, fields in theabsence of currents and charges. We now seek solutions of the homogeneouswave Equation (1.21). We introduce the four-vector kµ = (k, iω) where kis the wave vector and ω the corresponding angular frequency. Withxµ = (x, ict) we find that kx = k · x − ωt. Therefore we can write a travelingwave A(x, t) = A0exp(ik · x − ωt) as A(x) = A0exp(ikx). Traveling waveswith the vector potential of this form satisfy the wave equation providedthat k2 − ω2 = k2 = 0. This condition can be enforced in the solution byincluding a factor δ(k2) where δ is the Dirac δ function. The solution mustrepresent a three-dimensional vector so we make use of three mutuallyperpendicular unit vectors ελ(k) (λ = 1, 2, 3). The Lorentz condition in theCoulomb Gauge Equation (1.19) requires that k and A are perpendicular toeach other. If we choose ε1 and ε2 perpendicular to k (and to each other)and ε3 parallel to k, such that ε1, ε2, ε3 form a right-handed set of unitvectors, terms proportional to ε3 must be absent because of Equation (1.19).Because the wave Equation (1.21) is linear, its most general solution is alinear superposition of terms of the form aλ(k, ω) exp [i(kx − ωt)]ελ(k)δ(k2)λ = 1, 2. Our notation shows that the pre-factor aλ(k, ω) depends upon kand ω and that the unit vectors ελ(k) depend upon (the direction of) k. Themost general solution can thus be written as

A(x, t) =(

12π

)2 ∫d3k dω

2∑λ=1

aλ(k) ei(k·x−ωt)ελ(k) δ(k2) (1.22)

8 ELECTROMAGNETIC RADIATION AND MATTER

The ελ(k) are called polarization vectors. One can interpret this solu-tion as a four-dimensional version of the Fourier transform which isfamiliar from the solution of a wave equation in one dimension. The one-dimensional Fourier transform has a pre-factor 1/

√2π , hence the (1/2π )2

in Equation (1.22). This is the integral version of the Fourier transform.Substitution of Equation (1.22) in the wave Equation (1.21) shows thatEquation (1.22) is indeed a solution if the condition k2δ(k2) = 0 is satisfiedfor each term separately. This condition is satisfied for the Dirac-δ functionas in general x δ(x) = 0 because the Dirac-δ(x) function is even in x and it ismultiplied by the odd function x. The general solution in Equation (1.22)satisfies the condition in Equation (1.19) by construction. We know that Aand E are parallel, see Equation (1.17), so the direction of A as specified bya1(k, ω)ε1(k) and a2(k, ω)ε2(k) is the direction of polarization.

We can simplify Equation (1.22) by integrating over ω using the constraintprovided by the factor δ(k2) = 0. The condition k2 = k2 − ω2 = 0 leads tothe requirement that ω = ±|k|. This is also expressed by the property of theDirac-δ function

δ(k2) = δ(k2 − ω2) = 12ωk

[δ(|k| + ω) + δ(|k| − ω)] (1.23)

We introduce ωk = |k| and obtain

A(x, t)

=(

12π

)2 ∫ d3kdω2ωk

2∑λ=1

aλ(k, ω) ei(k·x−ωt)[δ(ωk + ω) + δ(ωk − ω)

]ελ(k)

=(

12π

)2 ∫ d3k2ωk

2∑λ=1

[aλ(k, −ωk) ei(k·x+ωkt) + aλ(k, ωk) ei(k·x−ωkt)

]ελ(k)

(1.24)We call the first term in Equation (1.24) with the negative value of ω thenegative energy solution, while we call the second term in Equation (1.24)with the positive value of ω the positive energy solution. It is conventionalto remove a factor 1/

√2π and a factor 1/

√2ωk and to add a factor

√4π in

Equation (1.24). This merely redefines the coefficients aλ. The reason for thefirst change is that the Fourier integral over d3k ought to be accompanied bya pre-factor 1/

√2π for each integration variable. The reason for the other

two changes will become clear in the next subsection.We would like to reinstate the compact notation exp(ikx) in Equation

(1.24). The positive energy term is already of this form but the negativeenergy term is not because the terms k · x and ωkt in the exponential donot have opposite signs. We can arrange for that by introducing a newintegration variable k′ = −k in the negative energy term only. This gives

HAMILTONIAN AND VECTOR POTENTIAL 9

three minus signs from d3k′ = −d3k and changes the signs of the integrationlimits in each of the three integrals. Exchanging the new upper and lowerintegration limits in each of the three integrals gives another three minussigns. The net result is that all minus signs cancel. We drop the prime in k′.The argument of aλ in the negative energy term will now be −k. Anotherway to see this is to consider that we are looking for solutions of the waveequation in a large box with periodic boundary conditions at its surface.Solutions with +k and −k have the same physical properties.

The expression for A becomes

A(x) =(1

2π

) 32 √

4π∫

d3k√2ωk

2∑λ=1

[aλ(−k, −ωk) e−ikxελ(−k) + aλ(k, ωk) eikxελ(k)

](1.25)

A is used to calculate E and B according to Equation (1.17). In the nextsection A, E and B become operators. The eigenvalues of E and B areobservable, so the operators E and B and therefore A must be Hermitianoperators. This means that A must be real quantity and we require thatA = A∗. We find thatA∗(x) =(

12π

) 32 √

4π∫

d3k√2ωk

2∑λ=1

[a∗λ(−k, −ωk) eikxε∗λ(−k) + a∗λ(k, ωk) e−ikxε∗λ(k)

](1.26)

Comparing Equation (1.26) with Equation (1.25) and using that the expo-nentials exp(±ikx) are orthogonal to each other, we find that∑

λ

aλ(−k, −ωk)ελ(−k) =∑

λ

a∗λ(k, ωk)ε∗λ(k) (1.27)

∑λ

aλ(k, ωk)ελ(k) =∑

λ

a∗λ(−k, −ωk)ε∗λ(−k) (1.28)

The two relations are each other’s complex conjugate, so there is justone condition on aλ and its complex conjugate. We will not attempt toformulate relations between individual terms in the sums over λ becausewe do not need those, and if we try it would lead to a left-handed set ofthree unit vectors ε or the appearance of minus signs in nasty places. UsingEquation (1.27) in Equation (1.25) we have the result

A(x) =(

12π

) 32 √

4π∫

d3k√2ωk

2∑λ=1

[aλ(k) eikxελ(k) + a∗λ(k) e−ikxε∗λ(k)

](1.29)