SOLUTIONS TO THE EXERCISES IN QUANTUM MECHANICS II: … · 2019-01-03 · in D-dimension, alternate perturbation theories, an asymptot ic method for slowly varying potentials, Klein

SOLUTIONS MANUAL FORSOLUTIONS TO THE EXERCISES

IN QUANTUM MECHANICS II:

ADVANCED TOPICS

S. Rajasekar and R. Velusamy

by

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Boca Raton London New York

CRC Press is an imprint of theTaylor & Francis Group, an informa business

SOLUTIONS MANUAL FORSOLUTIONS TO THE EXERCISES

IN QUANTUM MECHANICS II:

ADVANCED TOPICS

S. Rajasekar and R. Velusamy

by

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CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

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Contents

Preface ix

About the Authors xiii

Chapter 1 Quantum Field Theory 1

Chapter 2 Path Integral Formulation 15

Chapter 3 Supersymmetric Quantum Mechanics 23

Chapter 4 Coherent and Squeezed States 35

Chapter 5 Berry’s Phase, Aharonov–Bohm and Sagnac Ef-

fects 49

Chapter 6 Phase Space Picture and Canonical Transfor-

mations 57

Chapter 7 Quantum Computers 65

Chapter 8 Quantum Cryptography 75

vii

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viii Contents

Chapter 9 Some Other Advanced Topics 81

Chapter 10 Quantum Technologies 91

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Preface

Quantum mechanics is the study of the behaviour of matter and energy at themolecular, atomic, nuclear levels and even at sub-nuclear level. This book isintended to provide a broad introduction to fundamental and advanced topicsof quantum mechanics. Volume I is devoted for basic concepts, mathematicalformalism and application to physically important systems. Volume II coversmost of the advanced topics of current research interest in quantum mechanics.Both the volumes are primarily developed as a text at the graduate level andalso as reference books. In addition to worked-out examples, numerous col-lection of problems are included at the end of each chapter. Complete detailsof solutions of the problems are provided in the CD attached with each vol-ume. Some of the problems serve as a mode of understanding and highlightingthe significances of basic concepts while others form application of theory tovarious physically important systems/problems. Developments made in recentyears on various mathematical treatments, theoretical methods, their applica-tions and experimental observations are pointed out wherever necessary andpossible and moreover they are quoted with references so that readers canrefer them for more details.

The volume I consists of 21 chapters and 7 appendices. Chapter 1 sum-marizes the needs for the quantum theory and its early development (oldquantum theory). Chapters 2 and 3 provide basic mathematical framework ofquantum mechanics. Schrödinger wave mechanics and operator formalism areintroduced in these chapters. Chapters 4 and 5 are concerned with the ana-lytical solutions of bound states and scattering states respectively of certainphysically important microscopic systems. The basics of matrix mechanics,Dirac’s notation of state vectors and Hilbert space are elucidated in chapter6. The next chapter gives the Schrödinger, Heisenberg and interaction pic-tures of time evolution of quantum mechanical systems. Description of timeevolution of ensembles by means of density matrix is also described. Chap-ter 8 is concerned with Heisenberg’s uncertainty principle. A brief account ofwave function in momentum space and wave packet dynamics are presentedin chapters 9 and 10 respectively. Theory of angular momentum is coveredin chapter 11. Chapter 12 is devoted exclusively for the theory of hydrogenatom.

Chapters 13− 16 are mainly concerned with approximation methods suchas time-independent and time-dependent perturbation theories, WKB methodand variational method. The elementary theory of elastic scattering is pre-

ix

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x Preface

sented in chapter 17. Identical particles are treated in chapter 18. The nextchapter presents quantum theory of relativistic particles with specific empha-size on Klein–Gordon equation, Dirac equation and its solution for a freeparticle, particle in a box (Klein paradox) and hydrogen atom. Chapter 20examines the strange consequences of role of measurement through the para-doxes of EPR and a thought experiment of Schrödinger. A brief sketch of Bell’sinequality and the quantum mechanical examples violating it are given. Con-sidering the rapid growth of numerical techniques in solving physical problemsand significances of simulation studies in describing complex phenomena, thefinal chapter is devoted for a detailed description of numerical computation ofbound state eigenvalues and eigenfunctions, transmission and reflection prob-abilities of scattering potentials, transition probabilities of quantum systemsin the presence of external fields and electronic distribution of atoms. Somesupplementary and background materials are presented in the appendices.

The pedagogic features volume I of the book, which are not usually foundin text books at this level, are the presentation of bound state solutions ofquantum pendulum, Pöschl–Teller potential, solutions of classical counter partof quantum mechanical systems considered, criterion for bound state, scat-tering from a locally periodic potential and reflectionless potential, modifiedHeisenberg relation, wave packet revival and and its dynamics, hydrogen atomin D-dimension, alternate perturbation theories, an asymptotic method forslowly varying potentials, Klein paradox, EPR paradox, Bell’s theorem andnumerical methods for quantum systems.

The volume II consists of 10 chapters. Chapter 1 describes the basic ideasof both classical and quantum field theories. Quantization of Klein–Gordonequation and Dirac field are given. The formulation of quantum mechanicsin terms of path integrals is presented in chapter 2. Application of it to freeparticle and linear harmonic oscillator are considered. In chapter 3 some illus-trations and interpretation of supersymmetric potentials and partners are pre-sented. A simple general procedure to construct all the supersymmetric part-ners of a given quantum mechanical systems with bound states is described.The method is then applied to a few interesting systems. The next chapteris concerned with coherent and squeezed states. Construction of these stateand their characteristic properties are enumerated. Chapter 5 is devoted forBerry’s phase, Aharonov–Bohm and Sagnac effects. Their origin, properties,effects and experimental demonstration are presented. The features of Wignerdistribution function are elucidated in chapter 6. In a few decades time, it ispossible to realize a computer built in terms of real quantum systems thatoperate in quantum mechanical regime. There is a growing interest on quan-tum computing. Basic aspects of quantum computing is presented in chapter7. Deutsch–Jozsa algorithm of finding whether a function is constant or not,Grover’s search algorithm and Shor’s efficient quantum algorithm for inte-ger factorization and evaluation of discrete logarithms are described. Chapter8 deals with quantum cryptography. Basic principles of classical cryptogra-phy and quantum cryptography and features of a few quantum cryptographic

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Preface xi

systems are discussed. A brief introduction to other advanced topics suchas quantum gravity, quantum Zeno effect, quantum teleportation, quantumgames, quantum cloning, quantum diffusion and quantum chaos is presentedin chapter 9. The last chapter gives features of some of the recent technolog-ical applications of quantum mechanics. Particularly, promising applicationsof quantum mechanics in ghost imaging, detection of weak amplitude objects,entangled two-photon microscopy, detection of small displacements, lithogra-phy, metrology and teleportation of optical images are briefly discussed.

During the preparation of this book we have received great supportsfrom many colleagues, students and friends. In particular, we are grate-ful to Prof. N. Arunachalam, Prof. K.P.N. Murthy, Prof. M. Daniel,Dr. S. Sivakumar, Mr. S. Kanmani, Dr. V. Chinnathambi, Dr. P. Philom-inathan, Dr. K. Murali, Dr. S.V.M. Sathyanarayana, Dr. K. Thamilmaran,Dr.T. Arivudainambi and Dr.V.S. Nagarathinam for their suggestions andencouragement. It is a great pleasure to thank Dr. V.M. Gandhimathi,Dr. V. Ravichandran, Dr. S. Jeyakumari, Dr. G. Sakthivel, Dr. M. Santhiah,Mr.R. Arun, Mr. C. Jeevarathinam, R. Jothimurugan, Ms. K. Abirami andMs. S. Rajamani for typesetting some of the chapters. Finally, we thank ourfamily members for their unflinching support, cooperation and encouragementduring the course of preparation of this work.

Tiruchirapalli S. Rajasekar

May, 2014 R. Velusamy

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About the Authors

Shanmuganathan Rajasekar was born in Thoothu-kudi, Tamilnadu, India in 1962. He received his B.Sc.and M.Sc. in Physics both from the St. Joseph’s Col-lege, Tiruchirapalli. In 1987, he received his M.Phil.in Physics from Bharathidasan University, Tiruchi-rapalli. He was awarded Ph.D. degree in Physics(Nonlinear Dynamics) from Bharathidasan Univer-sity in 1992 under the supervision of Prof.M. Laksh-manan. He was a visiting scientist during 1992-93 atthe Materials Science Division, Indira Gandhi Cen-tre for Atomic Research, Kalpakkam and worked on

multifractals and diffusion under Prof.K.P.N. Murthy. In 1993, he joined asa Lecturer at the Department of Physics, Manonmaniam Sundaranar Uni-versity, Tirunelveli. In 2003, the book on Nonlinear Dynamics: Integrability,Chaos and Patterns written by Prof.M. Lakshmanan and the author was pub-lished by Springer. In 2005, he joined as a Professor at the School of Physics,Bharathidasan University. With Prof.M. Daniel he edited a book on Non-linear Dynamics published by Narosa Publishing House in 2009. His recentresearch focuses on nonlinear dynamics with a special emphasize on nonlinearresonances. He has authored or coauthored more than 80 research papers innonlinear dynamics.

Ramiah Velusamy was born in Srivilliputhur, Tamil-nadu, India in the year 1952. He received his B.Sc.degree in Physics from the Ayya Nadar Janaki Am-mal College, Sivakasi in 1972 and M.Sc. in Physicsfrom the P.S.G. Arts and Science College, Coimbat-ore in 1974. He worked as a demonstrater in theDepartment of Physics in P.S.G. Arts and ScienceCollege during 1974-77. He received M.S. Degree inElectrical Engineering at Indian Institute of Tech-nology, Chennai in the year 1981. In the same year,he joined in the Ayya Nadar Janaki Ammal College

as an Assistant Professor in Physics. He was awarded M.Phil. degree in Physicsin the year 1988. He retired in the year 2010. His research topics are quantumconfined systems and wave packet dynamics.

xiii

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C H A P T E R 1

Quantum Field Theory

1.1 If ψ = ψ1 + iψ2, show that the Lagrange equation obtained by indepen-dent variation of ψ and ψ∗ are equivalent to those obtained by variationsof ψ1 and ψ2.

Lagrange equations for ψ and ψ∗ are

∂L∂ψ

− ∂

∂t

(

∂L∂ψ

)

−∑ ∂

∂xµ

(

∂L∂(∂ψ/∂xµ)

)

= 0 , (1.1)

∂L∂ψ∗ − ∂

∂t

(

∂L∂ψ∗

)

−∑ ∂

∂xµ

(

∂L∂(∂ψ∗/∂xµ)

)

= 0 . (1.2)

Since

ψ = ψ1 + iψ2 and ψ∗ = ψ1 − iψ2 (1.3)

we have

∂

∂ψ=

∂

∂ψ1− i

∂

∂ψ2,

∂

∂ψ∗ =∂

∂ψ1+ i

∂

∂ψ2. (1.4)

Substituting (1.3)-(1.4) in (1.1)-(1.2) and adding and subtracting theresulting equations we get

∂L∂ψ1

− ∂

∂t

(

∂L∂ψ1

)

−∑ ∂

∂xµ

(

∂L∂(∂ψ1/∂xµ)

)

= 0 , (1.5)

∂L∂ψ2

− ∂

∂t

(

∂L∂ψ2

)

−∑ ∂

∂xµ

(

∂L∂(∂ψ2/∂xµ)

)

= 0 . (1.6)

1.2 Obtain the Euler–Lagrange equation for

L =1

2ψ2 − 1

2m2ψ2 − 1

2

(

∂ψ

∂x

)2

.

Also obtain the corresponding Hamiltonian density.

1

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2 Solutions to the Exercises in Quantum Mechanics II: Advanced Topics

We obtain

∂

∂t

∂L∂ψ

= ψ ,∂

∂x

(

∂L∂(∂ψ/∂x)

)

= −∂2ψ

∂x2,

∂L∂ψ

= −m2ψ .

Then the Euler–Lagrange equation

∂L∂ψ

− ∂

∂t

(

∂L∂ψ

)

− ∂

∂x

(

∂L∂(∂ψ/∂x)

)

= 0

becomes

ψ − ψxx +m2ψ = 0 .

Next, writing the canonical momentum density as π =∂L∂ψ

= ψ we ob-

tain

H = πψ − L

= ψ2 −[

1

2ψ2 − 1

2m2ψ2 − 1

2

(

∂ψ

∂x

)2]

=1

2ψ2 +

1

2m2ψ2 +

1

2

(

∂ψ

∂x

)2

.

1.3 Show that the Lagrangian density L = iψ∗ψ − (2/2m)∇ψ∗ · ∇ψ −V (r, t)ψ∗ψ leads to the Schrödinger equation.

The Euler–Lagrange equation is

∂L∂φ

− ∂

∂t

(

∂L∂φ

)

− ∂

∂x

(

∂L∂(∂φ/∂x)

)

= 0 .

With φ = ψ and in three-dimension this equation is

∂L∂ψ

− ∂

∂t

(

∂L∂ψ

)

−3

∑

k=1

∂

∂xk

(

∂L∂(∂ψ/∂xk)

)

= 0 .

Now, we calculate the various terms in the above equation with givenL. We get

∂L∂ψ

= −V ψ∗ ,∂L∂ψ

= iψ∗ ,∂

∂t

(

∂L∂ψ

)

= i

(

∂ψ∗

∂t

)

and∑ ∂

∂xk

(

∂L∂(∂ψ/∂xk)

)

= ∇ ∂L∂(∇ψ)

= ∇(

− 2

2m∇ψ∗

)

= − 2

2m∇2ψ∗ .

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Quantum Field Theory 3

Substituting these in the Euler–Lagrange equation we get

−i∂ψ∗

∂t= −

2

2m∇2ψ∗ + V ψ∗ .

Its complex conjugate is

i∂ψ

∂t= −

2

2m∇2ψ + V ψ

which is the Schrödinger equation.

1.4 Find the Euler–Lagrange equation corresponding to the Lagrangian den-sity

L = −1

2

(

∂φ

∂xµ

∂φ

∂xµ+m2φ

)

, µ = 1, 2, 3, 4.

Using the given L in the Euler–Lagrange equation we obtain

2φ−m2φ = 0 ,

2= ∇2 − 1

c2∂2

∂t2

which is the Klein–Gordon equation. Hence, the given L is the La-grangian density for the Klein–Gordon equation.

1.5 Rewrite the Euler–Lagrange field equation in terms of Lagrangian L.

The Euler–Lagrange equation in three dimensional space is

∂L∂φ

− ∂

∂t

(

∂L∂φ

)

−3

∑

k=1

∂

∂xk

(

∂L∂(∂φ/∂xk)

)

= 0 ,

where xk = x, y, z.

In three dimension L is given by

L =

∫

V

L(φ, φ,∇φ, t) dτ .

We divide the volume into infinitesimally small cells with dτi being thevolume of ith cell. Then we can replace the volume integral in the aboveequation by summation over all the cells. That is,

L = limδτi→0

∑

L(

φi, φi,∇φi, t)

δτi ,

where L = Li is the Lagrangian density of ith cell. The derivatives ofL with respect to φ and φ at a particular point are called functional

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4 Solutions to the Exercises in Quantum Mechanics II: Advanced Topics

derivatives and are denoted as∂L

∂φand

∂L

∂φ. Assume that δφi and δφi

are zero for all i except for one cell say, jth cell. Then we define

∂L

∂φ= lim

δτj→0

1

δτj

∂L

∂φj

=∂L∂φ

−∑

k

∂

∂xk

(

∂L∂(∂φ/∂xk)

)

or

∂L∂φ

=1

δτ

[

∂L

∂φ+∑

k

∂

∂xk

(

∂L∂(∂φ/∂xk)

)

]

.

Next,

∂L

∂φ= lim

δτj→0

1

δτj

∂L

∂φj

=∂L∂φ

or

∂L∂φ

=1

δτ

∂L

∂φ.

Substituting the above expressions in the Euler–Lagrange field equation

∂L∂φ

− ∂

∂t

(

∂L∂φ

)

−3

∑

k=1

∂

∂xk

(

∂L∂(∂φ/∂xk)

)

= 0

we obtain

1

δτ

[

∂L

∂φ− ∂

∂t

(

∂L

∂φ

)]

= 0 .

That is,

∂L

∂φ− ∂

∂t

(

∂L

∂φ

)

= 0

which is in the form of classical Lagrangian equaion of motion of amechanical system.

1.6 Given the H =∫

[(2/2m)∇ψ∗ · ∇ψ + V ψ∗ψ] dτ obtain the equation ofmotion of the operator ψ in the Heisenberg picture.

We obtain

idψ

dt= [ψ,H ]

=

[

ψ,

∫ (

2

2m∇′ψ′∗ · ∇′ψ′ + V ψ′∗ψ′

)

dτ ′]

.

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Quantum Field Theory 5

Expanding the commutator bracket we get

idψ

dt= −

2

2m

∫

[ψ, ψ∗]∇2ψ dτ +

∫

V ψδ3(r− r) dτ

= − 2

2m∇2ψ + V ψ

which is the Schrödinger equation.

1.7 Consider two Lagrangian densities L and L which differ by the diver-gence of some function of the fields as L = L+∂µF

µ(φ). Show that theequations of motion obtained from L and L would be identical.

Let S and S be the action integrals for L and L respectively. Whenthe fields are varied infinitesimally by δφ, then the changes in the actiondefined by these two Lagrangian densities will be

δS − δS =

∫

∂µ

(

∂Fµ

∂φδφ

)

dX dt .

Using Gauss theorem we can write the integral in the above equation asan integral over the dΩ as

δS − δS =

∫

Ω

(

∂Fµ

∂φδφ

)

dΩ .

Since δφ vanishes on the boundary, we have δS − δS = 0 , that is,δS = δS. Hence, using the equation

δS =

∫ L/2

−L/2

∫ t2

t1

[

∂L∂φ

− ∂

∂t

(

∂L∂φ

)

− ∂

∂X

(

∂L∂(∂φ/∂X)

)]

δφ

dX dt

we will get identical equation of motion for both L and L.

1.8 If ψ = (ψ1 + iψ2)/√2, find the relations between the canonically con-

jugate momenta π, π, π1 and π2 corresponding to ψ, ψ∗, ψ1 and ψ2

respectively.

We have ψ = (ψ1 + iψ2)/√2, ψ∗ = (ψ1 − iψ2)/

√2. Then

ψ =1√2

(

ψ1 + iψ2

)

, ψ∗ =1√2

(

ψ1 − iψ2

)

, (1.1)

ψ1 =1√2

(

ψ + ψ∗)

, ψ2 =1√2 i

(

ψ − ψ∗)

(1.2)

and

∂L∂ψ

=∂ψ

∂ψ1

∂ψ1

∂ψ+

∂ψ

∂ψ2

∂ψ2

∂ψ. (1.3)

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6 Solutions to the Exercises in Quantum Mechanics II: Advanced Topics

Further,

∂ψ1

∂ψ=

1√2,

∂ψ2

∂ψ=

1

i√2, (1.4a)

∂ψ1

∂ψ∗=

1√2,

∂ψ2

∂ψ∗= − 1

i√2. (1.4b)

Substituting (1.4) in (1.3) we get

∂L∂ψ

= π =1√2

∂L∂ψ1

+1

i√2

∂L∂ψ2

=1√2(π1 − iπ2) (1.5a)

∂L∂ψ∗

= π =1

i√2(π1 + iπ2) . (1.5b)

1.9 Prove the commutation relations [Ck, Cl] =[

C†k, C

†l

]

= 0 and[

Ck, C†l

]

= δkl.

We have

Ck =

∫

u∗k(r)ψ(r, t) dτ , Cl =

∫

u∗l (r

′)ψ′(r′, t) dτ ′ .

We obtain

[Ck, Cl] =

∫

τ

∫

τ ′

u∗ku

∗l (ψψ

′ − ψ′ψ) dτ ′ dτ = 0

since ψψ′ − ψ′ψ = 0. Similarly, with

C†k =

∫

uk(r)ψ†(r, t) dτ , C†

l =

∫

ul(r′)ψ′†(r′, t) dτ ′

we have[

C†k, C

†l

]

=

∫

τ

∫

τ ′

uk(r)ul(r′)(

ψ†ψ′† − ψ′†ψ†) dτ ′ dτ .

Since [π, π′] = [iψ†, iψ′†] = −2[ψ†, ψ′†] = 0 we get

[

C†k, C

†l

]

= 0.

Next,[

Ck, C†l

]

=

∫

τ

∫

τ ′

u∗k(r)ul(r

′)(

ψψ′† − ψ′†ψ)

dτ ′ dτ

=

∫

τ

u∗k(r)ul(r) dτ

because ψψ′† − ψ′†ψ = δ(r − r′). Since uj are orthogonal set we get[

Ck, C†l

]

= δkl.

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Quantum Field Theory 7

1.10 Determine the relation between the vacuum state |0 = |00 · · · and thestate |n1n2 · · · .

We have a†1|n1 =√n1 + 1 |n1 + 1, a†1|0 =

√1 |1 and (a†1)

2|0 =√1√2 |2. Repeating this process we get

(a†1)n1 |0 =

√

n1! |n1 , (a†2)n2 |0 =

√

n2! |n2 .

From the above equation we write

(a†1)n1(a†2)

n2 · · · |000 · · · =√

n1!n2! · · · |n1n2 · · · .

Hence,

|n1n2 · · · =1√

n1!n2! · · ·(a†1)

n1(a†2)n2 · · · |000 · · · .

1.11 Show that Nk and Nl commute.

We obtain

[Nk, Nl] = NkNl −NlNk

= C∗kCkC

∗l Cl − C∗

l ClC∗kCk

= C∗kCkC

∗l Cl − C∗

kC∗l CkCl + C∗

kC∗l CkCl − C∗

l ClC∗kCk

= C∗k [Ck, C

∗l ]Cl + C∗

l [C∗k , Cl]Ck

= [C∗kCk − C∗

l Cl] δkl

= 0 .

1.12 Assuming∫

f ′[π, ∂ψ′/∂x′j ] d3x′ = i∂f/∂xj find out [π,∫

(∇′ψ′)2 d3x′]and [π,

∫

m2ψ′2 d3x′]

We obtain[

π,∫

(∇′ψ′)2 d3x′]

=∑

j

∫

[

π, (∂′jψ

′)2]

d3x′

=∑

j

∫

∂′jψ

′ [π, ∂′jψ

′]+[

π, ∂′jψ

′] ∂′jψ

′d3x′

=

∑

j

(i∂j∂jψ + i∂j∂jψ)

= 2i∑

∂2jψ

= 2i∇2ψ .

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8 Solutions to the Exercises in Quantum Mechanics II: Advanced Topics

Next,

[

π,∫

m2ψ′2 d3x′]

= m2

[

π,∫

ψ′2d3x′]

= m2

∫

d3x′ ψ′ [π, ψ′] + [π, ψ′]ψ′

= m2

∫

d3x′ iψ′δ(r− r′) + iδ(r− r′)ψ′

= 2im2

∫

d3x′δ(r− r′)ψ′

= −2im2ψ .

1.13 For the electromagnetic field in vacuum show that [A, π2] = 2iδ3(r −r′)π′.

Consider the x-component of [A, π2]. We obtain

[

Ax, π2x

]

= Axπ′x2 − π′

x2Ax

= π′xAxπ′

x − π′xAxπ′

x +Axπ′xπ′

x − π′x2Ax

= π′xAxπ′

x − (πxAx −Axπ′x)π

′x − π′

x2Ax

= π′xAxπ′

x + [Ax, π′x]π

′x − π′

x2Ax .

We have the following commutation relations

[Aj(r, t), Aj′ (r′, t)] = [πj(r, t), πj′(r

′, t)] = 0 ,

[Aj(r, t), πj′(r′, t)] = iδjj′δ

3(r− r′) , j, j′ = x, y, z .

Using the above relations we get

[

Ax, π2x

]

= π′xAxπ′

x + iδ3(r− r′)π′x − π′

x2Ax

= π′xAxπ′

x − π′xπ′

xAx + π′xπ′

xAx

+iδ3(r− r′)π′x − π′

x2Ax

= π′x(Axπ′

x − π′xAx) + iδ3(r − r′)π′

x

= 2iδ3(r− r′)π′x .

We obtain similar equations for the y and z-components. Then

[

A, π2]

= 2iδ3(r− r′)π′ .

K24777_SM_Cover.indd 20 13/11/14 6:20 PM

Quantum Field Theory 9

1.14 Find the time dependence of the operator akλ in the Heisenberg rep-resentation and show that the operators of the electric field E and themagnetic induction HB are given by

E = i∑

k

2∑

λ=1

(

2πω(k)

V

)1/2

εkλ eik·r

(

akλ − a†−kλ

)

and

HB = i∑

k

2∑

λ=1

(

2πc2

V ω(k)

)1/2

(k× εkλ) eik·r

(

akλ − a†−kλ

)

.

We have in the Heisenberg representation

id

dtakλ = [akλ, H ] .

We have

H =∑

k′

2∑

λ=1

ω(k)(

ak′λa†k′λakλ − a

†k′λak′λakλ

)

.

Since[

akλ, a†k′λ′

]

= δkk′δλλ′ we get

id

dtakλ = ω(k)

[(

a†kλakλ + 1

)

akλ − a†kλakλakλ

]

= ω(k)akλ .

Integrating the above equation we get

akλ(t) = akλ(0)e−iωkt , a

†−kλ(t) = a−kλ(0)e

iωkt .

Then from

A(r, t) =∑

k

2∑

λ=1

(

2πc2

V ω(k)

)1/2

εkλ eik·r

(

akλ(t) + a†−kλ(t)

)

we obtain

∂A

∂t= −i

∑

k

2∑

λ=1

(

2πc2

V ω(k)

)1/2

ω(k)εkλ eik·r

(

akλ(t)− a†−kλ(t)

)

.

Hence,

E = −1

c

∂A

∂t= i

∑

k

2∑

λ=1

(

2πω(k)

V

)1/2

εkλ eik·r

(

akλ(t)− a†−kλ(t)

)

.

Since HB = ∇×A, we get

HB = i∑

k

2∑

λ=1

(

2πc2

V ω(k)

)1/2

(k× εkλ) eik·r

(

akλ(t)− a†−kλ(t)

)

.

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10 Solutions to the Exercises in Quantum Mechanics II: Advanced Topics

1.15 Show that φn|∇|φm = −(mωnm/)rnm.

We have

p = mdr

dt=

m

i[r, H0] .

Then

φn|p|φm = m

iφn|rH0 −H0r|φm

or

−iφn|∇|φm = m

i(φn|rH0|φm − φn|H0r|φm) .

From the above we obtain

φn|∇|φm =m

2(Emφn|r|φm − Enφn|r|φm) .

= −m

(En − Em)

φn|r|φm

= −mωnm

φn|r|φm

= −mωnm

rnm

1.16 Given the L = ψ†γ0(∂ − m)ψ obtain the Hamiltonian density of theDirac field.

We obtain

H =

4∑

n=1

pnψn − L

=∑

iψ∗nψn −

(

ψγ0∂0ψ + iγk∂kψ −mψ†γ0ψ)

= iψ†ψ −(

iψγ0∂0ψ + iψγk∂kψ −mψ†γ0ψ)

= iψ†ψ −(

iψ†ψ + iψ†γ0γk∂kψ −mψ†βψ)

= −iψ†γ0γk∂kψ +mψ†βψ

= ψ†i∂0ψ ,

where i∂0 = −iα · ∇+mβ.

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Quantum Field Theory 11

1.17 Express the total energy H of Dirac field in terms of ladder operators.

We obtain

H =

∫

ψ†i∂0ψ d3x

=

∫

d3x∑

p,s

(

a†p,sf†p,s + bp,sg

†p,s

)

∑

p′,s′

ω′(

ap′,s′fp′,s′ + b†p′,s′gp′,s′

)

=∑

p,s,p′,s′

ω′(

a†p,sap′,s′δp,p′δs,s′ − bp,sb†p′,s′δp,p′δs,s′

)

.

That is,

H =∑

p,s

ω(

a†p,sap,s − bp,sb†p,s

)

=∑

p,s

ω(

a†p,sap,s + b†p,sbp,s − 1)

.

1.18 For the electromagnetic field in vacuum determine H =∫

H d3x where

H = 2πc2π2 +1

8π(∇×A)2 − cπ · ∇φ.

We obtain

H =

∫

H d3x

=

∫ [

2πc2π2 +1

8π(∇×A)2 − cπ · ∇φ

]

d3x .

The last integral is

∫

π · ∇φd3x = πφ|∞−∞ −∫ ∞

−∞φ∇ · πd3x = 0

since ∇ · π = 0 and π vanishes rapidly at ∞. Then

H =

∫ [

2πc2π2 +1

8π(∇×A)2

]

d3x

=1

8π

∫

(

E2 +H2B

)

d3x

which is the usual expression for the total energy in the electromagneticfield.

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12 Solutions to the Exercises in Quantum Mechanics II: Advanced Topics

1.19 Obtain the values of the commutators [Ej , HBj′ ] where j, j′ = x, y, z.Then find the equations of motion of E and B.

We have [Ej , Ej′ ] = [HBj , HBj′ ] = 0. We find

[Ex, HBx] = [−4πcπx, (∇×A)x]

= −4πc [πx, (∇×A)x]

= −4πc

[

πx,

(

∂A′z

∂y′−

∂A′y

∂z′

)]

= 0 .

Similarly, [Ey, HBy] = [Ez , HBz] = 0. Next,

[Ex, HBy] = −4πc

[

πx,

(

∂A′x

∂z′− ∂A′

x

∂x′

)]

= 4πci∂

∂z′δ3(r− r′) .

Similarly, we can obtain the other commutation relations between theperpendicular components of E and HB.

The equation of motion of x-component of E is

i∂Ex

∂t= [Ex, H ]

=1

8π

∫

[

Ex, H′y2]

+[

Ex, H′z2]

d3x′ .

We obtain

[

Ex, H′y2]

=[

Ex, H′y

]

H ′y +H ′

y

[

Ex, H′y

]

= 4πci∂

∂z′H ′

y +H ′y4πci

∂

∂z′

= 8πci∂

∂z′δ3H ′

y

and

[

Ex, H′z2]

= 8πci∂

∂y′δ3H ′

z .

Then

i∂Ex

∂t=

1

8π8πci

∫ (

H ′y

∂

∂z′δ3 +H ′

z

∂

∂x′ δ3

)

d3x′

or

∂Ex

∂t= ci(∇×HB)x .

K24777_SM_Cover.indd 24 13/11/14 6:20 PM

Quantum Field Theory 13

That is,∂E

∂t= c∇×HB.

Next,

i∂HBx

∂t=

1

8π

∫

[

HBx, E′y2 + E′

z2]

d3x′

= −8π

8πci

(

∂

∂yEz −

∂

∂zEy

)

= −ic(∇×E)x .

That is,∂HB

∂t= −c∇×E.

1.20 For the E and HB of the previous problem find∂

∂t∇ ·E and

∂

∂t∇ ·HB.

We find i∂

∂t∇ ·E = [∇ · E, HB].

Consider [∇ ·E, HBx]. We obtain

[∇ · E, HBx] =

[

∂Ex

∂y,H ′

Bx

]

+

[

∂Ez

∂z,H ′

Bz

]

= 4πci

[

− ∂

∂y

∂

∂z′, δ3(r′ − r) +

∂

∂z

∂

∂y′δ3(r′ − r)

]

= 0

because∂

∂y′δ3 = − ∂

∂yδ3 and

∂

∂z′δ3 = − ∂

∂zδ3. That is, ∇ ·E commutes

with HBx and also with H. Then,∂

∂t∇ ·E = 0. In other words, ∇ ·E is

a constant of motion. We can choose ∇ ·E = 0. In a similar manner wecan show that ∇ ·HB is a constant of motion so that ∇ ·HB = 0 at allspace-time points.

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