Quantum Kinetic Theory and Applications · Quantum Kinetic Theory and Applications Electrons,...

Quantum Kinetic Theory and Applications

Electrons, Photons, Phonons

Quantum Kinetic Theoryand Applications

Electrons, Photons, Phonons

FEDIR T. VASKOOLEG E. RAICHEV

Institute of Semiconductor PhysicsNAS of Ukraine, Kiev

Fedir T. Vasko Oleg E. RaichevInstitute of Semiconductor Physics, NAS Institute of Semiconductor Physics, NAS45 Prospekt Nauki 45 Prospekt NaukiKiev 03028 Ukraine Kiev 03028 Ukraine

Library of Congress Control Number: 2005926337

ISBN-10: 0-387-26028-5 e-ISBN: 0-387-28041-3ISBN-13: 978-0387-26028-0

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Preface

Physical kinetics is the final section of the course of theoretical physicsin its standard presentation. It stays at the boundary between gen-eral theories and their applications (solid state theory, theory of gases,plasma, and so on), because the treatment of kinetic phenomena alwaysdepends on specific structural features of materials. On the other hand,the physical kinetics as a part of the quantum theory of macroscopicsystems is far from being complete. A number of its fundamental is-sues, such as the problem of irreversibility and mechanisms of chaoticresponses, are now attracting considerable attention. Other importantsections, for example, kinetic phenomena in disordered and/or stronglynon-equilibrium systems and, in particular, phase transitions in thesesystems, are currently under investigation. The quantum theory of mea-surements and quantum information processing actively developing inthe last decade are based on the quantum kinetic theory.

Because a deductive theoretical exposition of the subject is not con-venient, the authors restrict themselves to a lecture-style presentation.Now the physical kinetics seems to be at the stage of development when,according to Newton, studying examples is more instructive than learn-ing rules. In view of these circumstances, the methods of the kinetictheory are presented here not in a general form but as applications fordescription of specific systems and treatment of particular kinetic phe-nomena.

The quantum features of kinetic phenomena can arise for several rea-sons. One naturally meets them in strongly correlated systems, when itis impossible to introduce weakly interacting quasiparticles (for exam-ple, in a non-ideal plasma), or in more complicated conditions, such asin the vicinity of the phase transitions. Next, owing to complexity ofthe systems like superconductors, ferromagnets, and so on, the manifes-tations of kinetic phenomena change qualitatively. The theoretical con-

v

vi QUANTUM KINETIC THEORY

sideration of these cases can be found in the literature. Another reasonfor studying quantum features of transport and optical phenomena hasemerged in the past decades, in connection with extensive investigationof kinetic phenomena under strong external fields and in nanostructures.The quantum features of these phenomena follow from non-classical dy-namics of quasiparticles, and these are the cases the present monographtakes care of, apart from consideration of standard problems of quan-tum transport theory. Owing to intensive development of the physics ofnanostructures and wide application of strong external (both stationaryand time-dependent) fields for studying various properties of solids, thetheoretical methods presented herein are of current importance for anal-ysis and interpretation of the experimental results of modern solid statephysics.

This monograph is addressed to several categories of readers. First,it will be useful for graduate students studying theory. Second, the top-ics we cover should be interesting for postgraduate students of variousspecializations. Third, the researchers who want to understand the back-ground of modern theoretical issues in more detail can find a numberof useful results here. The phenomena we consider involve kinetics ofelectron, phonon, and photon systems in solids. The dynamical prop-erties and interactions of electrons, phonons, and photons are brieflydescribed in Chapter 1. Further, in Chapters 2−8, we present main the-oretical methods: linear response theory, various kinetic equations forthe quasiparticles under consideration, and diagram technique. The pre-sentation of the key approaches is always accompanied by solutions ofconcrete problems, to illustrate applications of the theory. The remain-ing chapters are devoted to various manifestations of quantum transportin solids. The choice of particular topics (their list can be found in theContents) is determined by their scientific importance and methodolog-ical value. The 268 supplementary problems presented at the end of thechapters are chosen to help the reader to study the material of the mono-graph. Focusing our attention on the methodical aspects and discussinga great diversity of kinetic phenomena in line with the guiding principle“a method is more important than a result,” we had to minimize bothdetailed discussion of physical mechanisms of the phenomena consideredand comparison of theoretical results to experimental data.

It should be emphasized that the kinetic properties are the impor-tant source of information about the structure of materials, and manypeculiarities of the kinetic phenomena are used for device applications.These applied aspects of physical kinetics are not covered in detail either.However, the methods presented in this monograph provide the theoret-ical background both for analysis of experimental results and for device

PREFACE vii

simulation. In the recent years, these theoretical methods were appliedfor the above-mentioned purposes so extensively that any comprehensivereview of the literature seems to be impossible in this book. For thisreason, we list below only a limited number of relevant monographs andreviews.

Fedir T. VaskoOleg E. RaichevKiev, December 2004

Monographs:1. J. M. Ziman, Electrons and Phonons, the Theory of Transport Phenomena in

Solids, Oxford University Press, 1960.2. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, W. A. Benjamin,

Inc., New York, 1962.3. A. A. Abrikosov, L. P. Gor’kov and I. E. Dzialoszynski, Methods of Quantum

Field Theory in Statistical Physics, Prentice-Hall, 1963.4. S. Fujita, Introduction to Non-Equilibrium Quantum Statistical Mechanics,

Saunders, PA, USA, 1966.5. D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultants Bu-

reau, New York, 1974.6. E. M. Lifshitz and L. P. Pitaevski, Physical Kinetics, Pergamon Press, Oxford,

1981.7. H. Bottger and V. V. Bryksin, Hopping Conduction in Solids, VCH Publishers,

Akademie-Verlag Berlin, 1985.8. V. L. Gurevich, Transport in Phonon Systems (Modern Problems in Condensed

Matter Sciences, Vol. 18), Elsevier Science Ltd., 1988.9. V. F. Gantmakher and Y. B. Levinson, Carrier Scattering in Metals and Semi-

conductors (Modern Problems in Condensed Matter Sciences, Vol. 19), Elsevier Sci-ence Ltd., 1987.

10. A. A. Abrikosov, Fundamentals of the Theory of Metals, North-Holland, 1988.11. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic

Properties of Semiconductors, World Scientific, Singapore, 1990.12. N. N. Bogolubov, Introduction to Quantum Statistical Mechanics, Gordon and

Breach, 1992.13. G. D. Mahan, Many Particle Physics, Plenum, New York, 1993.14. H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of

Semiconductors, Springer, Berlin, 1997.15. Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, 1997.16. D. K. Ferry and S. M. Goodnick, Transport in Nanostructures, Cambridge

University Press, New York, 1997.17. R. P. Feynmann, Statistical Mechanics, Addison-Wesley, 1998.18. A. M. Zagoskin, Quantum Theory of Many-Body Systems: Techniques and

Applications, Springer-Verlag, New York, 1998.19. F. T. Vasko and A. V. Kuznetsov, Electron States and Optical Transitions in

Semiconductor Heterostructures, Springer, New York, 1998.

viii QUANTUM KINETIC THEORY

20. J. Rammer, Quantum Transport Theory (Frontiers in Physics, Vol. 99), West-view Press, 1998.

21. T. Dittrich, P. Hanggi, G.-L. Ingold, B. Kramer, G. Schon, and W. Zverger,Quantum Transport and Dissipation, Wiley-VCH, Weinheim, 1998.

22. B. K. Ridley, Quantum Processes in Semiconductors, Oxford University Press,1999.

23. D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-mation, Springer, Berlin, Heidelberg, New York, 2000.

Reviews:1. D. N. Zubarev, Double-Time Green’s Functions, Sov. Phys. - Uspekhi 3, 320

(1960).2. R. N. Gurzhi and A. P. Kopeliovich, Low-Temperature Electrical Conductivity

of Pure Metals, Sov. Phys. - Uspekhi 133, 33 (1981).3. T. Ando, A. B. Fowler, and F. Stern, Electronic Properties of Two-Dimensional

Systems, Rev. Mod. Phys. 54, 437 (1982).4. J. Rammer and H. Smith, Quantum Field-Theoretical Methods in Transport

Theory of Metals, Rev. Mod. Phys. 58, 323 (1986); J. Rammer, Quantum TransportTheory of Electrons in Solids: A Single-Particle Approach, Rev. Mod. Phys. 63, 781(1991).

5. G. D. Mahan, Quantum Transport Equation for Electric and Magnetic Fields,Physics Reports 145, 251 (1987).

6. W. R. Frensley, Boundary Conditions for Open Quantum Systems Driven Farfrom Equilibrium, Rev. Mod. Phys. 62, 745 (1990).

7. B. Kramer and A. MacKinnon, Localization: Theory and Experiment, Rep.Prog. Phys. 56, 1469 (1993).

8. C. H. Henry and R. F. Kazarinov, Quantum Noise in Photonics, Rev. Mod.Phys. 68, 801 (1996).

9. C. W. J. Beenakker, Random-Matrix Theory of Quantum Transport, Rev. Mod.Phys. 69, 731 (1997).

10. Ya. M. Blanter and M. Buttiker, Shot Noise in Mesoscopic Conductors,Physics Reports 336, 1 (2000).

11. P. Lipavsky, K. Morawetz, and V. Spicka, Kinetic Equation for Strongly In-teracting Dense Fermi Systems, Annales de Physique 26, 1 (2001).

Contents

Preface v

1. ELEMENTS OF QUANTUM DYNAMICS 11. Dynamical Equations 12. S-Operator and Probability of Transitions 63. Photons in Medium 114. Many-Electron System 175. Electrons under External Fields 256. Long-Wavelength Phonons 35Problems 46

2. ELECTRON-IMPURITY SYSTEM 517. Kinetic Equation for Weak Scattering 518. Relaxation Rates and Conductivity 579. Quasi-Classical Kinetic Equation 6510. Multi-Photon Processes 7211. Balance Equations 7912. Conductance of Microcontacts 87Problems 94

3. LINEAR RESPONSE THEORY 9913. Kubo Formula 9914. Diagram Technique 10715. Bethe-Salpeter Equation 11516. Green’s Function as a Path Integral 12317. Dispersion of Dielectric Permittivity 130

ix

x QUANTUM KINETIC THEORY

18. Interband Absorption under External Fields 139Problems 147

4. BOSONS INTERACTINGWITH ELECTRONS 15519. Kinetic Equation for Boson Modes 15520. Spontaneous and Stimulated Radiation 16221. Phonon Instabilities 16922. Boson Emission by 2D Electrons 176Problems 184

5. INTERACTING PHONON SYSTEMS 18923. Phonon-Phonon Collisions 18924. Thermal Conductivity of Insulators 19725. Balance Equations for Phonons 20126. Relaxation of Long-Wavelength Phonons 20727. Polaritons and Dielectric Function of Ionic Crystals 215Problems 224

6. EFFECTS OF ELECTRON-ELECTRONINTERACTION 22928. Hartree-Fock Approximation 22929. Shift of Intersubband Resonance 23530. Exciton Absorption 24331. Electron-Electron Collision Integral 25032. Coulomb Drag Between 2D Electrons 25633. Dynamical Screening 262Problems 273

7. NON-EQUILIBRIUM ELECTRONS 28134. Electron-Boson Collision Integral 28135. Quasi-Isotropic and Streaming Distributions 28936. Diffusion, Drift, and Energy Balance 29937. Heating under High-Frequency Field 30938. Relaxation of Population 321Problems 331

8. NON-EQUILIBRIUM DIAGRAMTECHNIQUE 341

Contents xi

39. Matrix Green’s Function 34140. Generalized Kinetic Equation 34741. General Formulation of NDT 35542. NDT Formalism for Electron-Boson System 36443. Weak Localization under External Fields 374Problems 381

9. KINETICS OF BOUNDED SYSTEMS 39144. Boundary Conditions at Non-Ideal Surface 39145. Size-Dependent Conductivity 39846. Thermal Conductivity of Bounded Insulators 40547. Electron Relaxation by Near-Surface Phonons 411Problems 420

10. QUANTUM MAGNETOTRANSPORT 42548. Method of Iterations 42649. Green’s Function Approach 43450. Quasi-Classical Conductivity 44351. Quantum Hall Effect 45252. Magnetooptics 465Problems 476

11. PHOTOEXCITATION 48353. Photogeneration Rate 48354. Response to Ultrafast Excitation 49055. Partially Inverted Electron Distribution 49756. Photoinduced Interband Hybridization 50857. Excitation of Coherent Phonons 520Problems 528

12. BALLISTIC AND HOPPING TRANSPORT 53758. Quantized Conductance 53859. One-Dimensional Conductors 54960. Tunneling Current 56261. Coulomb Blockade 57462. Polaronic Transport 585Problems 596

xii QUANTUM KINETIC THEORY

13. MULTI-CHANNEL KINETICS 60763. Spin-Flip Transitions 60764. Spin Hydrodynamics 61765. Coupled Quantum Wells 62866. Auger Processes 63667. Kondo Effect 646Problems 654

14. FLUCTUATIONS 66368. Non-Equilibrium Fluctuations 66469. Quasi-Classical Approach 67470. Light Scattering 68571. Fluctuations in Mesoscopic Conductors 69872. NDT Formalism for Fluctuations 708Problems 720

Appendices 733Harmonic Oscillator 733Many-Band KP-Approach 737Wigner Transformation of Product 743Double-Time Green’s Functions 747Many-Electron Green’s Functions 751Equation for Cooperon 759Green’s Function in Magnetic Field 763Hamiltonian of Tunnel-Coupled Systems 771

Index779

Chapter 1

ELEMENTS OF QUANTUM DYNAMICS

The dynamical equations for quantum systems, the Schroedinger equation forpure states and the density-matrix equation for mixed states, form the theoreticalbackground for description of transport phenomena in systems with different kindsof elementary excitations (quasiparticles). Both single-particle formulation of theseequations and many-particle formalism, which is required for the cases of interactingquasiparticles, are presented below. This chapter is not a systematic introduction toquantum theory. It contains only the description of some basic equations and defini-tions (probability of transitions, second quantization, and so on). The properties ofconcrete quasiparticles (electrons, phonons, and photons in different materials) andtheir interaction are also discussed in order to use the corresponding results in thenext chapters.

1. Dynamical EquationsLet us start our consideration of the quantum dynamics with the sim-

plest case of a single particle propagating along the x direction. The evo-lution of such a particle is described by the time-dependent Schroedingerequation:

i�∂Ψ(δ)

xt

∂t= HΨ(δ)

xt , Ψ(δ)xt=t0

= Ψ(δ)x , (1)

where the initial state at t = t0 is determined by the wave functionΨ(δ)

x , which depends on the set of quantum numbers δ. The HamiltonianH can depend on time. A simple example of quantum evolution is aparticle moving in a one-dimensional potential. The Hamiltonian Hx forsuch a case is obtained from the classical expression for the energy afterreplacing the momentum by the operator proportional to the Planck

1

2 QUANTUM KINETIC THEORY

constant �:

Hx =p2

2m+ U(x) , p = −i�

∂

∂x, (2)

where m is the mass of the particle. The character of the dynamicsdepends essentially on the potential energy U(x). We mention, for ex-ample, formation of confined states in a potential well or tunneling pen-etration of the particle through a potential barrier. Different observablevalues (such as coordinate, velocity, and energy) of the system are de-termined by the quantum-mechanical average

Q(δ)

t =∫

dxΨ(δ)xt

∗QΨ(δ)

xt , (3)

where the operator Q corresponds to the classical expression for theobservable value. Note that Q

(δ)

t is expressed through a quadratic formof the Ψ-functions. Since Q

(δ)

t is real, any operator Q must be Hermitian.In particular, H = H+, because the Hamiltonian corresponds to theenergy of the system.

The operator nature of the characteristics of quantum systems makesit possible to rewrite Eqs. (1) and (3) in the integral representation. Weintroduce a kernel

H(x, x1) =[

p21

2mδ(x − x1)

]+ U(x1)δ(x − x1) (4)

containing Dirac’s δ-function, and transform the Schroedinger equation(1) to the following integral form:

i�∂Ψ(δ)

xt

∂t=∫

dx1H(x, x1)Ψ(δ)x1t, Ψ(δ)

xt=t0= Ψ(δ)

x . (5)

The kernel for the observable value, Q(x, x1) = [Q1δ(x − x1)], is in-troduced in the same way (here Q1 acts on the coordinate x1 of theδ-function), and we obtain

Q(δ)

t =∫

dx

∫dx1Ψ

(δ)xt

∗Q(x, x1)Ψ

(δ)x1t. (6)

In these formulations, the state with quantum numbers δ is described bythe wave function Ψ(δ)

xt and by the operators of physical values appearingin Eqs. (1) and (3), or by the x-dependent kernels in Eqs. (5) and (6).Such a description is called the coordinate (or x-) representation.

In many cases, the description of quantum dynamics can be simplifiedby using the Fourier-transformed wave function introduced according to

Elements of Quantum Dynamics 3

the relations

Ψ(δ)pt =

∫dxe−(i/�)pxΨ(δ)

xt , Ψ(δ)xt =

1L

∑p

e(i/�)pxΨ(δ)pt . (7)

In order to avoid the ambiguities due to δ-functions, the motion of theparticle is considered here for an interval of length L, with the use ofappropriate boundary conditions. In the limit L → ∞, the momentump in Eq. (7) is a quasi-discrete variable with values (2π�n/L), wheren is an integer. The substitution p → (2π�n/L) does not depend onthe type of the boundary conditions used (hard-wall, periodic, etc.),provided that n is a large number. A Fourier transformation of Eq. (1)with the Hamiltonian (2) leads to the Schroedinger equation

i�∂Ψ(δ)

pt

∂t=∑p1

H(p, p1)Ψ(δ)p1t, (8)

H(p, p1) ≡ 1L

∫dxe−(i/�)pxHxe(i/�)p1x,

which is similar to Eq. (5). The kernel H(p, p′) depends on a pair ofmomenta. The initial condition to Eq. (8) is determined by the Fouriertransformation of Ψ(δ)

x . In the above example of the particle in a one-dimensional potential, the Hamiltonian kernel is transformed to

H(p, p1) =p2

2mδp,p1 + U(p, p1), (9)

where δp,p1 is the Kronecker symbol (below we use two equivalent no-tations δa,b and δab for such symbols). The kinetic energy acquires itsclassical form, while the action of the potential is described by the kernelU(p, p1). The expression for an observable through Ψ(δ)

pt is written as

Q(δ)

t =1L2

∑p1p2

Ψ(δ)p1t

∗Q(p1, p2)Ψ

(δ)p2t, (10)

where the kernel Q(p1, p2) can be written in terms of Q in a similar wayas the Hamiltonian kernel in Eq. (8). The structure of Eqs. (8) and (10)is analogous to that of Eqs. (5) and (6). This description is called themomentum (or p-) representation of the problem under consideration.

Obviously, the nature of quantum dynamics does not depend on therepresentation used. For this reason, it is convenient to consider thewave function as a projection of the ket-vector |δ, t〉, which describes thestate defined by the quantum numbers δ, onto the bra-vector, 〈k|, which


determines the representation:

Ψ(δ)kt = 〈k|δ, t〉 k ↔ x, p, . . . . (11)

It should be noted that the above-introduced bra- and ket-vectors are notusual functions. They are Hermitian conjugate elements of the Hilbertspace satisfying the relations of orthogonality, normalization, and com-pleteness:

〈k| = |k〉+, 〈k|k′〉 = δk,k′ ,∑

k

|k〉〈k| = 1, (12)

where 1 is the unit operator. Using these notations, one may formulateany dynamical problem in the operator form.

The Schroedinger equation for the state δ in this representation takesthe following form:

i�∂ |δ, t〉

∂t= H |δ, t〉 , |δ, t = t0〉 = |δ〉 , (13)

with the initial condition determined by the ket-vector |δ〉. A similarequation for the Hermitian conjugate vector 〈δ, t| contains −〈δ, t|H onthe right-hand side. Using Eq. (11) and rewriting the kernel Q(k1, k2)as 〈k1|Q|k2〉, we define the observable Q

(δ)

t as follows:

Q(δ)

t =∑k1k2

〈δ, t|k1〉〈k1|Q|k2〉〈k2|δ, t〉 = 〈δ, t|Q|δ, t〉, (14)

so that the classical observable is expressed through the diagonal matrixelement. As a result, the dynamics of the system with a fixed initial state|δ, t = 0〉 (such a system is said to be in the pure state) is described byEqs. (13) and (14).

Transforming the double sum in Eq. (14) as∑

k1k2〈k1|Q|k2〉〈k2|δ, t〉

×〈δ, t|k1〉, it is convenient to separate the operator |δ, t〉〈δ, t| there. Thisoperator,

η(δ)t ≡ |δ, t〉〈δ, t|, (15)

known as the density matrix or as the statistical operator, describesthe quantum dynamics of the system. The quantity 〈k2|δ, t〉〈δ, t|k1〉 =〈k2|η(δ)

t |k1〉 is also called the density matrix in the |k〉-representation.One may consider, for example, x- or p-representation, or a representa-tion based upon discrete quantum numbers (problem 1.1). The descrip-tion of the quantum dynamics based on the density matrix formalism isconvenient for the cases when the initial state |δ, t = t0〉 of the quantum


system is not defined (for example, because of the quantum-mechanicaluncertainty).

Let us give a more general definition of the density matrix. Consideran ensemble of identical systems, which are distributed over the states δwith probabilities Pδ at the initial moment of time t = t0 (such a systemis called the mixed state, or the mixture of states). We introduce theobservable quantity Qt according to

Qt =∑

δ

PδQ(δ)

t . (16)

The probability for realization of δ-states is normalized as∑

δ Pδ = 1.Since the operator Q does not depend on the initial conditions, thedensity matrix for the mixed state is introduced as

ηt ≡∑

δ

Pδ |δ, t〉〈δ, t|, (17)

and the observable (16) is obtained from Eqs. (14) and (16) in the form

Qt =∑

k

〈k|Qηt|k〉 ≡ Sp(Qηt). (18)

Here and below Sp(A) (or, equivalently, SpA), where A is an arbitraryoperator, denotes the sum of the diagonal matrix elements of this oper-ator and is called the trace of the operator.

The equation of evolution for the density matrices (15) and (17) de-scribing dynamics of pure and mixed states, respectively, is obtainedin the following way. Let us take a derivative of the density matrixover time and use Eq. (13) together with the corresponding Hermitianconjugate equation. As a result,

i�∂

∂t|δ, t〉〈δ, t| = H|δ, t〉〈δ, t| − |δ, t〉〈δ, t|H. (19)

Now, let us multiply this equation by Pδ and calculate the sums over δof both its sides. Since H does not depend on δ, we obtain, accordingto Eq. (17), the operator equation

i�∂ηt

∂t= [H, ηt] (20)

describing the evolution of the quantum system. The right-hand side ofEq. (20) is written using the commutator defined as [A, B] = AB − BA,where A and B are arbitrary operators. The initial condition for Eq.(20) in the case of a pure state may be expressed as ηt=t0 = |δ〉〈δ|, whilefor a mixed state one needs additional physical restrictions removing theuncertainty of the initial state.


2. S-Operator and Probability of TransitionsThe evolution of the system with time-dependent Hamiltonian Ht is

described by the Schroedinger equation (1.13). The ket-vectors |δt〉 ≡|δ, t〉 at the instants t and t′ are connected through the evolution operatorS (also known as S-operator or scattering matrix):

|δt〉 = S(t, t′)|δt′〉. (1)

Equation (1.13) leads to the operator equation for S(t, t′), with the initialcondition at t = t′:

i�∂

∂tS(t, t′) = HtS(t, t′), S(t, t′)t=t′ = 1. (2)

For the case of a time-independent Hamiltonian, Ht = H, this equationis solved as

S(t, t′) = exp[− i

�H(t − t′)

]≡ S(t − t′), (3)

and the temporal evolution is determined only by the difference t − t′.If the initial value of the ket-vector, |δ, t = 0〉 = |δ〉, belongs to oneof the vectors determined by the eigenstate problem H|δ〉 = εδ |δ〉, theevolution is harmonic:

|δt〉 = exp(

− i

�εδt

)|δ〉. (4)

In the case of a mixed initial state, the evolution is described by a sumof oscillating factors with different energies εδ .

In the general case of the time-dependent Hamiltonian, it is convenientto transform Eq. (2) to the integral form:

S(t, t′) = 1 − i

�

∫ t

t′dτHτ S(τ, t′). (5)

The solution of this equation is obtained by iterations and is written as

S(t, t′) = 1 +∞∑

n=1

(− i

�

)n ∫ t

t′dt1 . . .

∫ tn−2

t′dtn−1

∫ tn−1

t′dtn

×Ht1 . . . Htn−1Htn . (6)

Introducing the operator of chronological ordering, T , we rewrite Eq.(6) as follows (problem 1.2):

S(t, t′) = 1+∞∑

n=1

(−i/�)n

n!

∫ t

t′dt1 . . .

∫ t

t′dtn−1

∫ t

t′dtnT

{Ht1Ht2 . . . Htn

},


T{

HtHt′}

={

HtHt′ , t > t′

Ht′Ht, t < t′. (7)

One can write S(t, t′) of Eqs. (6) and (7) as a chronologically orderedexponential operator

S(t, t′) = T{

exp[− i

�

∫ t

t′dτHτ

]}. (8)

This expression, together with Eq. (2), leads to the following propertiesof the evolution operator:

S(t, t′) = S+(t′, t), S+(t, t′)S(t, t′) = 1, S(t, t1)S(t1, t′) = S(t, t′), (9)

which can be checked by calculating the time derivatives (problem 1.3).Below we consider a system with time-independent Hamiltonian H

in the presence of a weak harmonic perturbation. In other words, wediscuss the evolution of the system with the Hamiltonian

H + (vωe−iωt + H.c.) ≡ H + Vt, (10)

where the operator vω is small. The letters H.c. in Eq. (10) indicatethe Hermitian conjugate contribution to the perturbation. A solution ofthis problem not only describes a response of the system to the harmonicperturbation, but also allows one to consider a modification of stationarystates under the time-independent perturbation vω + v+

ω , where ω = 0.It is convenient to use the interaction representation by introducing anew ket-vector |δt) according to |δt〉 = S(t)|δt), where S(t) is the S-operator (introduced by Eq. (3)) for the Hamiltonian H. Substituting|δt〉 = S(t)|δt) into Eq. (1.13), we multiply the latter by S+(t) from theleft and obtain the following Schroedinger equation in the interactionrepresentation:

i�∂|δt)∂t

= S+(t)VtS(t)|δt), |δ, t = 0) = |i〉. (11)

To solve Eq. (11) with the accuracy of the first order in the perturbationVt, we substitute the unperturbed ket-vector |i〉 to the right-hand sideof this equation. Since we assume that the unperturbed system is in theinitial state i, we have

|it) � |i〉 +1i�

∫ t

0dt′V (t′)|i〉, V (t) = S+(t)VtS(t). (12)

The probability of finding the system at the instant t in the state f(described by the ket-vector S(t)|f〉 of zero-order approximation), calcu-lated with the accuracy of the second order in the perturbation, is equal


to |〈f |S+(t)|it〉|2 = |〈f |it)|2. The probability of transition between thestates i and f is defined as a time derivative of this quantity:

Wif =d|〈f |it)|2

dt. (13)

We note that both |i〉 and |f〉 are the solutions of the eigenstate problemH|i〉 = εi|i〉. An explicit expression for Wif is determined after a simpleintegration over time:

Wif =1�2

d

dt

∣∣∣∣∣vfiei(ωfi−ω)t − 1i(ωfi − ω)

+ v∗if

ei(ωfi+ω)t − 1i(ωfi + ω)

∣∣∣∣∣2

, (14)

where ωfi = (εf −εi)/� is the frequency of transitions between the statesi and f of the unperturbed system and vfi = 〈f |vω |i〉. The probabilityof transitions has resonant behavior: at t → ∞ it is not equal to zeroonly when ω coincides with one of the transition frequencies. We notethat, on the small-time scale, a non-zero probability exists also for non-resonant conditions, owing to the energy-time uncertainty. In the caseof time-independent perturbations (ω = 0), the energy of the system isconserved, and the transitions occur between the degenerate states only.

Let us consider first the asymptotic behavior of Wif at large times un-der a time-independent perturbation V ≡ 2v (note that v is Hermitian).For this case, taking into account |vif |2 = |vfi|2, we obtain

Wif =4|vif |2

�2d

dt

2 − 2 cos ωfit

ω2f i

=8|vif |2

�2d

dt

[2 sin2(ωfit/2)

ω2f it

t

]. (15)

If ωfit � 1, the function 2 sin2(ωfit/2)/ω2f it goes to πδ(ωfi); see Fig.

1.1 and problem 1.4, where different presentations of Dirac’s δ-functionare discussed. As a result, the probability of transition becomes

Wif =2π

�|〈f |V |i〉|2δ(εf − εi). (16)

This important result is known as Fermi’s golden rule. We stress againthat the energy of the system is conserved, and only the states withεf = εi contribute into the probability (16) under a time-independentperturbation.

The probability of resonant transitions in the case of time-dependentperturbations is calculated in a similar way. If ωfi = 0, only the termscontaining ωfi −ω in the factor | . . . |2 of Eq. (14) are important at larget, and one obtains

Wif (ω) =2π

�|〈f |vω |i〉|2δ(εf − εi − �ω). (17)


Figure 1.1. Function F (ω) = sin2(ωt)/πω2t for t =3, 10, and 30 (dotted, dashed,and solid curves, respectively).

The energy conservation law

εf = εi + �ω (18)

is fulfilled for interlevel transitions excited by a harmonic perturbationwith the energy of quantum �ω.

Equations (16) and (17) can be derived in an alternative way, underthe assumption that the perturbation Vt is adiabatically turned on att = −∞. The first-order solution of the time-dependent Schroedingerequation (1.13) with the Hamiltonian H + Vt and boundary condition|δ, t = −∞〉 = |i〉 is written as

|it〉 � |i〉 +1i�

∫ t

−∞dt′eλt′S(t, t′)Vt′S

+(t, t′)|i〉 , (19)

where λ → +0 describes the adiabatic turning-on. Consider, for exam-ple, a time-independent perturbation V . The integral in Eq. (19) iseasily calculated by substituting τ = t′ − t. Since S(t, t′) = S(−τ), weobtain

〈f |it〉 = eλt 〈f |V |i〉εi − εf + i�λ

. (20)

The transition probability d|〈f |it〉|2/dt is reduced to Eq. (16) accordingto the first expression for the δ-function in problem 1.4. The case of


time-dependent perturbation is considered in a similar way, leading toEq. (17).

The probabilities of transitions, derived above in a pure quantum-mechanical approach, are the important characteristics determining ki-netic properties of different systems. Indeed, let us introduce the oc-cupation number njt for the state j, i.e., the average number of theparticles in the state j at the instant t, according to (see Eq. (1.17))

njt = 〈j|ηt|j〉 =∑

δ

Pδ |〈j|δt〉|2. (21)

One may expect that, under proper conditions (in the subsequent chap-ters this question will be considered in detail), the temporal evolutionof the occupation numbers is determined by the balance equation

∂njt

∂t=∑j′

Wjj′(nj′t − njt

), (22)

where Wjj′ is given by Eq. (16) for the case of time-independent per-turbations and Wjj′ = Wjj′(ω) + Wjj′(−ω), see Eq. (17), for time-dependent harmonic perturbations. The first and second terms on theright-hand side of Eq. (22) describe incoming (arrival) and outgoing (de-parture) contributions to the balance of occupation, respectively. Wenote that the arrival rate from the state j′ to the state j is equal toWjj′nj′t, while the departure rate from the state j to all other states isequal to njt

∑j′ Wjj′ . The balance equation (22) conserves the number

of the particles and, for the case of time-independent perturbations, theenergy of the system. In order to describe the temporal evolution of theother characteristics of the system (those which are sensitive to phasecorrelation), one has to consider quantum kinetic equations for ηt; seethe next chapters.

In the case of harmonic perturbations, one may express the powerabsorbed by the system through the transition probability (17). Theabsorbed power Uω is defined as the energy of the quantum, �ω, multi-plied by the difference between the rate of transition from the state j tothe state j′ (which corresponds to absorption of the quantum) and therate of emission of the quantum associated with the transitions from j′to j:

Uω = �ω∑jj′

[Wjj′(ω)nj(1 − nj′) − Wj′j(−ω)nj′(1 − nj)

]= �ω

∑jj′

Wjj′(ω)(nj − nj′). (23)


On the other hand, the absorbed power Uω for the system excited by anelectric field Et is determined by the electrodynamical expression It · Et,where It is the electric current density induced by the field and the lineover the expression denotes the averaging over the period 2π/ω. Withinthe accuracy of E2

t , which corresponds to the perturbation theory ap-plied above, one may consider It in the framework of the linear-responseapproximation and describe Uω through the frequency-dependent con-ductivity of the system; see Chapter 3.

3. Photons in MediumWe begin our consideration of the quantum dynamics of concrete phys-

ical systems with the case of electromagnetic field in the spatially inho-mogeneous medium described by the dielectric permittivity tensor εr.Starting from an expression for the energy of the electromagnetic fieldin the absence of free electric charges, we derive the Hamiltonian of thefield and, after a quantization procedure, describe the field as a set ofoscillators corresponding to elementary quasiparticles known as photons.The photons are an example of bosons, the particles with a symmetricwave function corresponding to the Bose-Einstein statistics.

The energy of the field is determined by the expression

Ef =18π

∫(V )

dr(Ert · εrErt + H2rt)

=18π

∫(V )

dr{

1c2

∂Art

∂t· εr

∂Art

∂t+ ([∇ × Art])2

}, (1)

where the integrals are taken over the normalization volume V . In Eq.(1) we assume a local relation between the electrostatic induction andthe field: εrErt. On the other hand, the magnetic induction is equal toHrt because the kinetic phenomena are considered in this book for non-magnetic materials only. The electric and magnetic field strengths, Ertand Hrt, which satisfy the Maxwell equations in medium, are expressedonly through the vector potential Art, since we have chosen the Coulombgauge ∇ · εrArt = 0 leading to zero scalar potential in the absence offree charges.

It is convenient to represent the electromagnetic field described by thevector potential Art as

Art =∑

ν

[qν(t)Aνr + q∗

ν(t)Aν∗r ], (2)


where the modes Aνr with frequencies ων are determined by the wave

equation following from the Maxwell equations:

[∇ × [∇ × Aνr ]] −

(ων

c

)2εrAν

r = 0. (3)

The modes satisfy the orthogonality and normalization conditions ac-cording to

∫(V )

drAν∗r · εrAν ′

r = 2πc2δνν ′ . The coefficients qν(t) in theexpression (2) can be considered as the generalized coordinates of ν-thmode. They satisfy the oscillator equation

d2qν(t)dt2

+ ω2νqν(t) = 0 (4)

corresponding to the harmonic oscillations with eigenfrequencies ων . Inthe presence of external sources described by the electric current densityIrt, one must add the term −c−1 ∫

(V )dr Irt · Art to the right-hand side

of Eq. (1) and (4π/c)Ir to the right-hand side of Eq. (3). Such acontribution describes the interaction of the modes Ar with externalcharges.

Introducing the generalized momentum pν(t) ≡ dqν(t)/dt, we applythe orthogonality and normalization conditions for the modes of Eq. (3)to rewrite the energy of the field given by Eq. (1) as a sum of oscillatorenergies:

Ef =12

∑ν

{|pν(t)|2 + ω2

ν |qν(t)|2}

. (5)

Since the solutions of Eq. (4) are proportional to exp(−iωνt), we havethe relation pν = −iωνqν . It is convenient to introduce the canonicallyconjugate variables

Qν(t) =qν(t) + q∗

ν(t)2

, Pν(t) = −iωνqν(t) − q∗

ν(t)2

, (6)

which are used here in order to rewrite the energy of the field as Ef =∑ν

{|Pν(t)|2 +ω2

ν |Qν(t)|2}

/2. The equations of motion acquire Hamil-tonian form: Pν = Qν = −∂Ef /∂Qν , Qν = ∂Ef /∂Pν .

In order to quantize the electromagnetic field, we have to replacethe canonically conjugate variables Qν(t) and Pν(t) by the operators ofgeneralized coordinate and momentum, Qν and Pν , which satisfy thecommutation relation

[Qν , Pν ′ ] = i�δνν ′ . (7)

Let us use the expression for the energy as a sum of the oscillatory con-tributions (5) and take into account the relation Pν = −i�∂/∂Qν . Then


we write the Hamiltonian of quantized field in the Q-representation:

Hph =12

∑ν

{−�

2 ∂2

∂Q2ν

+ ω2νQ

2ν

}. (8)

A solution of the eigenstate problem HphΨ{nν} = E{nν}Ψ{nν} deter-mines a set of occupation numbers, {nν}, for the given modes. Thesymmetrized wave function, corresponding to the Bose-Einstein statis-tics, is a product of the eigenfunctions of different modes, ψnν (Qν), whilethe total energy, E{nν}, is given by a sum of the oscillator energies:

Ψ{nν} =∏ν

ψnν (Qν), E{nν} =∑

ν

�ων

(nν +

12

). (9)

The occupation numbers nν are integers (nν ≥ 0). As follows from Eq.(9), the wave function Ψ{nν} is symmetric with respect to permutationsof each oscillatory function ψn(Q) (see Appendix A) with another oscil-latory function. The matrix elements of the generalized coordinate forthe transitions between the states with quantum numbers nν and n′

ν areequal to zero if n′

ν = nν ± 1, while for the transition between adjacentlevels these matrix elements are

〈n′ν |Qν |nν〉 =

√�

2ων

{ √nν + 1 ,√

nν ,n′

ν = nν + 1n′

ν = nν − 1 . (10)

The matrix elements of the generalized momentum are 〈n′ν |Pν |nν〉 =

±iων〈n′ν |Qν |nν〉, where the signs ± correspond to the transitions be-

tween the states with occupation numbers n′ν = nν ± 1 and nν . This

equation is consistent with the relation between the Fourier componentsof coordinate and momentum used in Eq. (6). Instead of a pair of canon-ically conjugate operators Qν and Pν , we introduce, by analogy to Eq.(A.11), two Hermitian conjugate creation and annihilation operators forthe mode ν:

bν =ωνQν + iPν√

2�ων, b+

ν =ωνQ

+ν − iP+

ν√2�ων

. (11)

Representing the contribution of the state ν in the Hamiltonian (8) as{. . .} = (ωνQ

+ν − iP+

ν )(ωνQν + iPν) + �ων , we rewrite Hph in the form

Hph =∑

ν

�ων

(b+ν bν +

12

). (12)

The Hamiltonian of the field is given as a sum of the contributions hosc

determined by Eq. (A.12), with the oscillator frequencies ων .


Therefore, the electromagnetic field in a medium is presented as asuperposition of quantized normal vibrations with frequencies ων andoccupation numbers nν . It is convenient to use a representation de-scribed by the ket-vector |{nν}〉 depending on the sets of occupationnumbers {nν}. Using these sets as independent variables of the problem(instead of the generalized coordinates Qν), one may define the creationand annihilation operators through their matrix elements

〈n′ν |b+

ν |nν〉 =√

nν + 1δn′ν ,nν+1,

〈n′ν |bν |nν〉 =

√nνδn′

ν ,nν−1, (13)

instead of using Eq. (11). This means that the operators b+ν and bν ,

while acting on the ket-vector |{nν}〉, change the occupation number ofthe photons of the mode ν by ±1, respectively:

b+ν |n1n2 . . . nν . . .〉 =

√nν + 1|n1n2 . . . nν + 1 . . .〉,

bν |n1n2 . . . nν . . .〉 =√

nν |n1n2 . . . nν − 1 . . .〉. (14)

The commutation rules for these operators are obtained by using eitherthe matrix elements (13) or the expressions of these operators throughQν and Pν , Eq. (11). For the Hermitian conjugate operators, one has

[bν , b+ν ′ ] = δnν ,nν′ , (15)

while the operators of the same kind (creation or annihilation) merelycommute with each other. It is the commutation rule (15) that leadsto the appearance of zero-field oscillation energy

∑ν �ων/2 in Eq. (12);

see also Eqs. (A.11) and (A.12). By analogy to the case of a singleoscillator, see Eq. (A.18), the set of ket-vectors |{nν}〉 is presented as

|{nν}〉 =∏ν

(bν)nν

√nν !

|{0}〉, (16)

where |{0}〉 describes the vacuum state where only zero-field oscillationsdue to quantum-mechanical uncertainty are present. The set of ket-vectors also satisfies the completeness, orthogonality, and normalizationconditions:∑

ν

|{nν}〉〈{nν}| = 1, 〈{nν}|{n′ν}〉 = δ{nν},{n′

ν}. (17)

The generalized Kronecker symbol δ{nν},{n′ν} is equal to unity only when

all the occupation numbers from the sets {nν} and {n′ν} coincide. The


description of the electromagnetic field given by Eqs. (13)-(17) is calledthe occupation number representation or the second quantization. It isanalogous to the description of a single oscillator given by Eqs. (A.16)-(A.19). In this representation, the sets of independent variables describ-ing the system are the numbers of quanta of the field in each mode.These quanta are called the photons in medium, i.e., the system is de-scribed in terms of quasiparticles. The operator of the photon numberfor the mode ν is introduced as nν = b+

ν bν . The justification of thisdefinition is the same as for a single oscillator, and the ket-vector |{nν}〉is the eigenvector of the operator nν corresponding to the eigenvalue nν ,according to nν |{nν}〉 = nν |{nν}〉; see Eq. (A.20).

Using the expansion (2) and expressing the amplitudes of vibrationsaccording to Eqs. (6) and (11) through the creation and annihilationoperators as qν =

√�/ων bν and q+

ν =√

�/ων b+ν , we get the quantized

operator of the vector potential

Ar =∑

ν

√�

ων

(Aν

r bν + Aν∗r b+

ν

), (18)

where the modes Aνr are determined from Eq. (3). The operators of

the second-quantized fields, Er and Hr, can be written by using Eq.(18) together with the relation qν(t) = −iωνqν(t) and by expressingthese fields through the vector potential according to Ert = −c−1Artand Hrt = [∇ × Art]. The classical vector of the radiation flux density(Poynting vector), Srt = (c/4π)[Ert × Hrt], is expanded in terms of themodes as follows:

Srt = − 14π

[∂Art

∂t× [∇ × Art]

]=

i

4π

∑νν ′

ων ′[(

qν ′(t)Aν ′r

−q∗ν ′(t)Aν ′∗

r

)× (qν(t)[∇ × Aν

r ] + q∗ν(t)[∇ × Aν∗

r ])]. (19)

In the second quantization representation, the operator of the radiationflux density, Sr, is written as

Sr =i�

4π

∑νν ′

√ων ′

ων

[(bν ′Aν ′

r − b+ν ′Aν ′∗

r

)

×(bν [∇ × Aν

r ] + b+ν [∇ × Aν∗

r ])]

(20)

after expressing the amplitudes qν(t) in Eq. (19) through the corre-sponding creation and annihilation operators.


In a homogeneous and isotropic medium with dielectric permittivityε, the modes Aν

r are the plane waves Aqµ exp(iq · r) with wave vector qand polarization µ. The amplitudes Aqµ are determined from the vectorequation

[q × [q × Aqµ]] +(ωqµ

c

)2εAqµ = 0 (21)

following from Eq. (3) and from the gauge condition (q · Aqµ) = 0.Equation (21) is equivalent to a system of three algebraic equations forthe components of the vector Aqµ. The requirement of orthogonalityand normalization for the amplitudes is written as

V ε

2πc2 (A∗qµ · Aqµ′) = δµµ′ , (22)

so that one can introduce the unit vectors of polarization, eqµ, accordingto Aqµ =

√2πc2/εV eqµ. These vectors have the properties of transver-

sity (following from the gauge conditions), orthogonality, and normal-ization (following from Eq. (22)), while Eq. (21) leads to a polarization-independent dispersion relation for the photon of frequency ωq :

(q · eqµ) = 0, (e∗qµ · eqµ′) = δµµ′ , ωq = cq, c =

c√ε. (23)

These relations describe propagation of the photons whose unit vectorsof polarization, eqµ=1 and eqµ=2, are directed in the plane perpendicularto the wave vector q. The dispersion of the photons is linear in q, and theproportionality coefficient c is the velocity of light in the medium. TheHamiltonian and the radiation flux density operator for the homogeneousand isotropic medium are expressed, according to Eqs. (12) and (20),through the photonic creation and annihilation operators for the statesν = (q, µ). The polarization vectors and the frequency of these statesare given by Eq. (23). The operator of electric field can be obtainedfrom Eq. (18):

Er = i∑qµ

√2π�ωq

εVeqµeiq·r

(bqµ − b+

−qµ

), (24)

where we assumed that e∗qµ = e−qµ. The magnetic-field operator is given

by a similar expression, which is obtained according to Hr = [∇ × Ar]and contains the polarization factor [q×eqµ] under the sum. The matrixelement of Sr for the case of plane waves is determined according toEqs. (20), (22), and (13) as 〈nν |Sr|nν〉 = (q/q)cV −1

�ωq(nν + 1/2). Inthis form, the Poynting vector has direct meaning of the flux of photonenergy density with velocity c in the direction of q. In non-homogeneousmedia, the description of the modes based upon Eqs. (3) and (18) is


more sophisticated, though relatively simple results exist for the case ofone-dimensional inhomogeneities (problems 1.5 and 1.6).

Finally, let us calculate the averaged occupation number of the modeν for the equilibrium distribution of photons with temperature Tph. Thisdistribution is described by the density matrix

ηeq = Z−1 exp(−Hph/Tph), Z = Sp exp(−Hph/Tph). (25)

The partition function Z is expressed through the photon energy (9)written as E{nν} =

∑ν �ωνnν + E0, where E0 is the energy of zero

vibrations, according to Z = exp(−E0/Tph)Z and

Z =∑{nν}

∏ν

e−�ωνnν/Tph =∏ν

∑n

e−�ωνn/Tph

=∏ν

(1 − e−�ων/Tph

)−1. (26)

The mean value of the occupation number of the mode ν is defined asnν = Spnν ηeq . It is expressed through Z as

nν = Z−1∑

{nν1}nνe

−E{nν1}/Tph = −Tph∂ lnZ

∂(�ων). (27)

Calculating the derivative in Eq. (27), we obtain the equilibrium Planckdistribution

nν =[e�ων/Tph − 1

]−1. (28)

This distribution allows one to describe various equilibrium propertiesof the boson gas (problems 1.7 and 1.8). It is valid for all kinds of thebosons whose number is not fixed.

4. Many-Electron SystemIn contrast to the case of photons, the dynamics of a system of elec-

trically charged particles depends on their interactions with externalelectric fields (created by different, with respect to the system underconsideration, charges) and externally applied magnetic fields (note thatwe consider non-magnetic materials only), as well as on the interactionof these particles with each other. The existence of the spin variableleads to a further sophistication of such dynamics. Below we discuss thequantum dynamics for electrons, charged particles with two differentspin states. The electrons are an example of fermions, the particles withan antisymmetric, with respect to particle permutation, wave functioncorresponding to the Fermi-Dirac statistics.


The Hamiltonian of the electron system in external fields is writtenas

He =∑

j

hj + Hf . (1)

Here hj is the one-electron (the index j numbers the electrons) operatorof the kinetic energy. It is given by the equation

hj =(pj − eAxjt/c)2

2m, (2)

where pj − eAxjt/c is the operator of kinematic momentum expressedthrough the canonical momentum p satisfying the ordinary commutationrelations [pα, xβ ] = −i�δαβ and through the vector potential Axjt. Thesecond term of Eq. (1), Hf , is the operator of the field energy Ef , thelatter is given by the first part of Eq. (3.1). Using the expressions

Ert = −1c

∂Art

∂t− ∇Φrt , Hrt = [∇ × Art] (3)

relating the electric and magnetic fields to the vector potential Art andscalar potential Φrt, we rewrite Ef as

Ef =∫

(V )

dr8π

{1c2

∂Art

∂t· εr

∂Art

∂t+ ([∇ × Art])2

}

+∫

(V )

dr4πc

∇Φrtεr∂Art

∂t+∫

(V )

dr8π

∇Φrtεr∇Φrt . (4)

This equation generalizes Eq. (3.1) to the case of non-zero gradient ofthe scalar potential. The tensor εr is assumed to be symmetric. Belowwe again employ the Coulomb gauge ∇ · (εrArt) = 0 and assume thatthe fields go to zero at the boundaries of the region V (one may alsouse the periodic boundary conditions). The first term of the expression(4) corresponds to the energy of transverse vibrations of the field anddescribes the photons in medium. After the quantization of the fielddone in the previous section, we can denote this term as Hph. Thesecond term on the right-hand side of Eq. (4) is equal to zero because

∇Φrtεr∂Art

∂t= ∇ ·

(Φrtεr

∂Art

∂t

)(5)

in the gauge used, and the integral over the volume V is reduced to asurface integral over an infinitely remote boundary where the fields areequal to zero. The third term of the expression (4) can be rewritten


according to ∇Φε∇Φ = ∇·(Φε∇Φ)−Φ∇·(ε∇Φ), and only the last termhere remains finite after integrating over the volume. Next, by using thePoisson equation ∇· (εr∇Φrt) = −4πρrt, where ρrt is the charge density,we obtain the following expression for this term:

−∫

(V )

dr8π

Φrt∇ · (εr∇Φrt) =12

∫(V )

drΦrtρrt. (6)

Now we see that the third term on the right-hand side of Eq. (4) de-scribes the interaction of electric charges with the longitudinal part ofthe electric field. We denote it below as Eint. In a homogeneous andisotropic medium with constant dielectric permittivity ε, one can easilysolve the Poisson equation as Φrt = ε−1 ∫

(V )dr′ρr′t/|r − r′| so that Eint

is expressed through the charge densities only:

Eint =12

∫(V )

∫(V )

drdr′ ρrtρr′t

ε|r − r′| . (7)

One should remember that both Φrt and ρrt include the contribu-tions of the external fields and charges. To extract these contributionsfrom Eint, it is convenient to separate the contributions coming fromthe internal (i) and external (e) charges under the integrals of Eq. (7).Then,

Eint =12

∫ ∫(V )

drdr′ ρ(i)rt ρ

(i)r′t

ε|r − r′| +12

∫ ∫(V )

drdr′ ρ(e)rt ρ(e)

r′tε|r − r′|

+∫ ∫

(V )

drdr′ ρ(e)rt ρ(i)

r′tε|r − r′| . (8)

The first term on the right-hand side of Eq. (8) is the energy of Coulombinteraction between the electrons of the system (the electrostatic en-ergy). The second term is the energy of interaction between the ex-ternal charges. It should be omitted in the following, because such acontribution is not relevant to the dynamics of the system under consid-eration. Finally, the last term is the energy of interaction of electronswith the longitudinal part of the external field. It can be rewritten ase−1 ∫

(V )drUrtρ

(i)rt , where Urt is the potential energy of an electron in the

external field. One may introduce the potential of the external field asUrt/e. Below we omit the index i in ρ(i)

rt .To transform Eint into the operator of the interaction, Hint, one should

use the charge density operator ρr = e∑

j δ(r − xj) instead of ρrt. As aresult, we obtain

Hint =12

∑jj′

′ e2

ε|xj − xj′ | +∑

j

Uxjt. (9)

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