Preface at first sight (one needs a quantum description of a huge number of degrees of freedom) we show that it can be reduced to a much simpler form,characterizing the environment in terms of its damping coefficient or dynamical susceptibility.After explaining this proce- dure for the damped oscillator in Chapter 11 and discussing dissipation and fluctuations. in Chapter 12 we extend the picture to a qubit(two-level system)in a dissipative envi- ronment t.We elucidate the roe the ent plays in transitions b ween the two qubit states,and,based on what we find,we provide a very general scheme to classify all possible types of environment. In the last part(and chapter)of the book,we give a short introduction to relativistic quantum mechanics.We explain how relativity is a fundamental symmetry of our world. ally leads us to the Dirac equation.Apart from obeying the relativistic symmetry.the Dirac equation predicted revolutionary new concepts,such as the existence of particles and anti-particles.Since the existence of anti- particles has been experimentally confirmed,just a few years after Dirac had put forward include them into our second quantization framework then explain ho particesanti-par icles and the elec magnetic field constitutes the basis of quantum electrodynamics.We briefly touch on this topic and show how a naive application of perturbation theory in the interaction between radiation and matter leads to divergences of almost all corrections one tries to calculate. The way to handle these divergences is given by the theory of renormalization,of which we discuss the basic idea in the last section of the chapte The book thus takes examples and applications from many different fields:we discuss the laser,the Cooper pair box,magnetism,positrons,vortices in superfluids,and many more examples.In this way,the book gives a very broad view on advanced quantum the- ory.It would be very well suited to serve as the principal required text for a master-level course on advanced quantum mechanics which is not exclusively directed toward elemen- tary particle physics.All m terial in the book could be covered in on depending on the amount of time available per week.The five parts of the book are also relatively self-contained,and could be used separately. All chapters contain many"control questions,"which are meant to slow the pace of the student and make sure that he or she is actively following the thread of the text.These ques- tions ofor instance be s during the At th there are four to ten larger exercis s,0e s,others present ing more interesting physical problems.We decided to provide in this book the solutions to one or two exercises per chapter,enabling students to independently try to solve a serious problem and check what they may have done wrong.The rest of the solutions are available online for teachers,and the corresponding exercises could be used as homework for the stud We hope that many students around the world will enjoy this book.We did our absolute best to make sure that no single typo or missing minus sign made it to the printed version, but this is probably an unrealistic endeavor:we apologize beforehand for surviving errors. If you find one,please be so kind to notify us,this would highly improve the quality of a possible next edition of this book
xiii Preface at first sight (one needs a quantum description of a huge number of degrees of freedom), we show that it can be reduced to a much simpler form, characterizing the environment in terms of its damping coefficient or dynamical susceptibility. After explaining this procedure for the damped oscillator in Chapter 11 and discussing dissipation and fluctuations, in Chapter 12 we extend the picture to a qubit (two-level system) in a dissipative environment. We elucidate the role the environment plays in transitions between the two qubit states, and, based on what we find, we provide a very general scheme to classify all possible types of environment. In the last part (and chapter) of the book, we give a short introduction to relativistic quantum mechanics. We explain how relativity is a fundamental symmetry of our world, and recognize how this leads to the need for a revised “relativistic Schrödinger equation.” We follow the search for this equation, which finally leads us to the Dirac equation. Apart from obeying the relativistic symmetry, the Dirac equation predicted revolutionary new concepts, such as the existence of particles and anti-particles. Since the existence of antiparticles has been experimentally confirmed, just a few years after Dirac had put forward his theory, we accept their existence and try to include them into our second quantization framework. We then explain how a description of particles, anti-particles, and the electromagnetic field constitutes the basis of quantum electrodynamics. We briefly touch on this topic and show how a naive application of perturbation theory in the interaction between radiation and matter leads to divergences of almost all corrections one tries to calculate. The way to handle these divergences is given by the theory of renormalization, of which we discuss the basic idea in the last section of the chapter. The book thus takes examples and applications from many different fields: we discuss the laser, the Cooper pair box, magnetism, positrons, vortices in superfluids, and many more examples. In this way, the book gives a very broad view on advanced quantum theory. It would be very well suited to serve as the principal required text for a master-level course on advanced quantum mechanics which is not exclusively directed toward elementary particle physics. All material in the book could be covered in one or two semesters, depending on the amount of time available per week. The five parts of the book are also relatively self-contained, and could be used separately. All chapters contain many “control questions,” which are meant to slow the pace of the student and make sure that he or she is actively following the thread of the text. These questions could for instance be discussed in class during the lectures. At the end of each chapter there are four to ten larger exercises, some meant to practice technicalities, others presenting more interesting physical problems. We decided to provide in this book the solutions to one or two exercises per chapter, enabling students to independently try to solve a serious problem and check what they may have done wrong. The rest of the solutions are available online for teachers, and the corresponding exercises could be used as homework for the students. We hope that many students around the world will enjoy this book. We did our absolute best to make sure that no single typo or missing minus sign made it to the printed version, but this is probably an unrealistic endeavor: we apologize beforehand for surviving errors. If you find one, please be so kind to notify us, this would highly improve the quality of a possible next edition of this book
Preface Finally we would like to thank our colleagues in the kavli institute of nanoscience at the Delft University of Technology and in the Dahlem Center for Complex Quantum Sys- mn understanding of everyone around us for this.J.D.would like to thank in particular Piet Brouwer and Dganit Meidan:they both were always willing to free some time for very helpful discussions about the content and style of the material in preparation. Yuli V.Nazarov Jeroen Danon
xiv Preface Finally, we would like to thank our colleagues in the Kavli Institute of Nanoscience at the Delft University of Technology and in the Dahlem Center for Complex Quantum Systems at the Free University of Berlin. Especially in the last few months, our work on this book often interfered severely with our regular tasks, and we very much appreciate the understanding of everyone around us for this. J.D. would like to thank in particular Piet Brouwer and Dganit Meidan: they both were always willing to free some time for very helpful discussions about the content and style of the material in preparation. Yuli V. Nazarov Jeroen Danon
PART I SECOND QUANTIZATION
PART I SECOND QUANTIZATION
Elementary quantum mechanics We assume that the reader is already acquainted with elementary quantum mechanics.An introductory course in quantum mechanics usually addresses most if not all concepts dis- cussed in this chapter.However.there are many ways to teach and learn these subiects.By including this chapter.we can make sure that we understand the basics in the same way.We advise students to read the first six sections(those on classical mechanics,the Schrodinge equation,the Dirac formulation,and perturbation theory)before going on to the advanced subjects of the next chapters,since these concepts will be needed immediately.While the other sections of this chapter address fundamentals of quantum mechanics as well,they do not have to be read right away and are referred to in the corresponding places of the follow- ing chapt ers.The te is meant to be con supply rigorou proofs or lengthy explan ons.The basics of quantum mechanics should be mastered at ar operational level:please also check Table 1.1 and the exercises at the end of the chapter. 1.1 Classical mechanics Let us start by considering a single particle of mass m,which is moving in a coordinate- dependent potential V(r).In classical physics,the state of this particle at a given moment of time is fully characterized by two ctors.its coordinate ()and its momentum p(). in the future-its position and momentum-can be unambiguously predicted once the initial state of the particle is known.The time evolution of the state is given by Newton's well-known equations 1.1) ar Here the force F acting on the particle is given by the derivative of the potential V(r),and momentum and velocity are related by p=mv. Classical mechanics can be formulated in a variety of equivalent ways.A commonly used alternative to Newton's laws are Hamilton's equations of motion (1.2) 3
1 Elementary quantum mechanics We assume that the reader is already acquainted with elementary quantum mechanics. An introductory course in quantum mechanics usually addresses most if not all concepts discussed in this chapter. However, there are many ways to teach and learn these subjects. By including this chapter, we can make sure that we understand the basics in the same way. We advise students to read the first six sections (those on classical mechanics, the Schrödinger equation, the Dirac formulation, and perturbation theory) before going on to the advanced subjects of the next chapters, since these concepts will be needed immediately. While the other sections of this chapter address fundamentals of quantum mechanics as well, they do not have to be read right away and are referred to in the corresponding places of the following chapters. The text of this chapter is meant to be concise, so we do not supply rigorous proofs or lengthy explanations. The basics of quantum mechanics should be mastered at an operational level: please also check Table 1.1 and the exercises at the end of the chapter. 1.1 Classical mechanics Let us start by considering a single particle of mass m, which is moving in a coordinatedependent potential V(r). In classical physics, the state of this particle at a given moment of time is fully characterized by two vectors, its coordinate r(t) and its momentum p(t). Since classical mechanics is a completely deterministic theory, the state of the particle in the future – its position and momentum – can be unambiguously predicted once the initial state of the particle is known. The time evolution of the state is given by Newton’s well-known equations dp dt = F = −∂V(r) ∂r and dr dt = v = p m. (1.1) Here the force F acting on the particle is given by the derivative of the potential V(r), and momentum and velocity are related by p = mv. Classical mechanics can be formulated in a variety of equivalent ways. A commonly used alternative to Newton’s laws are Hamilton’s equations of motion dp dt = −∂H ∂r and dr dt = ∂H ∂p , (1.2) 3
4 Elementary quantum mechanics where the Hamiltonian function H(r,p)of a particle is defined as the total of its kinetic and potential energy. H= m+V(r). (1.3) One advantage of this formalism is a clear link to the quantum mechanical description of the particle we present below. An important property of Hamilton's equations of motion is that the Hamiltonian Hitself is a constant of motion,i.e.dH/dt =0 in the course of motion. Control question.Can you prove thatH=directly follows from the equations of motion(1.2)? This is natural since it represents the total energy of the particle.and energy is conserved. This conservation of energy does not hold for Hamiltonians that explicitly depend on stance time-dependent force F gives an addition ext·rto the Hamiltonian.By changing the force,we can manipulate the particle and change its energy. 1.2 Schrodinger equation The quantum mechanical description of the same particle does not include any new param eters,except for a universal constant h.The dynamics of the particle are still determined by its mass m and the external potential V(r).The difference is now that the state of the particle is no longer characterized by just two vectors r(t)and p(r),but rather by a contin- ous function of coordinate)which is called the of the particle.The interpretation of this wave function is probabilistic:its modulus square(gives th probability density to find the particle at time t at the point r.For this interpretation to make sense,the wave function must obey the normalization condition dr lv(r.)2=1. (1.4) ie.the total probability to find the particle anywhere is 1.or in other words,the particle must be se mewhere at t any mo ent of time Since this is a probabilistic description,we never know exactly where the particle is.If the particle is at some time to in a definite state(r.to).then it is generally still impossible to predict at which point in space we will find the particle if we look for it at another time t. However.despite its intrinsically probabilistic nature.quantum mechanics is a determinis tic theory.Starting from the state.the state detemied ban volo o.the time-dependent Sompetely =w= 2m丽+ (1.5
4 Elementary quantum mechanics where the Hamiltonian function H(r, p) of a particle is defined as the total of its kinetic and potential energy, H = p2 2m + V(r). (1.3) One advantage of this formalism is a clear link to the quantum mechanical description of the particle we present below. An important property of Hamilton’s equations of motion is that the Hamiltonian H itself is a constant of motion, i.e. dH/dt = 0 in the course of motion. Control question. Can you prove that dH/dt = 0 directly follows from the equations of motion (1.2)? This is natural since it represents the total energy of the particle, and energy is conserved. This conservation of energy does not hold for Hamiltonians that explicitly depend on time. For instance, an external time-dependent force Fext gives an addition −Fext · r to the Hamiltonian. By changing the force, we can manipulate the particle and change its energy. 1.2 Schrödinger equation The quantum mechanical description of the same particle does not include any new parameters, except for a universal constant h¯. The dynamics of the particle are still determined by its mass m and the external potential V(r). The difference is now that the state of the particle is no longer characterized by just two vectors r(t) and p(t), but rather by a continuous function of coordinate ψ(r, t) which is called the wave function of the particle. The interpretation of this wave function is probabilistic: its modulus square |ψ(r, t)| 2 gives the probability density to find the particle at time t at the point r. For this interpretation to make sense, the wave function must obey the normalization condition dr |ψ(r, t)| 2 = 1, (1.4) i.e. the total probability to find the particle anywhere is 1, or in other words, the particle must be somewhere at any moment of time. Since this is a probabilistic description, we never know exactly where the particle is. If the particle is at some time t0 in a definite state ψ(r, t0), then it is generally still impossible to predict at which point in space we will find the particle if we look for it at another time t. However, despite its intrinsically probabilistic nature, quantum mechanics is a deterministic theory. Starting from the state ψ(r, t0), the state ψ(r, t) at any future time t is completely determined by an evolution equation, the time-dependent Schrödinger equation ih¯ ∂ψ ∂t = Hˆ ψ = − h¯ 2 2m ∂2 ∂r2 + V(r) ψ. (1.5)