Quantum mechanics lecture notes 2018

quantum mechanics lecture notes cambridge and advanced quantum mechanics lecture notes
CharlieNixon Profile Pic
CharlieNixon,United Kingdom,Researcher
Published Date:13-07-2017
Your Website URL(Optional)
Comment
Quantum Mechanics nd 2 term 2002 Martin Plenio Imperial College Version January 28, 2002 Office hours: Tuesdays 11am-12noon and 5pm-6pm Office: Blackett 622 Available at:Contents I Quantum Mechanics 5 1 Mathematical Foundations 11 1.1 The quantum mechanical state space . . . . . . . . . . . 11 1.2 The quantum mechanical state space . . . . . . . . . . . 12 1.2.1 From Polarized Light to Quantum Theory . . . . 12 1.2.2 Complex vector spaces . . . . . . . . . . . . . . . 20 1.2.3 Basis and Dimension . . . . . . . . . . . . . . . . 23 1.2.4 Scalar products and Norms on Vector Spaces . . . 26 1.2.5 Completeness and Hilbert spaces . . . . . . . . . 36 1.2.6 Dirac notation . . . . . . . . . . . . . . . . . . . . 39 1.3 Linear Operators . . . . . . . . . . . . . . . . . . . . . . 40 1.3.1 Definition in Dirac notation . . . . . . . . . . . . 41 1.3.2 Adjoint and Hermitean Operators . . . . . . . . . 43 1.3.3 Eigenvectors, Eigenvalues and the Spectral The- orem . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.3.4 Functions of Operators . . . . . . . . . . . . . . . 51 1.4 Operators with continuous spectrum . . . . . . . . . . . 57 1.4.1 The position operator . . . . . . . . . . . . . . . 57 1.4.2 The momentum operator . . . . . . . . . . . . . . 61 1.4.3 The position representation of the momentum operator and the commutator between position and momentum . . . . . . . . . . . . . . . . . . . 62 2 Quantum Measurements 65 2.1 The projection postulate . . . . . . . . . . . . . . . . . . 65 2.2 Expectation value and variance. . . . . . . . . . . . . . . 70 2.3 Uncertainty Relations . . . . . . . . . . . . . . . . . . . . 71 12 CONTENTS 2.3.1 The trace of an operator . . . . . . . . . . . . . . 74 2.4 The density operator . . . . . . . . . . . . . . . . . . . . 76 2.5 Mixed states, Entanglement and the speed of light . . . . 82 2.5.1 Quantum mechanics for many particles . . . . . . 82 2.5.2 How to describe a subsystem of some large system? 86 2.5.3 The speed of light and mixed states. . . . . . . . 90 2.6 Generalized measurements . . . . . . . . . . . . . . . . . 92 3 Dynamics and Symmetries 93 3.1 The Schr¨odinger Equation . . . . . . . . . . . . . . . . . 93 3.1.1 The Heisenberg picture . . . . . . . . . . . . . . . 96 3.2 Symmetries and Conservation Laws . . . . . . . . . . . . 98 3.2.1 The concept of symmetry . . . . . . . . . . . . . 98 3.2.2 Translation Symmetry and momentum conserva- tion . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.2.3 RotationSymmetryandangularmomentumcon- servation . . . . . . . . . . . . . . . . . . . . . . 105 3.3 General properties of angular momenta . . . . . . . . . . 108 3.3.1 Rotations . . . . . . . . . . . . . . . . . . . . . . 108 3.3.2 Group representations and angular momentum commutation relations . . . . . . . . . . . . . . . 110 3.3.3 Angular momentum eigenstates . . . . . . . . . . 113 3.4 Addition of Angular Momenta . . . . . . . . . . . . . . . 116 3.4.1 Two angular momenta . . . . . . . . . . . . . . . 119 3.5 Local Gauge symmetries and Electrodynamics . . . . . . 123 4 Approximation Methods 125 4.1 Time-independent Perturbation Theory . . . . . . . . . . 126 4.1.1 Non-degenerate perturbation theory . . . . . . . . 126 4.1.2 Degenerate perturbation theory . . . . . . . . . . 129 4.1.3 The van der Waals force . . . . . . . . . . . . . . 132 4.1.4 The Helium atom . . . . . . . . . . . . . . . . . . 136 4.2 Adiabatic Transformations and Geometric phases . . . . 137 4.3 Variational Principle . . . . . . . . . . . . . . . . . . . . 137 4.3.1 The Rayleigh-Ritz Method . . . . . . . . . . . . . 137 4.4 Time-dependent Perturbation Theory . . . . . . . . . . . 141 4.4.1 Interaction picture . . . . . . . . . . . . . . . . . 143CONTENTS 3 4.4.2 Dyson Series . . . . . . . . . . . . . . . . . . . . . 145 4.4.3 Transition probabilities . . . . . . . . . . . . . . . 146 II Quantum Information Processing 153 5 Quantum Information Theory 155 5.1 What is information? Bits and all that. . . . . . . . . . . 158 5.2 From classical information to quantum information. . . . 158 5.3 Distinguishingquantumstatesandtheno-cloningtheorem.158 5.4 Quantum entanglement: From qubits to ebits. . . . . . . 158 5.5 Quantum state teleportation. . . . . . . . . . . . . . . . 158 5.6 Quantum dense coding. . . . . . . . . . . . . . . . . . . . 158 5.7 Local manipulation of quantum states. . . . . . . . . . . 158 5.8 Quantum cyptography . . . . . . . . . . . . . . . . . . . 158 5.9 Quantum computation . . . . . . . . . . . . . . . . . . . 158 5.10 Entanglement and Bell inequalities . . . . . . . . . . . . 160 5.11 Quantum State Teleportation . . . . . . . . . . . . . . . 166 5.12 A basic description of teleportation . . . . . . . . . . . . 1674 CONTENTSPart I Quantum Mechanics 57 Introduction Thislecturewillintroducequantummechanicsfromamoreabstract point of view than the first quantum mechanics course that you took your second year. What I would like to achieve with this course is for you to gain a deeper understanding of the structure of quantum mechanics and of some of its key points. As the structure is inevitably mathematical, I will need to talk about mathematics. I will not do this just for the sake of mathematics, but always with a the aim to understand physics. At theendofthecourseIwouldlikeyounotonlytobeabletounderstand the basic structure of quantum mechanics, but also to be able to solve (calculate) quantum mechanical problems. In fact, I believe that the ability to calculate (finding the quantitative solution to a problem, or the correct proof of a theorem) is absolutely essential for reaching a real understanding of physics (although physical intuition is equally important). I would like to go so far as to state If you can’t write it down, then you do not understand it With ’writing it down’ I mean expressing your statement mathemati- cally or being able to calculate the solution of a scheme that you pro- posed. Thisdoesnotsoundlikeaveryprofoundtruthbutyouwouldbe surprised to see how many people actually believe that it is completely sufficient just to be able to talk about physics and that calculations are a menial task that only mediocre physicists undertake. Well, I can as- sure you that even the greatest physicists don’t just sit down and await inspiration. Ideas only come after many wrong tries and whether a try is right or wrong can only be found out by checking it, i.e. by doing some sorts of calculation or a proof. The ability to do calculations is not something that one has or hasn’t, but (except for some exceptional cases) has to be acquired by practice. This is one of the reasons why these lectures will be accompanied by problem sheets (Rapid Feedback System) and I really recommend to you that you try to solve them. It is quite clear that solving the problem sheets is one of the best ways to prepare fortheexam. SometimesI willaddsomeextraproblemstothe problem sheets which are more tricky than usual. They usually intend8 to illuminate an advanced or tricky point for which I had no time in the lectures. The first part of these lectures will not be too unusual. The first chapter will be devoted to the mathematical description of the quan- tummechanicalstatespace, theHilbertspace, andofthedescriptionof physical observables. The measurement process will be investigated in the next chapter, and some of its implications will be discussed. In this chapteryouwillalsolearntheessentialtoolsforstudyingentanglement, thestuffsuchweirdthingsasquantumteleportation,quantumcryptog- raphy and quantum computation are made of. The third chapter will presentthedynamicsofquantummechanicalsystemsandhighlightthe importance of the concept of symmetry in physics and particularly in quantum mechanics. It will be shown how the momentum and angular momentum operators can be obtained as generators of the symmetry groups of translation and rotation. I will also introduce a different kind ofsymmetrieswhicharecalledgaugesymmetries. Theyallowusto’de- rive’ the existence of classical electrodynamics from a simple invariance principle. This idea has been pushed much further in the 1960’s when people applied it to the theories of elementary particles, and were quite successful with it. In fact, t’Hooft and Veltman got a Nobel prize for it in 1999 work in this area. Time dependent problems, even more than time-independent problems, are difficult to solve exactly and therefore perturbation theoretical methods are of great importance. They will be explained in chapter 5 and examples will be given. Mostoftheideasthatyouaregoingtolearninthefirstfivechapters of these lectures are known since about 1930, which is quite some time ago. The second part of these lectures, however, I will devote to topics which are currently the object of intense research (they are also my main area of research). In this last chapter I will discuss topics such as entanglement, Bell inequalities, quantum state teleportation, quantum computation and quantum cryptography. How much of these I can coverdependsontheamountoftimethatisleft,butIwillcertainlytalk aboutsomeofthem. Whilemostphysicists(hopefully)knowthebasics of quantum mechanics (the first five chapters of these lectures), many of them will not be familiar with the content of the other chapters. So, aftertheselecturesyoucanbesuretoknowaboutsomethingthatquite a few professors do not know themselves I hope that this motivates9 you to stay with me until the end of the lectures. Before I begin, I would like to thank, Vincenzo Vitelli, John Pa- padimitrou and William Irvine who took this course previously and spotted errors and suggested improvements in the lecture notes and the course. These errors are fixed now, but I expect that there are more. If you find errors, please let me know (ideally via email so that the corrections do not get lost again) so I can get rid of them. Last but not least, I would like to encourage you both, to ask ques- tions during the lectures and to make use of my office hours. Questions areessentialinthelearningprocess, sotheyaregoodforyou, butIalso learnwhatyouhavenotunderstoodsowellandhelpmetoimprovemy lectures. Finally, it is more fun to lecture when there is some feedback from the audience.10Chapter 1 Mathematical Foundations Before I begin to introduce some basics of complex vector spaces and discuss the mathematical foundations of quantum mechanics, I would like to present a simple (seemingly classical) experiment from which we can derive quite a few quantum rules. 1.1 The quantum mechanical state space When we talk about physics, we attempt to find a mathematical de- scription of the world. Of course, such a description cannot be justified frommathematicalconsistencyalone, buthastoagreewithexperimen- tal evidence. The mathematical concepts that are introduced are usu- allymotivatedfromourexperienceofnature. Conceptssuchasposition and momentum or the state of a system are usually taken for granted in classical physics. However, many of these have to be subjected to a careful re-examination when we try to carry them over to quantum physics. One of the basic notions for the description of a physical sys- temisthatofits’state’. The’state’ofaphysicalsystemessentiallycan then be defined, roughly, as the description of all the known (in fact oneshouldsayknowable)propertiesofthatsystemanditthereforerep- resents your knowledge about this system. The set of all states forms whatweusuallycallthestatespace. Inclassicalmechanicsforexample thisisthephasespace(thevariablesarethenpositionandmomentum), which is a real vector space. For a classical point-particle moving in 1112 CHAPTER 1. MATHEMATICAL FOUNDATIONS one dimension, this space is two dimensional, one dimension for posi- tion, one dimension for momentum. We expect, in fact you probably know this from your second year lecture, that the quantum mechanical state space differs from that of classical mechanics. One reason for this can be found in the ability of quantum systems to exist in coherent superpositions of states with complex amplitudes, other differences re- late to the description of multi-particle systems. This suggests, that a good choice for the quantum mechanical state space may be a complex vector space. Before I begin to investigate the mathematical foundations of quan- tum mechanics, I would like to present a simple example (including some live experiments) which motivates the choice of complex vector spaces as state spaces a bit more. Together with the hypothesis of the existence of photons it will allow us also to ’derive’, or better, to make an educated guess for the projection postulate and the rules for the computation of measurement outcomes. It will also remind you of some of the features of quantum mechanics which you have already encountered in your second year course. 1.2 The quantum mechanical state space InthenextsubsectionIwillbrieflymotivatethatthequantummechan- ical state space should be a complex vector space and also motivate some of the other postulates of quantum mechanics 1.2.1 From Polarized Light to Quantum Theory Let us consider plane waves of light propagating along the z-axis. This light is described by the electric field vector E orthogonal on the di- rection of propagation. The electric field vector determines the state of light because in the cgs-system (which I use for convenience in this example so that I have as few andμ as possible.) the magnetic field 0 0 isgivenbyB =e×E. Giventheelectricandmagneticfield, Maxwells z equations determine the further time evolution of these fields. In the absence of charges, we know that E(r,t) cannot have a z-component,1.2. THE QUANTUM MECHANICAL STATE SPACE 13 so that we can write E (r,t) x E(r,t) =E (r,t)e +E (r,t)e = . (1.1) x x y y E (r,t) y The electric field is real valued quantity and the general solution of the free wave equation is given by 0 E (r,t) = E cos(kz−ωt+α ) x x x 0 E (r,t) = E cos(kz−ωt+α ) . y y y Here k = 2π/λ is the wave-number, ω = 2πν the frequency, α and α x y 0 0 are the real phases and E and E the real valued amplitudes of the x y field components. The energy density of the field is given by 1 2 2 (r,t) = (E (r,t)+B (r,t)) 8π h i 1 0 2 2 0 2 2 = (E ) cos (kz−ωt+α )+(E ) cos (kz−ωt+α ) . x y x y 4π Forafixedpositionr wearegenerallyonlyreallyinterestedinthetime- averagedenergydensitywhich, whenmultipliedwiththespeedoflight, determinestherateatwhichenergyflowsinz-direction. Averagingover one period of the light we obtain the averaged energy density¯(r) with h i 1 0 2 0 2 ¯(r) = (E ) +(E ) . (1.2) x y 8π For practical purposes it is useful to introduce thecomplex field com- ponents i(kz−ωt) i(kz−ωt) E (r,t) =Re(E e ) E (r,t) =Re(E e ) , (1.3) x x y y 0 iα 0 iα x y withE =E e andE =E e . Comparing with Eq. (1.2) we find x y x y that the averaged energy density is given by h i 1 2 2 ¯(r) = E +E . (1.4) x y 8π Usually one works with the complex field E x i(kz−ωt) i(kz−ωt) E(r,t) = (E e +E e )e = e . (1.5) x x y y E y14 CHAPTER 1. MATHEMATICAL FOUNDATIONS Thismeansthatwearenowcharacterizingthestateoflightbyavector with complex components. The polarization of light waves are described by E and E . In the x y general case of complex E and E we will have elliptically polarized x y light. There are a number of important special cases (see Figures 1.1 for illustration). 1. E = 0: linear polarization along the x-axis. y 2. E = 0: linear polarization along the y-axis. x 0 3. E =E : linear polarization along 45 -axis. x y 4. E =iE : Right circularly polarized light. y x 5. E =−iE : Left circularly polarized light. y x Figure 1.1: Left figure: Some possible linear polarizations of light, horizontally, vertically and 45 degrees. Right figure: Left- and right- circularly polarized light. The light is assumed to propagate away from you. In the following I would like to consider some simple experiments for which I will compute the outcomes using classical electrodynam- ics. Then I will go further and use the hypothesis of the existence of photons to derive a number of quantum mechanical rules from these experiments. Experiment I:Letusfirstconsideraplanelightwavepropagating inz-directionthatisfallingontoanx-polarizerwhichallowsx-polarized1.2. THE QUANTUM MECHANICAL STATE SPACE 15 light to pass through (but not y polarized light). This is shown in figure 1.2. After passing the polarizer the light is x-polarized and from Figure 1.2: Light of arbitrary polarization is hitting a x-polarizer. the expression for the energy density Eq. (1.4) we find that the ratio between incoming intensity I (energy density times speed of light) in and outgoing intensity I is given by out 2 I E out x = . (1.6) 2 2 I E +E in x y So far this looks like an experiment in classical electrodynamics or optics. Quantum Interpretation: Let us change the way of looking at this problem and thereby turn it into a quantum mechanical experi- ment. Youhaveheardatvariouspointsinyourphysicscoursethatlight comesinlittlequantaknownasphotons. Thefirsttimethisassumption had been made was by Planck in 1900 ’as an act of desperation’ to be able to derive the blackbody radiation spectrum. Indeed, you can also observe in direct experiments that the photon hypothesis makes sense. When you reduce the intensity of light that falls onto a photodetector, you will observe that the detector responds with individual clicks each triggered by the impact of a single photon (if the detector is sensitive enough). The photo-electric effect and various other experiments also confirm the existence of photons. So, in the low-intensity limit we have to consider light as consisting of indivisible units called photons. It is a fundamental property of photons that they cannot be split – there is no such thing as half a photon going through a polarizer for example. In this photon picture we have to conclude that sometimes a photon will be absorbed in the polarizer and sometimes it passes through. If the photon passes the polarizer, we have gained one piece of informa- tion, namely that the photon was able to pass the polarizer and that therefore it has to be polarized in x-direction. The probability p for16 CHAPTER 1. MATHEMATICAL FOUNDATIONS the photon to pass through the polarizer is obviously the ratio between transmitted and incoming intensities, which is given by 2 E x p = . (1.7) 2 2 E +E x y If we write the state of the light with normalized intensity E E x y q q E = e + e , (1.8) N x y 2 2 2 2 E +E E +E x y x y then in fact we find that the probability for the photon to pass the x- polarizer is just the square of the amplitude in front of the basis vector e This is just one of the quantum mechanical rules that you have x learned in your second year course. Furthermore we see that the state of the photon after it has passed the x-polarizer is given by E =e , (1.9) N x     E E x x ie the state has has changed from to . This transformation E 0 y of the state can be described by a matrix acting on vectors, ie   E E 1 0 x x = (1.10) 0 0 0 E y The matrix that I have written here has eigenvalues 0 and 1 and is therefore a projection operator which you have heard about in the sec- ondyearcourse,infactthisremindsstronglyoftheprojectionpostulate in quantum mechanics. Experiment II: Now let us make a somewhat more complicated experiment by placing a second polarizer behind the first x-polarizer. 0 The second polarizer allows photons polarized in x direction to pass through. If I slowly rotate the polarizer from the x direction to the y direction, we observe that the intensity of the light that passes through the polarizer decreases and vanishes when the directions of the two polarizers are orthogonal. I would like to describe this experiment mathematically. How do we compute the intensity after the polarizer1.2. THE QUANTUM MECHANICAL STATE SPACE 17 now? To this end we need to see how we can express vectors in the 0 basis chosen by the directionx in terms of the old basis vectorse ,e . x y 0 0 The new rotated basise ,e (see Fig. 1.3) can be expressed by the x y old basis by 0 0 e = cosφe +sinφe e =−sinφe +cosφe (1.11) x y x y x y and vice versa 0 0 0 0 e = cosφe −sinφe e = sinφe +cosφe . (1.12) x y x y x y 0 0 Note that cosφ =e ·e and sinφ =e ·e where I have used the real x y x x scalar product between vectors. Figure 1.3: The x’- basis is rotated by an angle φ with respect to the original x-basis. The state of the x-polarized light after the first polarizer can be rewritten in the new basis of the x’-polarizer. We find 0 0 0 0 0 0 E =E e =E cosφe −E sinφe =E (e ·e )e −E (e ·e )e x x x x x x x y x y x x y y Now we can easily compute the ratio between the intensity before 0 and after the x-polarizer. We find that it is I after 0 2 2 =e ·e =cos φ (1.13) x x I before or if we describe the light in terms of states with normalized intensity as in equation 1.8, then we find that 0 2 I e ·E after N x 0 2 =e ·E = (1.14) N x 0 2 0 2 I before e ·E +e ·E N N x y18 CHAPTER 1. MATHEMATICAL FOUNDATIONS where E is the normalized intensity state of the light after the x- N polarizer. This demonstrates that the scalar product between vectors plays an important role in the calculation of the intensities (and there- fore the probabilities in the photon picture). Varying the angleφ between the two bases we can see that the ratio oftheincomingandoutgoingintensitiesdecreaseswithincreasingangle o between the two axes until the angle reaches 90 degrees. Interpretation: Viewed in the photon picture this is a rather surprising result, as we would have thought that after passing the x- polarizer the photon is ’objectively’ in the x-polarized state. However, 0 uponprobingitwithanx-polarizerwefindthatitalsohasaqualityof 0 anx-polarized state. In the next experiment we will see an even more worryingresult. Forthemomentwenotethatthestateofaphotoncan be written in different ways and this freedom corresponds to the fact that in quantum mechanics we can write the quantum state in many different ways as a quantum superpositions of basis vectors. Let us push this idea a bit further by using three polarizers in a row. ExperimentIII:Ifafterpassingthex-polarizer,thelightfallsonto a y-polarizer (see Fig 1.4), then no light will go through the polarizer because the two directions are perpendicular to each other. This sim- Figure 1.4: Light of arbitrary polarization is hitting a x-polarizer and subsequently a y-polarizer. No light goes through both polarizers. ple experimental result changes when we place an additional polarizer between the x and the y-polarizer. Assume that we place a x’-polarizer between the two polarizers. Then we will observe light after the y- 0 polarizer (see Fig. 1.5) depending on the orientation of x. The light ˜ ˜ afterthelastpolarizerisdescribedbyE e . TheamplitudeE iscalcu- y y y lated analogously as in Experiment II. Now let us describe the (x-x’-y) experiment mathematically. The complex electric field (without the time dependence) is given by1.2. THE QUANTUM MECHANICAL STATE SPACE 19 Figure 1.5: An x’-polarizer is placed in between an x-polarizer and a y-polarizer. Now we observe light passing through the y-polarizer. before the x-polarizer: E =E e +E e . 1 x x y y after the x-polarizer: 0 0 E = (E e )e =E e =E cosφe −E sinφe . 2 1 x x x x x x x y after the x’-polarizer: 0 0 0 2 E = (E e )e =E cosφe =E cos φe +E cosφsinφe . 3 2 x x x x y x x x after the y-polarizer: ˜ E = (E e )e =E cosφsinφe =E e . 4 3 y y x y y y Therefore the ratio between the intensity before the x’-polarizer and after the y-polarizer is given by I after 2 2 = cos φsin φ (1.15) I before Interpretation: Again, if we interpret this result in the photon picture, then we arrive at the conclusion, that the probability for the photon to pass through both the x’ and the y polarizer is given by 2 2 cos φsin φ. This experiment further highlights the fact that light of one polarization may be interpreted as a superposition of light of other polarizations. This superposition is represented by adding vectors with complex coefficients. If we consider this situation in the photon picture we have to accept that a photon of a particular polarization can also be interpreted as a superposition of different polarization states.