Lecture Notes on Quantum Physics

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LectureNotesonQuantumPhysics MatthewFoulkes DepartmentofPhysics ImperialCollegeLondon SpringandSummerTerms,2011Chapter1 Introduction 1.1 WhatisQuantumPhysics? Quantumphysics is a catch-all term for the ideas, devices and technologies made possible by the development of quantum mechanics in the early part of the 20th century. This course concentrates on the ideas behind quantum mechanics itself, but the broader field of quantum physics encompasses everything from the sci- ence of electronic devices and lasers to the philosophical mysteries of quantum measurement theory. Quantum mechanics is our best current theory of matter and how it inter- acts. Matter in this context includes everything we normally think of as particles, waves, forces, and fields. In the quantum world, these are all (more or less) the same thing. Given an experimental set-up, quantum mechanics tells you: 1. What can be measured. 2. The possible results of any measurement. 3. The probability of obtaining each of the possible results. The rules used to calculate the probabilities, although abstract and mathematical, are precise and unambiguous. As a practical tool, quantum mechanics presents no difficulties and has been immensely successful. 34 CHAPTER1. INTRODUCTION 1.2 SuccessesandFailuresofQuantumPhysics To illustrate the extraordinary power and breadth of quantum theory, here are just a few of the phenomena it can explain: Atomic structure and spectra Radioactivity Properties and interactions of elementary particles Nucleosynthesis Semiconductor physics & devices Laser physics Superconductivity and superfluidity Chemical reactions The periodic table Density of matter Conductivity of copper Strength of steel Hardness of diamond Stability of matter Properties of neutron stars and white dwarfs Fisson/fusion Magnetism (The human brain?) Some of the items in the list may strike you as classical, but if you ask one or two “why” questions you soon find yourself running in to quantum mechanics. Take the density of matter as an example: this depends on the size of an atom, which depends on the radius of an electron orbit and hence on quantum theory. In fact, the radius of a Hydrogen atom, known as theBohrradiusa , is given by 0 2 4 0 10 a =  0:52910 m; 0 2 me 34 where := h=2  1:05 10 Js is Planck’s constant divided by 2. The appearance of Planck’s constant leaves no doubt that quantum theory is involved. The version of quantum theory covered in this course neglects relativistic ef- fects and is therefore an approximation, just as Newton’s laws are an approxima- tion to special relativity. The relativistic version of quantum mechanics, called quantum field theory, is very similar in outline but mathematically more difficult. Quantum theory as a whole (including quantum field theory) has never been known to fail. Its applications have been limited by the difficulty of solving the equations, which are only tractable for rather simple systems, so there is no guar- antee that problems will never be found; but even then quantum theory would remain useful, just as Newton’s laws remained useful after the advent of special relativity. There is, as yet, no good quantised theory of gravity, but whether this indicates a fundamental problem with quantum mechanics or a failure of human ingenuity is unclear.1.3. QUANTUMWEIRDNESS 5 1.3 QuantumWeirdness The most fascinating aspect of quantum mechanics is that it provides such a strange picture of the world. If you accept this picture — and given the practi- cal successes of the theory it is difficult not to — you are left with no choice but to make fundamental changes to your idea of reality. The first surprise is the wave-particle duality of the building blocks of mat- ter. The world is not made of waves and particles, as in classical physics, but of peculiar hybrid objects with aspects of both. Suppose, for example, that you find an electron at r at timet and then at r at a later timet . Since the electron is 1 1 2 2 supposed to be a particle, you might imagine that it travelled along some specific path r(t) from r = r(t ) to r = r(t ). According to Feynman’s path-integral 1 1 2 2 formulation of quantum mechanics, however, this is wrong. In a precise mathe- matical sense (only hinted at in this course), the electron took all possible paths from r to r at once. Even worse, the components arriving along different paths 1 2 interfered like waves. Wave-particle duality is not the only strange aspect of quantum theory. The physical state of a quantum mechanical particle-wave is described by awavefunc- tion, (x;t), analogous to the amplitude of a classical wave. Unlike a classical wave, however, (x;t) does not evolve according to the classical wave equation, 2 2 (x;t) 1 (x;t) = ; 2 2 2 x v t where v is the phase velocity, but according to the time-dependent Schrödinger equation, 2 2 (x;t) (x;t) +V(x) (x;t) =i ; 2 2m x t where m is the mass of the particle and V(x) is the potential through which it moves. The most striking feature of Schrödinger’s equation is that it has ani on the right-hand side, implying that the wave function is complex. Even if, by some fluke, (x;t) happened to be real at t = 0, it would not remain real. Complex waves are common in classical physics, of course, but the complex numbers are used only to simplify the mathematics and the physical waves remain real. In quantum theory, the wave function isreally complex. Perhaps the most puzzling aspect of quantum mechanics is that it predicts probabilities only. In classical physics, probabilities are used to describe our lack of knowledge of a physical system: if we know nothing about how a pack of cards has been shuffled, the probability of picking any particular card, say the three of6 CHAPTER1. INTRODUCTION spades, is 1=52; if we know where all the cards are in advance, we can find the three of spades every time and there is no need for probability theory. Even for a complicated system such as the air in the Albert Hall, we could, in principle, measure the positions and velocities of all the molecules and predict the future evolution using Newton’s laws; the probabilistic Maxwell-Boltzmann distribution is used only because the measurement is impractical and our knowledge incom- plete. It is tempting to imagine that the probabilistic nature of quantum theory arises in a similar way, and that quantum mechanics is just a rough statistical description of some more complicated underlying reality. As in the case of the air in the Albert Hall, we use a probabilistic description (there the Maxwell-Boltzmann equation; here quantum theory) only because our knowledge is incomplete. If we could discover the values of the hidden variables describing the underlying reality, we could dispense with probability theory altogether. Hidden-variable theories are not completely impossible, but Bell’s theorem shows that any such theory consistent with quantum mechanics must be non-local. This means, in effect, that every object in the universe has to be inter-dependent, and that we cannot interfere in one region without affecting everything else, no matter how far away. Most physicists find this idea so unsatisfactory that they prefer to think of nature as inherently probabilistic. These ideas are fun, but the right time to think about them (if ever) isafter you understand the workings of quantum theory. The aim of this course is to help you focus on the basics by making quantum mechanics as prosaic, straightforward and boring as possible If you are unwilling to wait and want to find out more about the philosophical issues now, readSpeakableandUnspeakableinQuantum Mechanics: Collected Papers on Quantum Philosophy by J.S. Bell. As well as inventing Bell’s theorem and helping demystify the philosophical mess left by Bohr and friends, Bell (who was born in Belfast in 1928 and died in 1990) was a very good writer. His book is readable and quite accessible, requiring only a minimum of mathematics. 1.4 CourseContent This course covers the experimental evidence that led to the development of quan- tum mechanics and provides an introduction to quantum mechanical concepts and wave mechanics. Concepts discussed include wave-particle duality, the wave function, the uncer-1.5. SOMEUSEFULNUMBERS 7 tainty principle, the Schrödinger equation, and the thorny question of mea- surement in quantum theory. Schrödinger’swavemechanics is one of several equivalent formulations of non- relativistic quantum theory. The others, Heisenberg’s matrix mechanics and Feynman’s path-integral theory, look very different mathematically but de- scribe the same physics and yield identical results. To keep the mathematics as simple as possible, the introduction to wave mechan- ics in the second half of the course considers only a single non-relativistic particle in one dimension. The emphasis is on quantitative understanding and the practi- cal application of physical principles rather than mathematical formalism (which is covered in some detail next year). 1.5 SomeUsefulNumbers When does quantum mechanics matter? The conventional answer is at or below atomic/molecular length/energy scales. This section discusses some of the most important length and energy scales associated with everyday matter. A less conventional answer, to which I subscribe, and which is supported by the long list of quantum phenomena in Sec. 1.2, is that almost everything is quantum mechanical. In defence of this position, one of the classworks near the end of the course is about the quantum mechanics of a house brick. 1.5.1 Lengths Distancebetweenairmolecules The ideal gas law V k T B PV =Nk T ) = : B N P If, for simplicity, every molecule is assumed to occupy a cube of sidea, so that 3 a =V=N, this gives 23 k T 1:3810 300 B 3 a =  5 P 1:0110 and hence 9 a 3:510 m:8 CHAPTER1. INTRODUCTION Boltzmann’sconstant The version of the ideal gas law taught in schools is PV =n RT ; m 1 wheren is the number of moles andR = 8:314 JK is the gas constant. m Sincen =N=N , whereN is the total number of molecules andN is m A A Avogadro’s number, this can be rewritten     N R PV = RT =N T =Nk T : B N N A A 23 1 The new constant k := (R=N )  1:38 10 JK is known as B A Boltzmann’s constant. On the whole, chemists prefer to work with moles andR, while physicists prefer molecules andk . B Distancebetweenatomsinmolecules/solids/liquids 10 A typical inter-atomic distance is a few10 m, otherwise known as a few Å. 10 (1 Ångstrom := 10 m.) Radiusofanatom A typical atomic radius is 1 Å. (The radius of a Hydrogen atom, the Bohr radiusa , is 0.529 Å.) 0 Radiusofanucleus 15 A typical nuclear radius is a few10 m. 1.5.2 Energies Thermalenergyatroomtemperature 23 21 k T 1:3810 300 4:1410 J (1=40) eV: B Theelectron-volt One eV is the kinetic energy gained by an electron falling through a po- 19 tential difference of 1 V: 1 eV =qV =e1 Joules = 1:610 J.1.5. SOMEUSEFULNUMBERS 9 Chemicalbond The energy of a typical covalent, ionic or metallic chemical bond is a few eV. (The van der Waals bonds between closed-shell atoms are much weaker.) Bindingenergiesofelectronsinatoms The energy required to strip an electron from an atom ranges from a few eV for the outermost “valence” electrons to thousands of eV for the innermost “core” electrons of heavy atoms.Chapter 2 Light is Waves 2.1 Evidence for the Wave-Like Nature of Light The waves with which we are most familiar — water waves, sound waves, the standing waves on a violin string — have several features in common. Superposition and interference: If several waves overlap, the total displacement is the sum of the displacements of each. Diffraction: Waves spread out after emerging from a narrow (.) opening. Refraction: Waves change direction at boundaries between regions where the wave speed differs. Light does all of these things, so light is a wave. Historical note The history of our understanding of light is interesting. Descartes and Newton, working in the 17th century, thought that light was a stream of particles, like bullets. It was not until the early 19th century that Thomas Young (born a Quaker in Somerset in 1773; learnt to read at 2; spoke a dozen languages; famous Egyptologist who helped decipher hieroglyph- ics; successful London physician) and others showed, apparently conclu- sively, that light was a kind of wave. For the next century or so, it was assumed that Newton and Descartes had been wrong. Following the ar- rival of quantum theory, it is now clear that Newton, Descartes and Young were all correct: light is both a particle and a wave. 1112 CHAPTER2. LIGHTISWAVES Light waves are special in that they can travel through a vacuum and do not require a medium such as water or air. Most waves travel at a fixed speed relative to the medium that supports them, but light waves in vacuum have no medium and hence no preferred frame of reference. This in part explains why they always travel at a constant speedc relative to the observer. The constancy of the speed of light underlies special relativity but does not play an important role in this course. As far as we are concerned, light waves are much like any other waves. The Schrödinger equation is a kind of wave equation and quantum mechanics is a theory of waves. To set the notation and establish a common starting point, the rest of this chapter revises some of the material from your Vibrations and Waves course. 2.2 Mathematical Description of Travelling Waves 2.2.1 Formula The formula for a travelling wave is (x;t) =a cos(kxt +): (2.1) At timet = 0, this wave is as shown in Fig. 2.1. Note that a cos(kxt +) =a cos(kx (=k)t); so there is a crest atx ==k whent = 0. Asx increases by 2=k at constantt,kx increases by 2 and (x;t) sweeps through one whole period. Hence   = 2=k; (2.2) k = 2=: Similarly, ast increases by 2= at constantx,t increases by 2 and (x;t) sweeps through one period. Hence 8 T = 2=;  = 1=T ==2; (2.3) : = 2 :2.2. MATHEMATICALDESCRIPTIONOFTRAVELLINGWAVES 13 1 0.8 0.6 0.4 0.2 0 φ /k -0.2 -0.4 -0.6 -0.8 -1 0 x Figure 2.1: The travelling wave of Eq. 2.1 at timet=0. 2.2.2 Phase velocity At timet, the wave has a crest where kxt + = 0 and hence where x ==k + (=k)t: This shows that the phase velocityv , which is the velocity of the wave crests, is p given by v = =: (2.4) p k 2.2.3 Group velocity In quantum mechanics, we are often interested in wave packets, since these are the closest thing we can find to classical particles. A typical wave packet is shown Ψ /a14 CHAPTER2. LIGHTISWAVES Figure 2.2: A wave packet. in Fig. 2.2. Although the crests inside the wave packet move at the phase velocity v , the envelope of the packet — indicated by the dashed line in Fig. 2.2 — moves p at thegroupvelocity d v = : (2.5) g dk This equation was discussed in your Vibrations and Waves course but may not have been derived there. In case you are interested, a derivation is included in Chapter 6 of these notes. As we shall see in Sec. 2.2.6, the phase velocity may be larger or smaller than the group velocity. If the phase velocity is larger than the group velocity, the crests within a wave packet travel more quickly than the envelope, appearing at the back, growing as they move forward, and then dying away at the front; if the phase velocity is smaller than the group velocity, the crests travel more slowly than the envelope, appearing at the front and dying away at the back.2.2. MATHEMATICALDESCRIPTIONOFTRAVELLINGWAVES 15 Sign conventions In this course, a right-going travelling wave is written as (x;t) =a cos(kxt +): Why not 0 (x;t) =a cos(tkx + ); as in the Vibrations and Waves course? The two forms are equivalent if 0  is set equal to: 0 a cos(tkx + ) =a cos(tkx) =a cos(kxt +): (The final step used the fact that cos = cos()). Why, then, do we use one form in V&W courses and another in QM courses? The reason is historical: the inventors of quantum theorychose to write the time depen- dence of the wave function as “t” and built their choice into the form of the Schrödinger equation itself. It would be too confusing to change this convention now. In fact, although the two forms are equivalent, one can argue that the “+t” version used in V&W is better than the “t” version used in 0 QM, because  =  is a more natural definition of the phase shift. 0 After all, as can be seen from Fig. 2.1, it is=k (==k), not=k, that gives the position of the maximum at timet = 0. 2.2.4 Amplitude Theamplitude of a wave at a pointx is the maximum displacement (of whatever it is that is waving) at that point. As shown in Fig. 2.3, the maximum displacement of a simple travelling wave, (x;t) =a cos(kxt +); is equal toa at all pointsx. 2.2.5 Complex representation i Sincee = cos +i sin, the travelling wave (x;t) =a cos(kxt +) can be written as  i(kxt+) (x;t) = Re ae16 CHAPTER2. LIGHTISWAVES Figure 2.3: The crests of the travelling wavea cos(kxt +) move steadily to the right, so the maximum displacement isa at all pointsx.  i i(kxt) = Re ae e  i(kxt) = Re Ae ; (2.6) where i A =ae (2.7) is known as thecomplexamplitude of (x;t). 2.2.6 Dispersion relations Any equation giving the angular frequency as a function of the wave vectork is called adispersionrelation. Light: The dispersion relation for light is = ck (or, equivalently,  = c=). Hence v = =c; (2.8) p k d v = =c: (2.9) g dk Sincev = v , the crests within a wave packet move at the same speed as p g the envelope.2.2. MATHEMATICALDESCRIPTIONOFTRAVELLINGWAVES 17 Quantum mechanical particle-waves: The dispersion relation for quantum me- 2 chanical particle-waves is = k =(2m). Hence k v = = ; (2.10) p k 2m d k v = = : (2.11) g dk m Sincev v , the crests within a wave packet move more slowly than the p g envelope. Large, gravity-dominated, deep-ocean waves: The dispersion relation for large p ocean waves is = gk. Hence r g v = = ; p k k r d 1 g v = = : g dk 2 k Sincev v , the crests within a wave packet move more quickly than the p g envelope. Small surface-tension-dominated water waves: The dispersion relation for small, p 3 surface-tension-dominated water waves is = k =, where is the sur- face tension and is the density. Hence s k v = = ; p k  s d 3 k v = = : g dk 2  Sincev v , the crests within a wave packet move more slowly than the p g envelope. Bath-time experiments called for here18 CHAPTER2. LIGHTISWAVES 2.2.7 Intensity Waves transmit energy. The energy density (energy per unit volume) atx is pro- portional to theintensity, defined here as the square of the amplitude, at that point. For example, if  i(kxt) =a cos(kxt +) = Re Ae ; then 2 i i  2 I =a =ae ae =AA =jAj : (2.12) The intensity of a simple travelling wave is therefore independent of position and time. For more complicated waves and interference patterns, this is no longer the case. Eyes and most optical instruments are sensitive to I and do not detect the phase directly. Other definitions of intensity In other courses, you may see intensity defined as the energy per unit 3 volume, measured in Jm , or the energy striking a unit area (oriented 2 perpendicular to the beam) per second, measured in Wm . To see how these two quantities are related, look at the following diagram showing a beam of light passing through a unit area. In time t, all of the light energy in the box of lengthct passes through the right-hand face of unit area. Hence, the energy striking a unit area in time t is uct, where u is the energy per unit volume in the box. Dividing by the time interval t gives the energy striking a unit area per 2 second, which is equal to uc. Since u/ a and c is a constant, both u 2 anduc are proportional toa . In this course, and in quantum physics in 2 general, intensity always means simplya . 2.3 Interference Because it is wave displacements that superpose, not intensities, the relative phases of the contributing waves matter:2.3. INTERFERENCE 19 pattern of relative phases (invisible) ) pattern of intensities (visible) In fact, if intensities added, there would be no interference. Section 2.2.7 showed that the intensity of a simple travelling wave is uniform, so no matter how many travelling waves were superposed, adding their intensities would give a uniform result. The fact that intensities, and thus energy densities, do not add is somewhat strange. You might wonder, for example, whether the total energy is conserved when two travelling waves overlap and an interference pattern is formed. Fortu- nately, it turns out (see question 7 on problem sheet 1 for an example) that the position average of the intensity is always equal to the sum of the intensities of the contributing waves. The total energy is therefore correct, even though the formation of the interference pattern redistributes that energy over space. The phenomenon of interference becomes even stranger in quantum theory, whereI(x;t)dx is the probability that a measurement of the position of a particle with wave function (x;t) yields a result betweenx andx +dx. 2.3.1 Example: the two-slit experiment The two-slit interference experiment is the standard example used to help under- stand the meaning of the quantum mechanical wave function and will play an important role later in this course. In fact, according to Feynman, the two-slit experiment contains “the only mystery” of quantum theory. (If you would like to read about this now, the first few pages of the Feynman Lectures on Physics: Quantum Mechanics v.3 are excellent.) To prepare for the later discussion, this section goes through the mathematics of the two-slit interference experiment for classical waves. The set-up is as illustrated in Fig. 2.4. Suppose that the wave emerging from the upper slit travels a distance =  before hitting a distant screen. The wave t emerging from the lower slit and hitting the same point on the screen travels a slightly longer distance, =  +d sin. (This formula assumes that the screen b is so far away that the rays from the two slits are effectively parallel; if the screen is close to the slits, the assumption of parallel rays is no good and the theory is harder.) Hence i(kt) i(kt) t b (;t) = Ae +Ae i(kt) i(k(+dsin)t) = Ae +Ae  ikdsin i(kt) = A 1 +e e :20 CHAPTER2. LIGHTISWAVES Figure 2.4: The two-slit experiment.  ikdsin Introducing a new complex amplitude,B =A 1 +e , this result becomes i(kt) (;t) = Be : The intensity emerging in the direction is   2 ikdsin  ikdsin I = jBj = A 1 +e A 1 +e  2 ikdsin ikdsin i = a 2 +e +e (becauseA =ae )  2 i i = 2a (1 + cos(kd sin)) because cos = (e +e )=2    kd sin 2 2 2 = 4a cos because 1 + cos = 2 cos ( =2) : 2 The diffraction pattern is as shown in Fig. 2.5. The first zero occurs where kd sin  = 2 2 and hence where   d sin = = : k 22.3. INTERFERENCE 21 Figure 2.5: The two slit diffraction pattern. Since d sin is the path-length difference, this is exactly what one might have expected: the zero of the interference pattern occurs when the waves from the two o slits are 180 out of phase. The interference pattern obtained in a real two-slit experiment is more compli- 2 cated because the slits are not infinitesimally wide. The cos (kd sin)=2 oscil- lation is still visible, but its amplitude is modulated by an envelope, the shape of which corresponds to the diffraction pattern of a single slit of finite width. Ques- tion 8 of problem sheet 1 asks you to work out the diffraction pattern of a single slit.22 CHAPTER2. LIGHTISWAVES

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