The Physics Of Quantum Mechanics - Physik Skripte

PrefacexiThe book formed the basis for lecture courses delivered in the academicyears 2008/9 and 2009/10. At the end of each year the text was revised inlight of feedback from both students and tutors, and insights gained whilstteaching. After the first set of lectures it was clear that students needed tobe given more time to come to terms with quantum amplitudes and Diracnotation. To this end some work on spin-half systems and polarised lightwas added to Chapter 1. The students found orbital angular momentumhard, and the way this is handled in what is now Chapter 7 was changed. Asection on the Heisenberg picture was added to Chapter 4. Chapter 10 wasrevised to correct a widespread misunderstanding about the singlet-tripletsplitting in helium, and Chapter 11 was revised to add thermodynamics tothe applications of the adiabatic principle.The major change between the first and second printings was a newChapter 6 on composite systems. This covers topics such as entanglement,Bell inequalities, quantum computing and density operators that are notnormally included in a first course on quantum mechanics. The discussionin this chapter of the measurement problem was rewritten at the secondrevision. We hope it now makes clear that quantum mechanics does notform a complete physical theory, and that it will inspire students to thinkhow it could be completed. It is most unusual for the sixth chapter of asecond-year physics textbook to be able to take students to the frontier ofhuman understanding, as this chapter does.The major change between the second and third printings was reworkingof §5.3 on one-dimensional scattering – this section now emphasises the rolesof parity and phase shifts and includes resonant scattering and the Breit–Wigner cross section.Students encountered considerable difficulty understanding the connection between spin and the gross structure of helium, so the treatment of thistopic was rewritten at the second revision.Problem solving is the key to learning physics and most chapters arefollowed by a long list of problems. These lists have been extensively revisedsince the first edition and printed solutions prepared. The solutions to starredproblems, which are mostly more-challenging problems, are now availableonline1 and solutions to other problems are available to colleagues who areteaching a course from the book. In nearly every problem a student will eitherprove a useful result or deepen his/her understanding of quantum mechanicsand what it says about the material world. Even after successfully solving aproblem we suspect students will find it instructive and thought-provokingto study the solution posted on the web.We are grateful to several colleagues for comments on the first twoeditions, particularly Justin Wark for alerting us to the problem with thesinglet-triplet splitting. Fabian Essler, John March-Russell and Laszlo Solymar made several constructive suggestions. We thank our fellow MertonianArtur Ekert for stimulating discussions of material covered in Chapter 6 andfor reading that chapter in draft form.July 20101James BinneyDavid mesBinney/QBhome.htm

1Probability and probabilityamplitudesThe future is always uncertain. Will it rain tomorrow? Will Pretty Lady winthe 4.20 race at Sandown Park on Tuesday? Will the Financial Times AllShares index rise by more than 50 points in the next two months? Nobodyknows the answers to such questions, but in each case we may have information that makes a positive answer more or less appropriate: if we are inthe Great Australian Desert and it’s winter, it is exceedingly unlikely to raintomorrow, but if we are in Delhi in the middle of the monsoon, it will almostcertainly rain. If Pretty Lady is getting on in years and hasn’t won a race yet,she’s unlikely to win on Tuesday either, while if she recently won a couple ofmajor races and she’s looking fit, she may well win at Sandown Park. Theperformance of the All Shares index is hard to predict, but factors affectingcompany profitability and the direction interest rates will move, will makethe index more or less likely to rise. Probability is a concept which enablesus to quantify and manipulate uncertainties. We assign a probability p 0to an event if we think it is simply impossible, and we assign p 1 if wethink the event is certain to happen. Intermediate values for p imply thatwe think an event may happen and may not, the value of p increasing withour confidence that it will happen.Physics is about predicting the future. Will this ladder slip when Istep on it? How many times will this pendulum swing to and fro in anhour? What temperature will the water in this thermos be at when it hascompletely melted this ice cube? Physics often enables us to answer suchquestions with a satisfying degree of certainty: the ladder will not slip provided it is inclined at less than 23.34 to the vertical; the pendulum makes3602 oscillations per hour; the water will reach 6.43 C. But if we are pressedfor sufficient accuracy we must admit to uncertainty and resort to probabilitybecause our predictions depend on the data we have, and these are alwayssubject to measuring error, and idealisations: the ladder’s critical angle depends on the coefficients of friction at the two ends of the ladder, and thesecannot be precisely given because both the wall and the floor are slightlyirregular surfaces; the period of the pendulum depends slightly on the amplitude of its swing, which will vary with temperature and the humidity ofthe air; the final temperature of the water will vary with the amount of heattransferred through the walls of the thermos and the speed of evaporation

2Chapter 1: Probability and probability amplitudesfrom the water’s surface, which depends on draughts in the room as well ason humidity. If we are asked to make predictions about a ladder that is inclined near its critical angle, or we need to know a quantity like the period ofthe pendulum to high accuracy, we cannot make definite statements, we canonly say something like the probability of the ladder slipping is 0.8, or thereis a probability of 0.5 that the period of the pendulum lies between 1.0007 sand 1.0004 s. We can dispense with probability when slightly vague answersare permissible, such as that the period is 1.00 s to three significant figures.The concept of probability enables us to push our science to its limits, andmake the most precise and reliable statements possible.Probability enters physics in two ways: through uncertain data andthrough the system being subject to random influences. In the first case wecould make a more accurate prediction if a property of the system, such as thelength or temperature of the pendulum, were more precisely characterised.That is, the value of some number is well defined, it’s just that we don’tknow the value very accurately. The second case is that in which our systemis subject to inherently random influences – for example, to the draughtsthat make us uncertain what will be the final temperature of the water.To attain greater certainty when the system under study is subject to suchrandom influences, we can either take steps to increase the isolation of oursystem – for example by putting a lid on the thermos – or we can expand thesystem under study so that the formerly random influences become calculableinteractions between one part of the system and another. Such expansionof the system is not a practical proposition in the case of the thermos – theexpanded system would have to encompass the air in the room, and thenwe would worry about fluctuations in the intensity of sunlight through thewindow, draughts under the door and much else. The strategy does workin other cases, however. For example, climate changes over the last tenmillion years can be studied as the response of a complex dynamical system– the atmosphere coupled to the oceans – that is subject to random externalstimuli, but a more complete account of climate changes can be made whenthe dynamical system is expanded to include the Sun and Moon becauseclimate is strongly affected by the inclination of the Earth’s spin axis to theplane of the Earth’s orbit and the Sun’s coronal activity.A low-mass system is less likely to be well isolated from its surroundingsthan a massive one. For example, the orbit of the Earth is scarcely affectedby radiation pressure that sunlight exerts on it, while dust grains less than afew microns in size that are in orbit about the Sun lose angular momentumthrough radiation pressure at a rate that causes them to spiral in from nearthe Earth to the Sun within a few millennia. Similarly, a rubber duck leftin the bath after the children have got out will stay very still, while tinypollen grains in the water near it execute Brownian motion that carriesthem along a jerky path many times their own length each minute. Giventhe difficulty of isolating low-mass systems, and the tremendous obstaclesthat have to be surmounted if we are to expand the system to the point atwhich all influences on the object of interest become causal, it is natural thatthe physics of small systems is invariably probabilistic in nature. Quantummechanics describes the dynamics of all systems, great and small. Ratherthan making firm predictions, it enables us to calculate probabilities. If thesystem is massive, the probabilities of interest may be so near zero or unitythat we have effective certainty. If the system is small, the probabilisticaspect of the theory will be more evident.The scale of atoms is precisely the scale on which the probabilistic aspectis predominant. Its predominance reflects two facts. First, there is no suchthing as an isolated atom because all atoms are inherently coupled to theelectromagnetic field, and to the fields associated with electrons, neutrinos,quarks, and various ‘gauge bosons’. Since we have incomplete informationabout the states of these fields, we cannot hope to make precise predictionsabout the behaviour of an individual atom. Second, we cannot build measuring instruments of arbitrary delicacy. The instruments we use to measure

1.1 The laws of probability3atoms are usually themselves made of atoms, and employ electrons or photons that carry sufficient energy to change an atom significantly. We rarelyknow the exact state that our measuring instrument is in before we bring itinto contact with the system we have measured, so the result of the measurement of the atom would be uncertain even if we knew the precise state thatthe atom was in before we measured it, which of course we do not. Moreover, the act of measurement inevitably disturbs the atom, and leaves it in adifferent state from the one it was in before we made the measurement. Onaccount of the uncertainty inherent in the measuring process, we cannot besure what this final state may be. Quantum mechanics allows us to calculateprobabilities for each possible final state. Perhaps surprisingly, from the theory it emerges that even when we have the most complete information aboutthe state of a system that is is logically possible to have, the outcomes ofsome measurements remain uncertain. Thus whereas in the classical worlduncertainties can be made as small as we please by sufficiently careful work,in the quantum world uncertainty is woven into the fabric of reality.1.1 The laws of probabilityEvents are frequently one-offs: Pretty Lady will run in the 4.20 at SandownPark only once this year, and if she enters the race next year, her form andthe field will be different. The probability that we want is for this year’srace. Sometimes events can be repeated, however. For example, there isno obvious difference between one throw of a die and the next throw, soit makes sense to assume that the probability of throwing a 5 is the sameon each throw. When events can be repeated in this way we seek to assignprobabilities in such a way that when we make a very large number N oftrials, the number nA of trials in which event A occurs (for example 5 comesup) satisfiesnA pA N.(1.1)In any realistic sequence of throws, the ratio nA /N will vary with N , whilethe probability pA does not. So the relation (1.1) is rarely an equality. Theidea is that we should choose pA so that nA /N fluctuates in a smaller andsmaller interval around pA as N is increased.Events can be logically combined to form composite events: if A is theevent that a certain red die falls with 1 up, and B is the event that a whitedie falls with 5 up, AB is the event that when both dice are thrown, the reddie shows 1 and the white one shows 5. If the probability of A is pA and theprobability of B is pB , then in a fraction pA of throws of the two dice thered die will show 1, and in a fraction pB of these throws, the white diewill have 5 up. Hence the fraction of throws in which the event AB occurs is pA pB so we should take the probability of AB to be pAB pA pB . In thisexample A and B are independent events because we see no reason whythe number shown by the white die could be influenced by the number thathappens to come up on the red one, and vice versa. The rule for combiningthe probabilities of independent events to get the probability of both eventshappening, is to multiply them:p(A and B) p(A)p(B)(independent events).(1.2)Since only one number can come up on a die in a given throw, theevent A above excludes the event C that the red die shows 2; A and C areexclusive events. The probability that either a 1 or a 2 will show is obtainedby adding pA and pC . Thusp(A or C) p(A) p(C)(exclusive events).(1.3)In the case of reproducible events, this rule is clearly consistent with theprinciple that the fraction of trials in which either A or C occurs should be

4Chapter 1: Probability and probability amplitudesthe sum of the fractions of the trials in which one or the other occurs. Ifwe throw our die, the number that will come up is certainly one of 1, 2, 3,4, 5 or 6. So by the rule just given, the sum of the probabilities associatedwith each of these numbers coming up has to be unity. Unless we know thatthe die is loaded, we assume that no number is more likely to come up thananother, so all six probabilities must be equal. Hence, they must all equal16 . Generalising this example we have the rulesWith just N mutually exclusive outcomes,NXpi 1.i 1If all outcomes are equally likely, pi 1/N.(1.4)1.1.1 Expectation valuesA random variable x is a quantity that we can measure and the value thatwe get is subject to uncertainty. Suppose for simplicity that only discretevalues xi can be measured. In the case of a die, for example, x could be thenumber that comes up, so x has six possible values, x1 1 to x6 6. If piis the probability that we shall measure xi , then the expectation value ofx isXpi xi .(1.5)hxi iIf the event is reproducible, it is easy to show that the average of the valuesthat we measure on N trials tends to hxi as N becomes very large. Consequently, hxi is often referred to as the average of x.Suppose we have two random variables, x and y. Let pij be the probability that our measurement returns xi for the value of x and yj for the valueof y. Then the expectation of the sum x y ishx yi Xijpij (xi yj ) Xijpij xi Xpij yj(1.6)ijPxi regardless of what weButj pij is the probability that we measurePmeasure for y, so it must equal pi . Similarly i pij pj , the probability ofmeasuring yj irrespective of what we get for x. Inserting these expressionsin to (1.6) we findhx yi hxi hyi .(1.7)That is, the expectation value of the sum of two random variables is thesum of the variables’ individual expectation values, regardless of whetherthe variables are independent or not.A useful measure of the amount by which the value of a random variablefluctuates from trial to trial is the variance of x:ED2(1.8)(x hxi)2 x2 2 hx hxii hxi ,where we have made use of equation (1.7). The expectation hxi is not a2randomE variable, but has a definite value. Consequently hx hxii hxi andD22hxi hxi , so the variance of x is related to the expectations of x andx2 by2 2x (x hxi)2 x2 hxi .(1.9)

1.2 Probability amplitudes5Figure 1.1 The two-slit interference experiment.1.2 Probability amplitudesMany branches of the social, physical and medical sciences make extensiveuse of probabilities, but quantum mechanics stands alone in the way that itcalculates probabilities, for it always evaluates a probability p as the modsquare of a certain complex number A:p A 2 .(1.10)The complex number A is called the probability amplitude for p.Quantum mechanics is the only branch of knowledge in which probability amplitudes appear, and nobody understands why they arise. Theygive rise to phenomena that have no analogues in classical physics throughthe following fundamental principle. Suppose something can happen by two(mutually exclusive) routes, S or T , and let the probability amplitude for itto happen by route S be A(S) and the probability amplitude for it to happenby route T be A(T ). Then the probability amplitude for it to happen by oneroute or the other isA(S or T ) A(S) A(T ).(1.11)This rule takes the place of the sum rule for probabilities, equation (1.3).However, it is incompatible with equation (1.3), because it implies that theprobability that the event happens regardless of route isp(S or T ) A(S or T ) 2 A(S) A(T ) 2 A(S) 2 A(S)A (T ) A (S)A(T ) A(T ) 2 p(S) p(T ) 2ℜe(A(S)A (T )).(1.12)That is, the probability that an event will happen is not merely the sumof the probabilities that it will happen by each of the two possible routes:there is an additional term 2ℜe(A(S)A (T )). This term has no counterpartin standard probability theory, and violates the fundamental rule (1.3) ofprobability theory. It depends on the phases of the probability amplitudesfor the individual routes, which do not contribute to the probabilities p(S) A(S) 2 of the routes.Whenever the probability of an event differs from the sum of the probabilities associated with the various mutually exclusive routes by which itcan happen, we say we have a manifestation of quantum interference.The term 2ℜe(A(S)A (T )) in equation (1.12) is what generates quantuminterference mathematically. We shall see that in certain circumstances theviolations of equation (1.3) that are caused by quantum interference are notdetectable, so standard probability theory appears to be valid.How do we know that the principle (1.11), which has these extraordinaryconsequences, is true? The soundest answer is that it is a fundamentalpostulate of quantum mechanics, and that every time you look at a digitalwatch, or touch a computer keyboard, or listen to a CD player, or interactwith any other electronic device that has been engineered with the helpof quantum mechanics, you are testing and vindicating this theory. Ourcivilisation now quite simply depends on the validity of equation (1.11).

6Chapter 1: Probability and probability amplitudesFigure 1.2 The probability distributions of passing through each of thetwo closely spaced slits overlap.1.2.1 Two-slit interferenceAn imaginary experiment will clarify the physical implications of the principle and suggest how it might be tested experimentally. The apparatusconsists of an electron gun, G, a screen with two narrow slits S1 and S2 ,and a photographic plate P, which darkens when hit by an electron (seeFigure 1.1).When an electron is emitted by G, it has an amplitude to pass throughslit S1 and then hit the screen at the point x. This amplitude will clearlydepend on the point x, so we label it A1 (x). Similarly, there is an amplitudeA2 (x) that the electron passed through S2 before reaching the screen at x.Hence the probability that the electron arrives at x isP (x) A1 (x) A2 (x) 2 A1 (x) 2 A2 (x) 2 2ℜe(A1 (x)A 2 (x)). (1.13) A1 (x) 2 is simply the probability that the electron reaches the plate afterpassing through S1 . We expect this to be a roughly Gaussian distributionp1 (x) that is centred on the value x1 of x at which a straight line from Gthrough the middle of S1 hits the plate. A2 (x) 2 should similarly be a roughlyGaussian function p2 (x) centred on the intersection at x2 of the screen andthe straight line from G through the middle of S2 . It is convenient to write Ai Ai eiφi pi eiφi , where φi is the phase of the complex number Ai .Then equation (1.13) can be writtenp(x) p1 (x) p2 (x) I(x),where the interference term I ispI(x) 2 p1 (x)p2

mechanics, it is no less important to understand that classical mechanics is just an approximation to quantum mechanics. Traditional introductions to quantum mechanics tend to neglect this task and leave students with two independent worlds, classical and quantum. At every stage we try to explain how classical physics emerges from quantum .