[ebook] the mystery of the quantum world 2ed

8/2/2019 [eBook] the Mystery of the Quantum World 2ed

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Chapter OneReality in the QuantumWorld

1.1 The quantum revolutionsQuantum mechanics, created early this century in response tocertain experimental facts which were inexplicable according topreviously held ideas (conveniently summarised by the titleclassical physics), caused three great revolutions. In the first placeit opened up a completely new range of phenomena to which themethods of physics could be applied: the properties of atoms andmolecules, the complex world of chemical interactions, previouslyregarded as things given from outside science, became calculable interms of a few fixed parameters. The effect of this revolution hascontinued successfully through the physics of atomic nuclei, ofradioactivity and nuclear reactions, of solid-state properties, torecent spectacular progress in the study of elementary particles. Inconsequence all sciences, from cosmology to biology, are, at theirmost fundamental level, branches of physics. Through physics theycan, at least in principle, be understood. Indeed, on contemplatingthe success of physics, it is easy to be seduced into the belief thateverything is physics-a belief that, if it is intended to imply thateverything is understood, is certainly false, since, as we shall see,the very foundation of contemporary theoretical physics ismysterious and incomprehensible.The second revolution was the apparent breakdown of deter-minism, which had always been an unquestioned ingredient and aninescapable prediction of classical physics. Note that we are using


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2 Reality in the quantum worldthe word determinism solely with regard to physical systems,without at this stage worrying about which systems can be sodescribed; that is, we are not here concerned with such concepts asfree will. In a deterministic theory the future behaviour of anisolated physical system is uniquely determined by its present state.If, however, the world is correctly described by quantum theory,then, even for simple systems, this deterministic property is notvalid. The outcome of any particular experiment is not, even inprinciple, predictable, but is chosen at random from a set ofpossibilities; all th at c an be predicted is the probability of particularresults when th e experiment is repeated man y times. It is im po rta ntto realise that t he probability aspects that enter here do so for a dif-ferent reason th an , fo r example, in the tossing of a coin, or throwof a dice, or a horse race; in these cases they enter because of ourlack of precise knowledge of the orginal state of the system,whereas in quan tum theory, even if we had complete knowledge ofthe initial state, the outcome would still only be given as aprobability.

Naturally, physicists were reluctant to accept this breakdown ofa cherished dogma-Einsteins obje ction to the idea of G od playingdice with the universe is the most familiar expression of thisreluctance-and it was suggested th at th e ap pa ren t failure of deter-minism in the theory w as due t o a n incompleteness in the descrip-tion of the system. Many attempts to remedy this incompleteness,by introducing what ar e referred t o as hidden variables, have beenmade. These attempts will form an important part of our laterdiscussion.W e are accustomed t o regarding the behaviour, at least of simplemechanical systems, as being completely deterministic, so if thebreakdown of determinism implied by quantum mechanics isgenuine, it is a n im po rtan t discovery which must affect ou r view o fthe physical world. Nevertheless, our belief in determinism arisesfrom experience rather than logic, and it is quite possible to con-ceive of a certain degree of random ness entering into mechanics; noobvious violation of co m mon sense is involved. Such is not thecase with the third revolution brought about by quantummechanics. This challenged the basic belief, implicit in all sciencean d indeed in almost the whole of hu m an thinking, th at there existsa n objective reality, a reality tha t does no t depend for its existenceon its being observed. It is because of this challenge that all who


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The quantum revolutions 3endeavour to study, or even take an interest in, reality, the natureof what is, be they philosophers or theologians or scientists,unless they are content to study a phantom world of their owncreation, should know about this third revolution.To provide such knowledge, in a form accessible to non-scientists, is the aim of this book. It is not intended for those whowish to learn the practical aspects of quantum mechanics. Manyexcellent books exist to cover such topics; they convincinglydemonstrate the power and success of the theory to make correctpredictions of a wide range of observed phenomena. Normallythese books make little reference to this third revolution; they omitto mention that, at its very heart, quantum mechanics is totallyinexplicable. For their purpose this omission is reasonable becausesuch considerations are not relevant to the success of quantummechanics and do not necessarily cast doubt on its validity. In1912, Einstein wrote to a friend, The more success the quantumtheory has, the sillier it looks. [Letter to H Zangger, quoted onp 399 of the book Subtle is the Lord by A Pais (Oxford: Clarendon1982).] If it is true that quantum mechanics is silly, then it is sobecause, in the terms with which we are capable of thinking, theworld appears to be silly. Indeed the recent upsurge of interest inthe topic of this book has arisen from the results of recentexperiments; results which, though they beautifully confirm thepredictions of quantum mechanics, are themselves, quiteindependent of any specific theory, at variance with what anapparently convincing, common-sense, argument would predict(see Chapter 5 , especially $85.4 and 5 . 5 , for a complete discussionof these results).We can emphasise the essentially observational nature of theproblem we are discussing by returning to the experimental facts wementioned at the start of this section, and which gave birth to quan-tum mechanics. Although, by abandoning some of the principles ofclassical physics, quantum theory predicted these facts, it did notexplain them. The search for an explanation has continued and weshall endeavour in this book to outline the various possibilities. Allinvolve radical departures fr om our normal ways of thinking aboutreality.On almost all the topics which we shall discuss below there is alarge literature. However, since this book is intended to be apopular introduction rather than a technical treatise, I have given


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4 Reality in the quantum worldvery few references in the text but have, instead, added a detailedbibliography. For the same reason various ifs and buts andqualifying clauses, that experts might have wished to see insertedat various stages, have been omitted. I hope that these omissionsdo not significantly distort the argument.I have tried to keep the discussion simple and non-technical,partly because only in this way can the ideas be communicated tonon-experts, but also because of a belief that the basic issues aresimple and that highly elaborate and symbolic treatments onlyserve to confuse them, or, even worse, give the impression thatproblems have been solved when, in fact, they have merely beenhidden. The appendices, most of which require a little moreknowledge of mathematics and physics than the main text, givefurther details of certain interesting topics.Finally, I conclude this section with a confession. For over thirtyyears I have used quantum mechanics in the belief that the prob-lems discussed in this bo ok were of no great interest and could, inany case, be sorted out with a few hours careful thought. I thinkthis attitud e is shared by mo st who learned the subject when I di d,or later. Maybe we were influenced by remarks like th at with whichMax Born concluded his marvellous book on modern physics[A tom ic Physics (London: Blackie 1935)] For what lies withinthe limits is knowable, and will become known; it is the world ofexperience, wide, rich enough in changing hues and patterns toallure us to explore it in all directions. What lies beyond, the drytracts of metaphysics, we willingly leave to speculative philosophy.It was only when, in the course of writing a book on elementaryparticles, I found it necessary to do this sorting out, that Idiscovered how far from the truth such an attitude really is. Thepresent book has arisen from my attempts to understand thingsthat I mistakenly thought I already understood, to venture, if youlike, into speculative philosophy, and to discover what progresshas been m ade in the task of incorporating t he stran ge phenomenaof the quantum world into a rational and convincing picture ofreality.1.2 External realityAs I look around the ro om where I am now sitting I see various


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External reality 5objects. That is, through the lenses in my eyes, through the struc-ture of the retina, through assorted electrical impulses received inmy brain, etc, I experience sensations of colour and shape whichI interpret as being caused by objects outside m yself. These objectsform part of what I call the real world or the external reality.That such a reality exists, independent from my observation of it,is an assumption. The only reality that I know is the sensations ofwhich I am conscious, so I make an assumption when I introducethe concept that there a re real external objects tha t cause these sen-sations. Logically there is no need for me t o d o this; my consciousmind could be all that there is. Many philosophers and schools ofphilosophy have, indeed, tried to take this point very seriouslyeither by denying the existence of an external reality, o r by claimingthat , since the concept canno t be properly defined, proved to exist,or proved no t to exist, then it is useless and should not be discussed.Such views, which as philosophic theories ar e referred to by wordssuch as idealism or positivism, are logically tenable, but aresurely unacceptable on aesthetic grounds. It is much easier for meto understand my observations if they refer to a real world, whichexist even when not observed, than if the observations are infact everything. Thus, we all have an intuitive feeling that outthere a real world exists and that its existence does not dependupo n us. We can observe it, interact with it, even change it, but wecannot make it go away by not looking at it. Although wecan give no p roof , we do not really dou bt tha t full many a floweris born to blush unseen, and waste its sweetness on the desertair.It is imp ortan t that we should try to understand why we have thisconfidence in the existence of an external reality. Presumably onereason lies in selective evolution which has built in to our geneticmake-up a predisposition towards this view. It is easy to see whya tendency to think in terms of an external reality is favourable tosurvival. The man who sees a tree, and goes on to the idea thatthere is a tree, is more likely to avoid running into it, and therebykilling himself, than the man who merely regards the sensation ofseeing as something wholly contained w ithin his mind. The fact ofthe built-in prejudice is evidence that the idea is at least useful.However, since we are, to som e extent, thinking beings, we shouldbe able to find rational arguments which justify our belief, andindeed there are several. These depend on those aspects of our


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6 Reality in the quantum worldexperience which are naturally understood by the existence of anexternal reality and which do not have any natural explanationwithout it. If, for example, I close my eyes and, for a time, ceaseto observe the objects in the room, then, on reopening them, I see,in general, the same objects. This is exactly what would be expectedon the assumption that the objects exist and are present even whenI do not actually look at them. Of course, some could have moved,or even been taken away, but in this case I would seek, andnormally find, an explanation of the changes. Alternatively I coulduse different methods of observing, e.g. touch, smell, etc, and Iwould find that the same set of objects, existing in an externalworld, would explain the new observations. Thirdly, I am awarethrough my consciousness of other people. They appear to besimilar to me, and to react in similar ways, so, from the existenceof my conscious mind, I can reasonably infer the existence of realpeople, distinct from myself, also with conscious minds. Finally,these other people can communicate to me their observations, i.e.the experiences of their conscious minds, and these observationswill in general be compatible with the same reality that explains myown observations.In summary, it is the consistency of a vast range of differenttypes of observation that provides the overwhelming amount ofevidence on which we support our belief in the existence of anexternal reality behind those observations. We can contrast thiswith the situation that occurs in hallucinations, dreams, etc, wherethe lack of such a consistency makes us cautious about assumingthat these refer to a real world.We turn now to the scientific view of the world. At least priorto the onset of quantum phenomena this is not only consistentwith, but also implicitly assumes, the existence of an externalreality. Indeed, science can be regarded as the continuation of theprocess, discussed above, whereby we explain the experiences ofour senses in terms of the behaviour of external objects. We havelearned how to observe the world, in ever more precise detail, howto classify and correlate the various observations and then how toexplain them as being caused by a real world behaving accordingto certain laws. These laws have been deduced from our experience,and their ability to predict new phenomena, as evidenced by theenormous success of science and technology, provides impressive


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The breakdown of determinism 7support for their validity and for the picture of reality which theypresent.

This beautifully consistent picture is destroyed by quantumphenomena. Here, we are amazed to find that one item, crucial tothe whole idea of an external reality, appears to fail. It is no longertrue that different methods of observation give results that are con-sistent with such a reality, or at least not with a reality of the formthat had previously been assumed. No reconciliation of the resultswith an acceptable reality has been found. This is the major revolu-tion of quantum theory, and, although of no immediate practicalimportance, it is one of the most significant discoveries of scienceand nobody who studies the nature of reality should ignore it.It will be asked at this stage why such an important fact is notimmediately evident and well known. (Presumably if it had beenthen the idea of creating a picture of an external reality wouldnot have arisen so readily.) The reason is that, on the scaleof magnitudes to which we are accustomed, the new, quantumeffects are too small to be noticed. We shall see examples of thislater, but the essential point is that the basic parameter ofquantum mechanics, normally denoted by f~ ( h bar) has thevalue 0.OOO OOO000OOO OOO OOO OOO OOO 001 (approximately) whenmeasured in units such that masses are in grams, lengths incentimetres and times in seconds. (Within factors of a thousand orso, either way, these units represent the scale of normal experi-ence.) There is no doubt that the smallness of this parameter ispartially responsible for our dimculty in understanding quantumphenomena-our thought processes have been developed in situa-tions where such phenomena produce effects that are too small tobe noticed, too insignificant for us to have to take them intoaccount when we describe our experiences.

1.3 The potential barrier and thebreakdown of determinism

We now want to describe a set of simple experiments whichdemonstrate the crucial features of quantum phenomena. To beginwe suppose that we have a flat table on which there is a smooth


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8 Reality in the quantum worldhill, Th is is illustrated in figure 1. If we roll a small ball, fro m theright, towards the hill then, for low initial velocities, the ball willroll up the hill, slowing down as it does so, until it stops and thenrolls back down again. In this case we say that the ball has beenreflected. For larger velocities, however, the ball will go right overthe hill and will roll down the other side; it will have beentransmitted.

TableFigure 1 A simple exam ple of a potential barrier experiment,in which a ball is rolled up a hill. The ball will be reflected ortransmitted by the hill according t o w hether the initial velocityis less or greater than some critical value.

By repeating this experiment several times we readily find thatthere is a critical velocity, which we shall call V, such that, if theinitial velocity is smaller than V then the ball will be reflected,whereas if it is greater than V then it will be transmitted. We canwrite this symbolically as

v < V:reflectionv > V : ransmission

where v denotes the initial velocity, and the symbols < , > meanis less than, is greater than, respectively.Th e force tha t causes the b all to slow dow n as it rises u p the hillis the gravitational force, an d it is possible to calculate V f rom thelaws of classical physics (details are given in Appendix 1). Similarresults would be obtained with any other type of force. What isactually happening is that the energy of motion of the ball (called


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The breakdown of determinism 9kinetic energy) is being changed into energy due to the force(called potential energy). The ball will have slowed to zero velocitywhen all the kinetic energy has turned into potential energy.Transmission happens when the initial kinetic energy is greater th anthe maximum possible potential energy, which occurs at the top ofthe hill. In the general case we shall refer t o this type of experimentas reflection or transmission by a potential barrier.Now we introduce quan tum physics. The simple result expressedby equation ( l . l ) , which we obtained from experiment and whichis in agreement with the laws of classical mechanics, is not in factcorrect. For example, even when v < Vthere is a possiblity tha t theparticle will pass through the barrier. This phenomenon is some-times referred to as quantum tunnelling. The reason why wewould not see it in our simple laboratory experiment is that withobjects of normal sizes (which we shall refer to as macroscopicobjects), i.e. things we can hold and see, the effect is far too smallto be noticed. Whenever v is measurably smaller than V theprobability of transmission is so small that we can effectively sayit will never happen. (Some appropriate numbers are given inAppendix 4.)With microscopic objects, i.e. those with atomic sizes andsmaller, the situation is very different and equation (1.1) does notdescribe the results except for sufficiently small, or sufficientlylarge, velocities. For velocities close to V we find, to our surprise,tha t the value of v does not tell us whether or not the particle willbe transmitted. If we repeat the experiment several times, alwayswith a fixed initial velocity ( v ) we would find tha t in some cases theparticle is reflected and in some it is transmitted. The value of vwould no longer determine precisely the fate of the particle whenit hits the barrier; rather it would tell us the probability of a particleof that velocity passing through. For low velocities the probabilitywould be close to zero, and we would effectively be in the classicalsituation; as the velocity rose towards V the probability oftransmission would rise steadily, eventually becoming very close tounity for v much larger than V , thus again giving the classicalresult.Before we comment on the implications of these results, it isworth considering a more readily appreciated situation which is insome ways analogous. On one of the jetties in the lake of Genevathere is a large fountain , the Jet deau. The water from this tends


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10 Reality in the quantum worldto fall onto the jetty, in amounts that vary with the direction of thewind. On any day in summer people walk along the jetty andeventually they reach the barrier of the falling water. At this stagesome are reflected, they look around for a while and then turnback; others however are transmitted and, ignoring the possibilityof getting wet, carry on to the end of the jetty. By observing fora time, on any particular afternoon, it would be possible tocalculate the probability that any given person would pass thebarrier. This probability would depend on the direction of the windat the time of observation-the direction would therefore play ananaloguous role to that of the initial velocity in our previous experi-ment. There would, however, be nothing in any way surprisingabout our observations at Geneva, no breakdown of determinismwould be involved, people would behave differently becausethey are different. Indeed it might be possible to predict some ofthe effects: the better dressed, the elderly, the female (?). . would,perhaps, be more likely to be reflected. The more information wehad, the better would we be able to predict what would happenand, indeed, leaving aside for the moment subtle questions aboutfree will which inevitably arise because we are discussing thebehaviour of people, we might expect that if we knew everythingabout the individuals we could say with certainty whether or notthey would pass the barrier. In this sense the probability aspectswould arise solely from our ignorance of all the facts-they wouldnot be intrinsic to the system. In all cases where probability entersclassical physics this is the situation.

We must contrast this perfectly natural happening with thepotential barrier experiment. Here the particles are, apparently,identical. What then determines which are reflected and whichtransmitted? Attempts to answer this question fall into two classes:Orthod ox theories. In such theories it is accepted that the particlesgenuinely are identical, so there is nothing available with which toanswer the question except the statement that it is a random choice,subject only to the requirement that when the same experiment isrepeated many times the correct proportion have been reflected.Quantum theory, as normally understood, is a theory of this type.If such theories are correct then determinism, as defined in 0 1.1, isnot a property of our world; probability enters physics in anintrinsic way and not just through our ignorance. The situation is


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The breakdown of determinism 11thus different in nature from that of people passing the Jet deauin Geneva. Herein lies the second revolution of quan tum physics towhich we referred in the opening section. The physical world is notdeterministic. It is worth noting here that, although quantumphenomena are readily seen only on the microscopic scale, this lackof determinism can easily manifest itself on any macroscopic scaleone might choose. We give a simple example in Appendix 2.Hidden variable theories. In such theories the particles reaching thebarrier are not identical; they possess other variables in additionto their velocities and, in principle, the values of these variablesdetermine the fate of each particle as it reaches the barrier; nobreakdown of determinism is required and the probability aspectonly enters through our ignorance of these values, exactly as inclassical physics. At this stage of our discussion readers are prob-ably thinking that hidden variable theories surely contain the tru th ,and that we have not yet given any good reasons for abandoningdeterminism. They are right, but this will soon change and we shallsee that hidden variable theories, which are discussed more fully inChapter 5 , have many difficulties.

Before proceeding we shall look a little more carefully at ourpotential barrer experiment. Since we are interested in whether ornot particles pass through the barrier we must have detectors whichrecord the passage of a particle, e.g. by flashing so that we can seethe flash. We shall assume that our detectors are perfect, i.e. theynever miss a particle. Then if we have a detector on the left of thebarrier it will flash when a particle is transmitted, whereas one onthe right will flash for a reflected particle. Suppose N particles, allwith the same velocity, are sent and suppose we see R flashes in theright-hand detector and T i n the left-hand detector. Because everyparticle must go somewhere, we will findR + T = N . (1 .2 )

Provided N is large, the probability of transmission is defined to beT divided by N and the probability of reflection R divided by N,i.e.and ( 1 . 3 )

(1.4)


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12 Reality in the quantum worldwhere PT and PR denote the probabilities of transmission andrelfection, respectively.

If we were to repeat the experiments, using N further particles,then we would not obtain exactly the same values for R and T.(Compare the fact that in 100 tosses of a coin we would not alwaysobtain exactly 50 heads.) These differences are statistical fluctua-tions and their effect on the values of PTand PRcan be made assmall as we desire by making N large enough. In fact, the error isproportional to the inverse of the square root of N. In all the subse-quent discussion we shall assume that N is sufficiently large forstatistical fluctuations to be ignored.At this stage everything in our experiment appears to be inaccordance with the concept of external reality. Indeed we have asimple picture of what happens: each particle moves freely until itreaches the potential barrier, at which stage it makes a choice,either through a hidden variable procedure or with some degree ofrandomness, as to whether to pass through or not. Such a choicewould be made regardless of whether the detectors were present.After a suitable lapse of time we would have either a particletravelling to the right or one travelling to the left. This would bethe external reality. If the detectors were present one of them wouldflash, thereby telling us which of the two possibilities had occurred.The detectors however would only observe the reality, they wouldnot create it.This simple picture of reality is, as we shall now show, false. Itis not compatible with another method of observing the samesystem and therefore fails one of the consistency tests for realitygiven in 41.2. In the next section we shall describe this othermethod of observation and see why it is so devastating to the ideaof external reality.

1.4 The experimental challenge to realityWe continue with our experiment in which particles are directed ata potential barrier but now, instead of having detectors to tell uswhether a particle has been reflected or transmitted, we havemirrors which deflect both sets of particles towards a commondetector. There are many ways of constructing such mirrors, par-


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The experimental challenge to reality 13ticularly if our particles are charged , e.g. if they are electrons, w henwe could use suitable electric fields. For this experiment we mustalso allow the particles t o follow slightly different paths, which caneasily be arranged if there is some degree of variation in the initialdirection. To be specific, we suppose that the source of particlesgives a uniform distribution over some small angle. Then the finaldetector must cover a region of space sufficiently large to see par-ticles following all possible paths. In fact, we split it into severaldetectors, denoted by A, B, C, etc, so that we will be able toobserve how the particles are distributed among them. In figure 2we give a plan of the experiment. This plan also show s two separateparticle paths reaching the detector labelled C.

Detectors, A . B , C , 0 , E ,

,\

!ight-handiirrorLeft- handmirror

Figure 2 A plan of the modified potential barrier experi-ment. The mirrors can be put in place to deflect the reflectedand transmitted particles to a common set of detectors. Twopossible particle paths to detector C are shown.

, \/ \,

\\ \

\ /\\

\', Potential ,, barr ier ,. '

Source 0; particles


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14 Reality in the quantum worldWe now do three separate sets of experiments. For the first set

we only have the right-hand mirror. Thus only the particles that arereflected by the barrier will be able to reach the detectors. When wehave sent N particles, where N is large, the detectors will haveflashed R times. These R flashes will have some particular distribu-tion among the various detectors. A possible example of such adistribution, for five detectors, is shown in figure 3 ( a ) .

23B C D E


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The experimental challenge to reality 15Next, we repeat these experiments with the right-hand mirrorremoved and the left-hand mirror in place. This time only the

transm itted particles will reach the detectors, so, when we have sentN particles, we will have T flashes. In figure 3 (b )we show apossible distribution of these among the same five detectors.For our third set of experiments we have both mirrors inposition. Thus all particles, whether reflected or transmitted by thebarrier, will be detected. When N particles have been sent, therewill have been N flashes. Can we predict the distribution of theseamong the various detectors? Surely, we can. We know whathappens to the transmitted particles, e.g. figure 3 (b ) , nd also tothe reflected particles, e.g. figure 3 ( a ) .We also know th at the par-ticles are sent separately so they cannot collide or otherwise get ineach others way. We therefore expect to obta in the sum of the twoprevious distributions. This is shown in figure 3 (c) for ourexample. The world, however, is not in accord with this expecta-tion. The distribution seen when both mirrors a re present is not thesum of the distributions seen with the two mirrors separately.Indeed, it is quite possible for some detectors to receive fewerparticles when both mirrors are present than when either one ispresent. A typical possible form showing this effect is given infigure 3 ( d ) .Can we understand these results? Can we understand, forexample, why there ar e paths for particles t o reach detector B wheneither mirror is present but such paths are not available if bothmirrors are present? The only possibility is that in the latter caseeach individual particle knows about, i.e. is influenced by, bothmirrors. This is not compatible with the view of reality, discussedin the previous section, in which a particle either passes through oris reflected. On the contrary, the reality suggested by the experi-ments of this section is that each particle somehow splits into twoparts, one of which is reflected by one mirror and one by the other.Such a picture is, however, not compatible with the results of thedetector experiments in which each individual particle is seen to goone way or the other and never to split into tw o particles. Thus thesimple pictures of reality suggested by these two sets of experimentsare mutually contradictory.Clearly we should not accept this perplexing situation withoutexaming very carefully the steps that have led to it. The first thingwe would want to check is that the experimental results are valid,


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16 Reality in the quantum worldHere I have to make an apology. Contrary to what has been impliedin the above discussion, the experiments that have been describedhave not actually been done. For a variety of technical reasons noreal experiment can ever be made quite as simple as a thoughtexperiment. The apparent incompatibility we have met does occurin real experiments, but the discussion there would be much morecomplicated and the essential features would be harder to see. Theresults of our simple experiments actually come from theory, inparticular from quantum theory, but the success of that theory inmore complicated, real, situations means that we need have nodoubt about regarding them as valid experimental results.As another possibility for rescuing the picture of reality given inthe previous section, we might ask whether we abandoned it tooreadily in the face of the evidence from the mirror experiments. Onexamining the argument we see that a key step lay in the statementthat a reflected particle, for example, could not know about theleft-hand mirror. Behind this statement lay the assumption thatobjects sumciently separated in space cannot influence each other.Is this assumption true and, if so, were our mirrors sufficiently wellseparated? With regard to the second question one answer is that,according to quantum mechanics, which provided our results, thedistance is irrelevant. Perhaps more important, however, is the factthat the irrelevance of the distance scale seems to be experimentallysupported in other situations. The only hope here, then, is to ques-tion the assumption; maybe the belief that objects can be spatiallyseparated so that they no longer influence each other is false. If thisis so , then it is already a serious criticism of the normal picture ofreality, in which the idea that objects can be localised plays acrucial role. We shall return to this topic later.Are there any other alternatives? Certainly some rather bizarrepggsibilities exist. The decision to put the second mirror in placewas made prior to the experiment with two mirrors being per-formed. Maybe this process somehow affected the particles used inthe experiment and hence led to the observed results. Alternatively,it could in some way have affected the first mirror, so that the twomirrors knew about each other and therefore behaved differently.Such things could be true, but they seem unlikely. We mentionthem here to emphasise how completely the results we havediscussed in this chapter violate our basic concept of reality, andalso because they are, in their complexity, in stark contrast to the


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Summary 17elegant simplicity of the quantum theoretical description of theseexperiments. It is this description that forms the topic of the nextchapter.1.5 Summary of Chapter OneIn this chapter we have discussed two separate sets of experimentsassociated with the passage of a particle through a potentialbarrier. The experiments measure different things, so the resultsobtained are not directly comparable and clearly cannot in them-selves be contradictory. However, we have tried to justify ourinterest in what actually happens in addition to what is seen, andwhen we use the experiments to tell us what happens we obtainincompatible information. The first experiment tells us thatparticles are either transmitted or reflected by the barrier. We cantherefore consider, for example, a particle that is reflected andremains always to the right of the barrier. The second experimentthen tells us that in some cases the subsequent behaviour of thisparticle can depend on whether or not the left-hand mirror is

Mir ro r -wh ichmay be presentor not

Path of particle, eflec ted by thebar r ie rI'

//

\\ \Question How can the ref lecte d parti cle 'know'whenthe mirror is present 7

Figure 4 A pictorial representationof the challenge to realitygiven by the experiments we have described.


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18 Reality in the quantum worldpresent, regardless of how far away it might be. Readers should beconvinced that this is crazy-because it is crazy. It also happensto be true. This is the challenge to reality which is a consequenceof quantum phenomena. We illustrate it, pictorially, in figure 4.How this challenge is being met, the extent to which we canunderstand what is actually happening, the possible forms ofreality to which quantum phenomena lead us, are the subjects thatwill occupy us throughout the remainder of this book.


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Chapter TwoQuantum Theory

2.1 The description of a particle inquantum theoryThe fam iliar, classical, description of a particle requires that, at alltimes, it exists a t a particular position. Indeed, the rules of classicalmechanics involve this position and allow us to calculate how itvaries with time. According to quantum mechanics, however, theserules are only an approximation to the truth and are replaced byrules that do not refer explicitly to this position b ut , instead,predict the time variation of a quantity fr om which it is possible t ocalculate the probability of the particle being in a particular place.We shall indicate below the circumstances in which the classicalapproximation is likely to be valid.The probability will be a positive num ber (any probability has tobe positive) which, in general, will vary with time and with thespatial point considered. As an example, figure 5 is a graph of sucha probability, and shows how it varies with the distance, denotedby x , along a straight line from some fixed point 0. This graphrepresents a particle which is close to the point labelled P. Thewidth of the distribution, shown in the figure as U,, ives some ideaof the uncertainty in the true position of the particle. There areprecise methods of defining this uncertainty but these are notimportant fo r our purpose. Clearly a very narrow peak correspondsto accurate knowledge of the position of the particle and, con-versely, a wide peak to inaccurate knowledge.


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20 Quantum theory

Figure 5 A typical probability graph for a particle which isclose to a point P. he probability of finding the particle in theneighbourhood of any point is proportional to the height ofthe curve at that point. If we measure area in units such thatthe total area under the curve is one, then the probability thatthe particle is in the interval from QI o QZ s equal to theshaded area. For a simple peak of this form the uncertainty inposition is the width of the peak, denoted here by .Ux.

At this stage it might be thought that we can always use theclassical approximation, where particles have exact positions, byworking w ith sufficiently na rrow peaks. How ever, if we d o this welose something else. It tu rns o ut th at th e width of th e peak is alsorelated to the uncertainty in the velocity of the particle, moreprecisely t he velocity in the direction of th e line between th e points0 and P , only here the relation is the opposite way round: thenarrower the peak, the larger the uncertainty. In consequence,although there is no limit to the accuracy with which either theposition or the velocity can be fixed, the price we have to pay formaking one m ore definite is loss of in form ation on the othe r. Thisf a a is known as the Heisenberg uncertainty principle.Quantitatively, this principle states that the product of theposition uncertainty an d the velocity uncertainty is at least as largeas a certain fixed num ber divided by the mass of th e particle beingconsidered. T he fixed num ber is, in fact, the constant +z introduced


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The description of a particle in quantum theory 21earlier. We can then write the uncertainty principle in the form

U,U,>& /m (2.1)where U, s the uncertainty in the velocity and m is the mass of theparticle.

The quantity + is Plancks constant. We quote again its value,this time in SI units:

4 = 1.05x kgm2s-.This is a very small number! We can now see why quantum effectsare hard to see in the world of normal sized, i.e. macroscopic,objects. For example, we consider a particle with a mass of onegram (about the mass of a paper clip). Suppose we locate this toan accuracy such that U, s equal to one hundredth of a centimetre(10-4m). Then, according to equation (2.1), the error in velocitywill be about 10-m per year. Thus we see that the uncertaintyprinciple does not put any significant constraint on the position andvelocity determinations of macroscopic objects. This is whyclassical mechanics is such a good approximation to the macro-scopic world.

We contrast this situation with that which applies for an electroninside an atom. The uncertainty in position cannot be larger thanthe size of the atom, which is about 10-m. Since the electronmass is approximately kg, equation (2.1) then yields avelocity uncertainty of around lo6 ms-. This is a very largevelocity, as can be seen, for example, by the fact that it correspondsto passage across the atom once every 10-l6s. Thus we guess,correctly, that quantum effects are very important inside atoms.

Nevertheless, readers may be objecting on the grounds that, evenin the microscopic world, it is surely possible to devise experimentsthat will measure the position and velocity of a particle to a higheraccuracy than that allowed by equation (2.1), and therebydemonstrate that the uncertainty principle is not correct. Suchobjections were made in the early days of quantum theory and wereshown to be invalid. The crucial reason for this is that themeasuring apparatus is also subject to the limitations of quantumtheory. In consequence we find that measurement of one of thequantities to a particular accuracy automatically disturbs the otherand so induces an error that satisfies equation (2.1). As a simpleexample of this, let us suppose that we wish to use a microscope


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22 Quantum theoryto measure the position of a particle, as illustrated in figure 6 . Themicroscope detects light which is reflected from the particle. Thislight, however, consists of pho tons , each of w hich carries mom en-tum . T hu s the velocity of the particle is continuously being alteredby the light that is used to measure its position. It is not possiblet o calculate these changes since they de pen d on t he directions of thepho tons aft er collision. T he resulting uncertainty can be show n tobe that given by the uncertainty relation. The caption to figure 6explains this more fully. Most textbooks of quantum theory, e .g.those mentioned in the bibliography ($6.5) , include a detailedanalysis of this experiment and of other similar thoughtexperiments.

Aperture

ObjectIni t ial direction of photon :,kwrth wavelength I-igure 6 Showing how the uncertainty principle is operativewhen a microscope is used to fix a position. For an accuratemeasurement of position the aperture should be large, but thisleads to a large uncertainty in the direction of the photon, andhence to a large uncertainty in the momentum of the object.In fact, the error in position is given by IJsincr and that inmomentum by p sin a where p is the photon momentum,related to its wavelength by I = 27rAJp [cf equation (2 .4 ) ] .Hence the product of the errors is equal to 27rh, as required.Note that a crucial part of the argument here is that light isquantised, i.e. light of a given wavelength comes in quantawith a fixed momentum.

So far in this section we have taken the probability to dependupon just one variable, namely the distance x alon g some line. I ngeneral, of course, it will depend upon position in three-dimensional space. Nothing in the above discussion is greatly


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The wavefunction 23affected. The position uncertainty in any particular direction isalways related by the uncertainty principle, equation (2.1), to thevelocity uncertainty in the same direction.Since we are considering one particle, which has to besomewhere, the probabilities of finding it in a particular region ofspace, when added over all such regions, must give unity. Becausethe points of space are not discrete but rather continuous, thisaddition is performed by an integral. Most readers will probablynot wish to be troubled by such technicalities so , since they are notessential for understanding the subsequent discussion, we relegatefurther details of this and a few other matters connected with theprobability t o Appendix 3. One fact will be useful for us to know.In the one-dimensional case the probability of finding the particlein any interval is equal to the area under the graph of the prob-ability curve, bounded by that interval. This is illustrated in figure5 . Of course, in order that the total probability should be unity itis important that the area is measured in units such that the totalarea under the probability graph is equal to one.

To proceed we must now go beyond the probability and considerthe quantity from which it is obtained. This is called the wave-function and, being the basic quantity which is calculated byquantum mechanics, it will play an important part in the develop-ment of ou r story. What th e wavefunction means is, as we shall see,very unclear; what it is, however, is really quite simple. Since itinvolves ideas that will be new to some readers we devote the nextsection to it.

2.2 The wavefunctionWe consider a system of a single particle acted upon by someforces. I n classical mechanics the state of the system at any time isspecified by the position and velocity of the particle at that time.The subsequent motion is then uniquely determined for all futuretimes by solution of Newtons second law of motion, which tells usthat the acceleration is the force divided by the mass.In quantum theory the state of the system is specified by awavefunction. Instead of Newtons law we have Schrodingersequation. This plays an analogous role because it allows thewavefunction to be uniquely determined a t all times if it is known


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24 Quantum theoryat some initial time. Thus quantum mechanics is a deterministictheory of wavefunctions, just as classical mechanics is of positions.

The wavefunction of a particle exists at all points of space. Itconsists of two numbers, whose values, in general, vary with thepoint considered. We shall find it convenient later to picture thesetwo numbers by regarding the wavefunction as a line on a plane,like that shown in figure 7 . The two numbers are then the lengthof the line and the angle it makes with some fixed line. We shallrefer to these numbers as the magnitude and the angle of thewavefunction. Readers who wish to use the proper technicallanguage should refer to Appendix 4.

This l ine i s the

4 7ixed Line This IS thereal p a r t o fthe wavefunctionFigure 7 Showing how a wavefunction at a particular pointin space can be represented by a line on a plane.

As mentioned in the previous section, the wavefunction at agiven point determines the probability for the particle to be at thatpoint. In fact, the relation between the wavefunction and the pro-bability is very simple: the probability is proportional to the squareof the magnitude of the wavefunction. It does not depend in anyway on the angle of the wavefunction.The classical notion of a particles position is therefore related tothe magnitude of the wavefunction. What about the classicalvelocity? Not surprisingly, this is related to the angle. In fact, thevelocity is proportional to the rate at which the angle of thewavefunction varies with the point of space, i.e. with x . The reason


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The wavef nction 25for this is discussed in Appendix 4 (but only for readers with thenecessary mathematical knowledge). Note that here we are speak-ing of the actual velocity, not the uncertainty in the velocity which,as discussed earlier, is proportional to the width of the peak in theprobability.For easier visualisation of what is happening it is useful tosimplify the idea of a wavefunction by thinking about its so-calledreal p a rt , which is the projection of the wavefunction along somefixed line, as shown in figure 7 . For example, the real part of thewavefunction corresponding to the probability distribution offigure 5 might look like figure 8. The dashed line in this figure isthe magnitude of the wavefunction. The rate of oscillation of thereal part is proportional to the velocity of the particle.

part

p s . This is th e magnitude

,

1 I 1 1 I -Thisisminusthe magnitude

Figure 8 A typical wave packet. The broken curve indicatesthe magnitude of the wavefunction and the solid curve givesthe real part. The rate of oscillation is proportional to theaverage velocity of the particle.


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26 Quantum theory

I

We shall see later that it is necessary to have a method ofadding wavefunctions. The method we use can be understood byreference to figure 9. We wish to add the wavefunctions representedby the lines in figures 9 (a )and (b) .To do this we join the beginningof the first line to the end of the second; then the line joining thebeginning of the second to the end of the first is the line thatrepresents the sum of the two wavefunctions. This is illustrated infigure 9(c) . It is not hard to show that, with this definition, it isirrelevant which line is called the first and which the second. Wenow notice the important fact that this definition is not the sameas using ordinary addition to add the numbers associated with eachwavefunction. In particular, the magnitude of the sum of twowavefunctions is not the same as the sum of the magnitudes of thewavefunctions. As an example of this, whereas, since magnitudesare always positive, the sum of two magnitudes is always greaterthan either, this is not necessarily the case for the magnitude of thesum, as is seen in figure 10. Note, however, that the real parts ofwavefunctions do add just like ordinary numbers.

I First

YSecondavefunction

- - - - - -

Q3econaIavefunction

\ Sum o f l a ) and Ib lI o ) I b ) Ic 1

Figure 9 Showing how two wavefunctions, (a) an d ( b ) ,together to give a new wavefunction (c). dd

Readers who wish to know further mathematical detailsregarding wavefunctions, their addition, etc, should consultAppendix 4. Such details will not be essential for what follows.We are now in a position to understand the quantum mechanicaltreatment of the two types of potential barrier experiment intro-duced earlier. These topics will be our concern in the next twosections.


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The potential barrier in quantum mechanics 27

Iio) ( b ) ( C )Figure 10 Another example of addition of two wavefunc-tions. We note, in particular, that the magnitude of the sumof the wavefunctions is smaller than the magnitudes of eitherof the two wavefunctions.

2.3 The potential barrier according toquantum mechanicsWe require for this problem an initial state which corresponds asclosely as possible to the classical situation, i.e. a particle on theright of the barrier and moving towards it with a velocity v. To thisend we take a wavefunction with a magnitude that is peaked in theneighbourhood of the initial position and with an angular varia-tion such that the average velocity is equal to v. There will of coursebe an uncertainty in both the position and the velocity, accordingto equation (2 .1 ) .A possible form for the square of the magnitude,which we recall is proportional to the probability, is shown in figurel l ( a ) . Since we are dealing with one particle the area under thispeak will be equal to one.The Schrodinger equation now determines the subsequentbehaviour of this wavefunction. We shall not discuss the methodof solving the equation but merely state the results. The peak in thewavefunction moves towards the barrier with a velocity approx-imately v-this is very similar to the classical motion of a particlewhere there are no forces. There is, in addition, a small increase inthe width of the peak, so the situation at a later time is shown infigure 1 l(b). When the peak reaches the barrier, where the effect ofthe force begins to be felt, it spreads out more rapidly and thensplits into two peaks, as seen in figure 1l(c).These two peaks thenmove away from the barrier in opposite directions, so a little later


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28 Quantum theorywe have the situation shown in figure 1l(d). O ur wavefunction hasseparated into two peaks, one reflected and one transmitted by thebarrier.It is a consequence of the Schrodinger equation that, throughoutthe motion, the total area under the graph of the square of the

Pr b i l i t y

Poslt'lon ofbarrierFigure 11 Showing how the wavefunction for a particleincident on a potential barrier varies with time. The graphs( a ) - ( d ) show the square of the magnitude of the wavefunc-tion at fo ur successive times. Th e pictures correspond to a casewhere the probability of transmission is greater than that ofreflection.


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Interference 29length of the wavefunction remains equal to one. In fact we knowthat this has to be true for consistency with the probabilityinterpretation-the particle always has to be somewhere. The prob-ability that it is on the right of the barrier, i.e. that it has beenreflected, is given by the area under the right-hand peak, whereasthe probability fo r transmission is given by the a rea under the peakon the left. Thus the calculation allows us to predict theseprobabilities and to compare with the results of experiments asdiscussed in $1.3. In all cases where calculations using theSchrodinger equation have been compared with experiment theagreement is perfect. In particular, it is worth mentioning that weobtain agreement with the classical result for a very high or verylow potential barrier, namely almost 100% reflection or transmis-sion respectively.We must now look more closely a t what ou r calculation fo r thepotential barrier experiment really tells us. After collision with thebarrier the wavefunction, and hence the probability, is the sum oftwo pieces. H ere we are ignoring the fact tha t the two parts a re inpractice joined because the wavefunction is never quite zero, justvery small, between them. W ha t, then, happens when we make anobservation which tells us whether the particle has been reflected?Clearly, in some sense, we select one of the two peaks in the wave-function. In other words, we might say that the wavefunction hasjumped fro m having two peaks t o having only one. This process isreferred to as reduction of the wave packet. What it means, whetherit happens and, if so, how, are topics to which we shall return.

To close this section we emphasise that the wavefunction isdetermined from the initial conditions in a completely deterministicway. Knowing the initial wavefunction exactly (e.g. figure 1 ( ~ ) ) ,we can calculate, without any uncertainty, the wavefunction at alllater times and hence the probability of transmission or reflection.The non-deterministic, probabilistic, aspects of the potentialbarrier experiment arise because we do not observe wavefunctionsbut rather particles; in particular, we can observe the position ofan individual particle after it has interacted with the barrier.2.4 InterferenceWe shall next consider the quantum theoretical description of the


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30 Quantum theorysecond type of barrier experiment discussed in Chapter One. Inthis, we recall, there were mirrors which could bring both thereflected and the transmitted particles to the same set of detectors.We begin then with the same initial state as before (figure l l ( a ) )and follow the wavefunction to the situation shown in figure 1 l ( d ) .Here, to a good approximation, the wavefunction can be regardedas a sum of two wavefunctions, one giving the left-hand peak andthe other the right-hand peak. Note that the operation of addingthe two wavefunctions is rather trivial at this stage since, at anygiven point of space, at most one of the two wavefunctions whichare added is different from zero. In the subsequent motion each ofthe two peaks will change independently; in fact they will move ina manner closely resembling the classical motion of a free particle.(It is irrelevant here that the area under each peak is not actuallyequal to one.)Eventually, if the mirrors are present, the peaks will cometogether in the neighbourhood of the detectors. At this stage theaddition is no longer trivial since both wavefunctions are differentfrom zero at the same place. This means that the feature mentionedat the end of 52.2 becomes relevant, and the probability resultingfrom the two wavefunctions is not equal to the sum of the prob-abilities associated with the separate wavefunctions.We have here an example of an extremely importantphenomenon known as interference. It occurs in a wide range ofphysical situations even where quantum effects are not relevant. Asan example, we can think of two pebbles being dropped onto thesurface of a still pond. Ripples will spread out from the points ofimpact. At some positions on the pond the ups and the downsfrom the two circular wave patterns will always come at the sametime and the wave will therefore be enhanced. At others they willbe out of phase, i.e. an up from one will arrive at the same timeas a down from the other, in which case they will cancel eachother and the water will remain still. Figure 12 illustrates thissituation.

In our quantum mechanics problem the situation is rather morecomplicated since we are not just adding numbers, which can bepositive or negative, but adding lines, and we recall that the resultdepends on the angle between the lines. On the other hand, if wethink just of the real parts of the wavefunctions, then what happensis very similar to the case of water waves, The precise forms of the


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Interference 31two wavefunctions to be added will depend on the length of thepath to any particular detector (see figure 4, for example). Itfollows that the nature of the interference observed will depend onwhich detector is considered. Certainly, in general, the probabilityresulting from the sum of the two wavefunctions will be differentfrom the sum of the probabilities coming from each separately.

Figure 12 Illustrating the way that waves interfere. The thin linesrepresent the contributions of two different sources, and the heavylines their sum, all plotted as functionsof time. In ( a ) he two con-tributions almost exactly cancel, whereas in ( b ) they have similarphase and add to produce a larger effect.


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1creen with twonarrow shts Screen whereinterferencepattern i s seen

Figure 13 ( a )An experiment which shows interference effects forelectromagnetic radia tion, e.g. light. T he radiatio n from th e sourcecan reach the right-hand screen through either slit. At the point Pth e radiation will arrive in phase because the two pa th lengths ar e thesam e, whence the re is constructive interference. A t points away fromP he path lengths are different an d destructive interference is poss-ible. A pattern of intensity like that shown in (b) emerges on thescreen.


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Other applications of quantum theory 33This is in accordance with the observations which we found sosurprising in 41.4.

Detailed calculations yielding precise results are, of course,possible. Similar calculations can be done for other situations inwhich quantum mechanical interference occurs, and where theresults can be verified by experiments. Of particular importance a reexperiments where electrons are scattered off crystals. Here theinterference is between parts of the wavefunction scattered offdifferent sites in the crystal. Comparison of the results withcalculated predictions reveals information on the structure of thecrystal.A brief historical note is of interest here. The long-standingconflict between a corpuscular theory of light (favoured by IsaacNewton) and a wave theory was generally believed to have beensettled in favour of the latter by observation of interference effectswhen light was passed through two slits (see figure 13). Interferenceimplied waves. It was therefore a shock when electrons, longestablished as particles, were also found to show interferenceeffects. This schizophrenic behaviour became known as particle-wave duality. The same duality applies to electromagneticradiation, of which light is an example. The particles of light arecalled photons. In our potential barrier example, the particle natu reis seen most naturally in the first set of experiments where theparticle is observed either t o be transmitted or reflected. The wavenature is seen in the second set, where there is evidence forinterference effects.

Quantum theory successfully incorporates both features andenables us to calculate correctly all microscopic phenom ena that d onot involve relativistic effects. A brief review of some of thesuccesses of the theory is given in the next section, with which weconclude this chapter. The big question of what the quantumtheoretical calculations actually mean is left to Chapter Three.

2.5 Other applications of quantumtheoryIn this section we shall outline som e of the most important applica-tions of quantum theory to various areas of physics, applications


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34 Quantum theorywhich ensured that, in spite of its problems, it rapidly gained accep-tance. Nothing in the remainder of our discussion will depend onthis section, so it may be omitted by readers who are in a hurry.The section is also somewhat more demanding with regard tobackground knowledge of physics than most.The understanding of electricity and magnetism, besides beingthe prerequisite for the scientific and technological revolutions ofthis century, was the great culminating triumph of nineteenth cen-tury, classical, physics. By combining simple experimental laws,deduced from laboratory experiments, into a mathematically con-sistent scheme, Maxwell unified electric and magnetic phenomenain his equations of electromagnetism. These equations predictedthe existence of electromagnetic waves capable of travellingthrough space with a calculable velocity. Visible light, radio waves,ultraviolet light, heat radiation, x-rays, etc, are all examples, differ-ing only in frequency and wavelength, of such waves.The first hint of any inadequacy within this scheme of classicalphysics came with the calculation of the way in which the intensityof electromagnetic radiation emitted by a black body (i.e. a bodythat absorbs all the radiation falling upon it at a particulartemperature) varies with the frequency of the radiation. Theassumptions which went into the calculation were of a very generalnature and were part of the accepted wisdom of classical physics;the results, however, were clearly incompatible with experiment. Inparticular, although there was agreement at low frequency, thecalculated distribution increased continuously at high frequencyrather than decreasing to zero as required.Max Planck, in 1900, realised that one simple modification to theassumptions would put everything right, namely, that emission andabsorption of radiation by a body can only occur in finite sizedpackets of energy equal to h times the frequency. The constant ofproportionality introduced here, and denoted by h , is the originalPlancks constant. For various reasons it is usual now to workinstead with the quantity h , which we quoted in equation (2.2), andwhich is equal to h divided by 27r.The packets of energy, introduced by Planck, are the quantawhich gave rise to the name quantum theory. Each such quantumis now known to be a photon, i.e. a particle of electromagneticradiation, but such a concept was a heresy at the time of Plancksoriginal suggestion; electromagnetic radiation (e.g. light, radio


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Other applications of quantum theory 35waves, etc) was known to be waves! The quantisation was there-fore assumed t o be simply something to do with the processes ofemission and absorption.Such a view was shown to be untenable by the observation ofthe photoelectric effect, in which electrons are knocked out ofatoms by electromagnetic radiation. If we assume that the energyin a uniform beam of light, incident upon a plate, is distributeduniformly across the plate, then it is possible to calculate the timerequired for sufficient energy to fall on one atom to knock out anelectron. This is normally of the order of several seconds, incontrast to the observation that the effect starts immediately.Further, the energy of the emitted electrons is, apart from aconstant, proportional to the frequency of the radiation. Einstein,in 1905 , showed that all the observations were in perfect agreementwith the assumption that the radiation travelled as photons, eachcarrying the energy E appropriate t o its frequency according to therelation previously used by Planck:

E = hf (2.3)where f s the frequency.The final confirmation of the idea of photons came from theobservation, in 1922, of the C om pton effect, in which radiation wasseen to decrease in frequency when it was scattered by electrons.This can be explained very simply as being due to the loss of energyin the photon-electron collision, a loss that can be exactly calculatedfrom the laws of conservation of energy and momentum.

Although quantum theory began with its application to radia-tion, the ideas were soon applied to particles. In 191l , de Brogliesuggested that, if waves can have particle properties, then it isreasonable to expect particles to have wave properties. Heintroduced the relation:I = h / m v (2.4)

between the wavelength I , the velocity v , and the mass m of aparticle. The major achievements of quantum mechanics havebeen, following this relation, in its application to matter, inparticular to the structure of atoms.The experimental work of Rutherford, early this century,showed that an atom consists of a small, positively charged,nucleus, which contains most of the mass of the ato m, surrounded


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36 Quantum theoryby a number o f negatively charged electrons which are boun d to th enucleus by the attractive electric force. Each atom was thereforelike a miniature solar system, with th e electrons playing th e role ofplanets, orbiting th e nuclear sun. Prior to the adven t of qu an tumthe ory there were, however, serious problems with this picture: whydid the orbiting electrons not radiate electromagnetic waves,thereby losing energy so tha t they would fall int o the nucleus? W hywere the energies available to a given atom only a set of discretenumbers, rather than a continuum as would be expected fromclassical mechanics?

Q ua ntu m theo ry provides a complete answer to these questions.All th e energy levels of ato ms can b e calculated fro m . th eSchrodinger equation, in perfect agreement with experiment. Theinteractions between atoms, as observed in molecules, chemicalprocesses a nd atom ic scattering experiments can also be unders toodfrom this equation. As we mentioned in 81.1, quantum theorysuccessfully brought a whole new range of phenomena into thedomain of calculable physics.The details of all this are outside the scope of this particularbo ok , bu t it is worthwhile to give a simple picture of w hy th e wavena tur e of the electron helps us to understand the qua ntum answersto the problems mentioned above with the classical picture of theatom. If we consider a wave on a string with fixed end points,then only certain wavelengths are allowed, because an integralmu ltiple of th e wavelength m ust fit exactly into the string. A conse-quence is that the string can only vibrate with a particular set of

frequencies; a fact which is crucial to many musical instruments.The frequencies which occur can be altered by changing either thelength or the tension of the string. In an atom the situation issimilar, except tha t, instead of having a wave on a string with fixedend po ints, we have a wave on a circle (the orb it), which mu st joinsmoothly on to itself. Th us the circumference of th e circle has to bean exact integral multiple of the wavelength. As we show inAppendix 5 , this condition yields the energy levels of the simplesta tom.T he transition from one energy level in an atom to another, bythe emission of a photon, i.e. by electromagnetic radiation, is anexample of an important class of very typically quantumph en om en a, in which one particle spon taneously decays into (say)two othe rs. Calling the first particle A an d the others B and C , we


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Other applications of quantum theory 37can write this as

A + B + C .If we start with a large number of A particles then, after a giventime, some of them will have decayed. It is usual to define ahalf-life as the time taken for half of the particles in a large initialsample to have decayed. The half-life depends on the process con-sidered and values ranging from tiny fractions of a second to timesbeyond the age of the universe are known.Even though the half-life for the decay of a certain type ofparticle, e.g. the A particle above, might be known, it will not bepossible to say when a particular A particle will decay. This israndom; like, for example, the choice of transmission or reflectionin the potential barrier experiment. Indeed, one can think of sometypes of decays as being rather like a particle bouncing backwardsand forwards between high potential barriers; eventually theparticle passes through a barrier and decay occurs. In general, if westart with a wavefunction describing only identical A particles, thenit will change into a sum of a wavefunction describing A particles,

I

Figure 14 A comparison of the data obtained in a two-slitinterference experiment, like that shown in figure 13, usingneutrons. The curve is the calculated prediction of quantumtheory and the experimental results are shown as dots. Theagreem ent is perfect. (After an original figure in the paper ofZeilinger, Gaehler, Shull and Treimer Symposium on neutronscattering Am. Inst . Phys. 1981.)


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38 Quantum theorywhich will have a magnitude decreasing with time, and one describ-ing (B + C), which will have an increasing magnitude.

Finally, we mention the recent, very accurate, experiments whichshow that neutrons passing through a double slit, as in figure 13,interfere exactly as predicted by quantum theory. An example isshown in figure 14. These experiments were carried out in responseto the recent upsurge of interest in checking carefully the validityof quantum theoretical predictions in as many circumstances aspossible. We shall later mention other such tests. In all cases so farthe theory is completely satisfactory.

2.6 Summary of Chapter TwoWe have shown how the classical description of a particle,involving its position and velocity, is replaced by a description interms of a wavefunction. If this wavefunction is known at someinitial time, for an isolated system, then it is completely determinedfor all future times by the solution of the Schrodinger equation.The relation of the wavefunction to experimental observationintroduces the non-causal aspects into the problem since thewavefunction only predicts the probability of obtaining a givenexperimental result. For macroscopic objects the range of prob-abilities is effectively so small that the classical approximation isnormally adequate, This, however, is certainly not true in themicroscopic world, where the quantum effects are important.

We have seen in particular how quantum mechanics predicts thepreviously discussed results of the potential barrier experiment andhave noted especially the importance of interference effects inobtaining these results. Such interference effects are also importantin the many successes of quantum theory which we have discussed.Any measurement on a system described by quantummechanics chooses one of certain possible results, i.e. it selects partof the wavefunction. This process, known as reduction of thewavefunction, will need to be considered further in the nextchapter.


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Chapter ThreeQuantum Theory andExternal Reality

3.1 Review of the problemIn Chapter One we saw that there are aspects of the observed worldwhich appear to be mutually contradictory, at least when they areinterpreted in terms of our normal pictures of reality. Since suchpictures are necessarily developed from experience of things that wecan see and feel, that is, from the microscopic world, whereas theapparent contradictions only occur when microscopic objects areinvolved, we should perhaps not be too disturbed by this discovery.Our pictures of reality, the words and metaphors we use, are notnecessarily appropriate for the world of the very small. Theevidence of Chapter One suggests that we need new pictures.In the second chapter we discussed quantum theory, which,developed in response to the strange phenomena seen in themicroscopic world, very beautifully predicts such phenomena. Ithas proved to be the most successful and comprehensive theoryknown in physics. We are therefore naturally encouraged to expectthat it might help us to understand the reality that underlies theobservations of the microscopic world. How far such an expect-ation is realised is the topic of this and the following chapter.We shall ask whether quantum theory has merely allowed us tocalculate the results of our experiments or whether it has, in addi-tion, answered the problems we met in the first chapter. We recallthat these problems did not lie in the results themselves but arosewhen we asked what was actually happening. Does quantum theory


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40 Quantum theory and external realitytell us what happens when a particle hits a potential barrier? Whatdoes it tell us about the external reality which is present before wemake our observations? Is it, indeed, even compatible with theexistence of such a reality? These are some of the questions we shalltry to answer.

3.2 The ensemble interpretation ofquantum mechanicsAs we have seen, quantum theory deals with a wavefunction, whichit states is causally determined from some initial conditions. Thepassage from this wavefunction to experimental observation usesthe assumption that the wavefunction gives probabilities formeasurements to yield particular values. In order to test the predic-tions of the theory it is necessary to prepare a large number ofidentical systems and perform the same measurement on each. Werecall that we used this procedure to define the probabilities oftransmission and reflection in $1.3. Of course, the word identicalnow must refer to the wavefunction, i.e. identical systems aredefined to be systems with the same initial wavefunction (andtherefore the same wavefunction for all future times).The large number of identical systems is referred to as anensemble. For any such ensemble the predictions of quantumtheory are precise and deterministic. For example, quantum theorytells us what percentage of a given (large) number of particles willpass through a potential barrier. What it cannot tell us, of course,is whether any particular one of the particles will pass through.Some writers on this topic have therefore adopted the view thatquantum theory is a theory of ensembles and as such tells usanything about individual systems. This is a perfectly reasonableview and it may be the correct one to take. There are then nofurther difficulties in the interpretation of quantum theory, andthe subject does not cause any philosophical problems. We mustnot, however, go on from this to claim that we have solved theproblems met in the first chapter. We have merely ignored them.We do not only have experimental results for ensembles. Individualsystems exist and the problems arise when we observe them. It ispossible to argue that quantum theory says nothing about such


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The wavefunction as a measure of our knowledge 41individual systems but , even if this is true, the prob lems do not goaway.

We shall, in this chapter, adopt a more positive view andcontinue to hope that the theory which predicts our results mightalso help to explain them.3.3 The wavefunction as a measure ofour knowledgeIn 52.2 we tended to regard the wavefunction as describing aparticular system (in fact, just a single particle). Suppose, however,we take the view that the wavefunction instead describes ourknowledge of the system. By implication the system might the n bethought of to have other properties of which, at the time, we areignorant. For example, the particle may actually be at a specificposition, whereas we know only that it has a certain probability ofbeing in some region. This, at first sight, appears to be a veryreasonable view. It is indeed the situation that occurs wheneverprobability aspects arise in non-quantum situations.T o have a trivial example of this, let us suppose that I am in aroom with 10oO other people. Assume also that I know the lo00 ismade up of 498 French men, 2 French girls, 200 Norwegian menand 300 Norwegian girls. With this information I would know thatthe probability of the person immediately behind me being Frenchwas one in two. Now suppose that I looked at the person behindme and saw that she was female. The probability of the personbeing French would immediately change to one in fifty.If the situation in quantum theory is of a similar nature then theissue of the reduction of the wavefunction, raised in 52.3, im-mediately goes away. When the wavefunction is just an expressionof ou r knowledge of the tru th , then it is not surprising, and is evenexpected, that is should suddenly change to something else when ameasurement is made. A measurement has simply changed ourknowledge (this of course is normally the purpose of makingmeasurements).Superficially attractive though this view of the wavefunction maybe, it is in one very impor tant respect inadequate. It cann ot explainthe phenomenon of interference. We remind ourselves here thatthere is abundant experimental evidence for interference effects


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42 Quantum theory and external realityand, contrary to what appears to happen in some discussions.of theinterpretation problems of quantum theory, they cannot be ignored.Wavefunctions which merely represent our knowledge of a systemcannot interfere. We can see this immediately in the case of thepotential barrier experiment. There we require that somethingfollows both routes to the detector. That something cannot be ourknowledge, which, if it is anywhere, is in our brain. If the particlereally has followed one route then we are back with the problemas to how its motion can be influenced by the presence of the othermirror. It is not an answer to this to say that we know about theother mirror; the behaviour of the particles surely cannot dependupon the information contained in the brains of particularindividuals.We can therefore be sure that, if interference actually occurs, thisinterpretation of the wavefunction must be wrong. However, theform of the qualification used here is important. What we know isthat the results of our observation can be predicted from thecalculation of the interference effect. It looks as though in-terference is actually happening but it is possible that this is not so,but that, instead, the calculation just happens to give the rightanswer. A simple analogy might help here. An umpire at a cricketmatch counts the number of balls that have been bowled by placingpebbles in his pocket, one for each ball. When six pebbles are inthe pocket he calls over and play changes ends. Now the reasonfor this change is not directly anything to do with pebbles in theumpires pocket, it is because six balls have been bowled and therules say that play changes ends every six balls. The pebbles can beused by the umpire to make the calculation because of the rules ofarithmetic which ensure that the right answer will be obtained. Itcould be that a similar thing is happening with the interferencecalculation; it gives the right answer but the real reason for theexperimental facts lies elsewhere.Where? Clearly we must look at the hidden information-at theproperties not contained in our knowledge of the system, andtherefore not in the wavefunction. We are then in the domain ofhidden variable theories which we discuss in detail in Chapter Five.However, to complete this section we should look ahead and notethat such theories do not in fact eliminate the need for an interferingwavefunction. Indeed, it is inconceivable that any theory couldsuccessfully reproduce all the correct effects of interference unless


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The wavefunction as part of external reality 43the interference actually happens. Thus, although it was importantto mention the reservations of the previous paragraph, I believethey can now be forgotten.

3.4 The wavefunction as part of externalrealityWe now want to consider the possibility that the wavefunctionshould be treated rather more seriously than in the preceding twosections, so that we can use it to tell us something about theexternal reality. W e shall try t o regard the wavefunction not as justa description of a statistical ensemble, as in 53.1, or as a catalogueof our information about a system, as in 53.2, but as somethingthat really exists, something that is, indeed, part of the externalreality which we observe.There are a t least three good reasons why we should w ant t o con-sider this assumption. First, since the classical picture of a singleparticle, always having a precise position and following a specificpa th , is not compatible with the observations described in 01.4, wedo not have any other object available for our representation ofreality. Secondly, the evidence that wavefunctions can interferestrongly suggests that they are real, e.g. just like ripples on the sur-face of a pond. In o rder to understand the third reason we need toknow about certain symmetry properties that have to be imposedon wavefunctions describing more than on e particle. If we have twoidentical particles, e.g. two electrons, then in classical mechanicswe could distinguish them, for example, by their positions. Inquantum theory, on the other hand, they are described by awavefunction which tells us the probability of finding an electrona t one place and a n electron a t another place; in no way are the twoelectrons distinguished. This means that the wavefunction must besymmetrical in the two electrons, i.e. it must not change if we inter-change them. Actually, the truth is a little different from thisbecause in some particular cases the wavefunction has to change itssign. Such a change, however, does not alter any of the physics,which is determined by the square of the magnitude of thewavefunction. A more detailed discussion of this is given inAppendix 4. Here we merely note that the symmetry properties give


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44 Quantum theory and external realityrise to important, testable, predictions, which have been verifiedand which would be very hard to understand without the assump-tion that wavefunctions have a real existence.Our tentative picture of the potential barrier experiment istherefore that of a wavefunction which has a value that varies withthe point of space being considered. We are familiar with quantitiesof this type, e.g. the temperature of the air at different points ofa room, or the number of flies per unit volume in a field of cattle.Actually the wavefunction is a little different since, as we recall, itis a line or, alternatively, two numbers at each point of space. Thisfact, however, does not affect the present discussion, so we shallcontinue to refer simply to the value of the wavefunction.As is illustrated for example in figure 11, the wavefunction'is ingeneral not constant but changes with time. Again this is a conceptwith which we are familiar; the temperatures at various points ina room, for example, will similarly change with time, e.g. when theheating has been switched off. We therefore have a simple pictureof reality, with the wavefunction describing something that actuallyhappens.There are, however, two difficulties associated with this picture.The first of these is due to the fact that the world does not consistof just one particle. We remember that the wavefunction we haveused so far was specifically designed to treat only one particle. Howdo we generalise this to accommodate additional particles?Consider a world of two particles, which we shall call A and B.As a first guess we might try having a wavefunction for particle Aand a separate and independent one for particle B. Then theprobability of finding A at some point would not depend on theposition of B. This is reasonable for particles that are genuinelyindependent, i.e. not interacting. It is, however, quite unreason-able, and is indeed false, for particles that are interacting. In thiscase the wavefunction must depend on rwo positions. It will thentell us the probability for finding particle A at one position and par-tide B at the other. (Some further details are given in Appendix 4.)One can express this by saying that the wavefunction does not existin the usual space of three dimensions but in a space of two-times-three dimensions. It is no longer true to say that at a particularpoint of space the wavefunction has a particular value. Rather wehave to say that, associated with every two points of space (or, if


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The wavefunction as a measure of our knowledge 45we prefer to express it this way, with every point of a six-dimensional space) there is a particular value fo r the wavefunction.

Of course, we cannot stop at two particles and must go on toinclude 3 , 4 , etc, with the wavefunction depending on the corre-sponding number of points, 9, 12, etc, in space. At this stage thewavefunction starts to look more like a mathematical device thansomething that is part of the real world. Certainly it is not now ofthe form of the familiar quantities mentioned earlier. These arelocal, i.e. at a single point of space there is a number which is thetemperature. The wavefunction, on the contrary, is non-local; inorder to establish its value we need to give many positions in space.We shall find this non-locality occurring again in our discussion.It should be noted here that the two-particle wavefunction isnot, in general, simply a product of two one-particle wavefunc-tions. T o understand this distinction we recall that the square of themagnitude of the wavefunction gives the probability of findinga particle at each of the two points. If the particles are quiteindependent, and not in any way correlated in position, then theprobability of finding a particle a t a point P will not depend on theposition of the other. In such a case the wavefunction will be asimple product of two wavefunctions, each depending upon oneposition. In most real situations, however, particles interact andtherefore their positions are correlated. The wavefunction is thennot of the product type but is, rather, o ne funct

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