CHAPTER 4 THE UNCERTAINTY PRINCIPLE

The success of scientific theories, particularly Newton’s theoryof gravity, led the French scientist the Marquis de Laplace atthe beginning of the nineteenth century to argue that theuniverse was completely deterministic. Laplace suggested thatthere should be a set of scientific laws that would allow us topredict everything that would happen in the universe, if only weknew the complete state of the universe at one time. Forexample, if we knew the positions and speeds of the sun andthe planets at one time, then we could use Newton’s laws tocalculate the state of the Solar System at any other time.
Determinism seems fairly obvious in this case, but Laplace wentfurther to assume that there were similar laws governingeverything else, including human behavior.
The doctrine of scientific determinism was strongly resistedby many people, who felt that it infringed God’s freedom tointervene in the world, but it remained the standard assumptionof science until the early years of this century. One of the firstindications that this belief would have to be abandoned camewhen calculations by the British scientists Lord Rayleigh and SirJames Jeans suggested that a hot object, or body, such as astar, must radiate energy at an infinite rate. According to thelaws we believed at the time, a hot body ought to give offelectromagnetic waves (such as radio waves, visible light, or Xrays) equally at all frequencies. For example, a hot body shouldradiate the same amount of energy in waves with frequenciesbetween one and two million million waves a second as inwaves with frequencies between two and three million millionwaves a second. Now since the number of waves a second isunlimited, this would mean that the total energy radiated wouldbe infinite.
In order to avoid this obviously ridiculous result, the Germanscientist Max Planck suggested in 1900 that light, X rays, andother waves could not be emitted at an arbitrary rate, but onlyin certain packets that he called quanta. Moreover, eachquantum had a certain amount of energy that was greater thehigher the frequency of the waves, so at a high enoughfrequency the emission of a single quantum would require moreenergy than was available. Thus the radiation at highfrequencies would be reduced, and so the rate at which thebody lost energy would be finite.
The quantum hypothesis explained the observed rate ofemission of radiation from hot bodies very well, but itsimplications for determinism were not realized until 1926, whenanother German scientist, Werner Heisenberg, formulated hisfamous uncertainty principle. In order to predict the futureposition and velocity of a particle, one has to be able tomeasure its present position and velocity accurately. Theobvious way to do this is to shine light on the particle. Someof the waves of light will be scattered by the particle and thiswill indicate its position. However, one will not be able todetermine the position of the particle more accurately than thedistance between the wave crests of light, so one needs to uselight of a short wavelength in order to measure the position ofthe particle precisely. Now, by Planck’s quantum hypothesis, onecannot use an arbitrarily small amount of light; one has to useat least one quantum. This quantum will disturb the particleand change its velocity in a way that cannot be predicted.
moreover, the more accurately one measures the position, theshorter the wavelength of the light that one needs and hencethe higher the energy of a single quantum. So the velocity ofthe particle will be disturbed by a larger amount. In otherwords, the more accurately you try to measure the position ofthe particle, the less accurately you can measure its speed, andvice versa. Heisenberg showed that the uncertainty in theposition of the particle times the uncertainty in its velocity timesthe mass of the particle can never be smaller than a certainquantity, which is known as Planck’s constant. Moreover, thislimit does not depend on the way in which one tries tomeasure the position or velocity of the particle, or on the typeof particle: Heisenberg’s uncertainty principle is a fundamental,inescapable property of the world.
The uncertainty principle had profound implications for theway in which we view the world. Even after more than seventyyears they have not been fully appreciated by manyphilosophers, and are still the subject of much controversy. Theuncertainty principle signaled an end to Laplace’s dream of atheory of science, a model of the universe that would becompletely deterministic: one certainly cannot predict futureevents exactly if one cannot even measure the present state ofthe universe precisely! We could still imagine that there is a setof laws that determine events completely for some supernaturalbeing, who could observe the present state of the universewithout disturbing it. However, such models of the universe arenot of much interest to us ordinary mortals. It seems better toemploy the principle of economy known as Occam’s razor andcut out all the features of the theory that cannot be observed.
This approach led Heisenberg, Erwin Schrodinger, and PaulDirac in the 1920s to reformulate mechanics into a new theorycalled quantum mechanics, based on the uncertainty principle.
In this theory particles no longer had separate, well-definedpositions and velocities that could not be observed, Instead,they had a quantum state, which was a combination of positionand velocity.
In general, quantum mechanics does not predict a singledefinite result for an observation. Instead, it predicts a numberof different possible outcomes and tells us how likely each ofthese is. That is to say, if one made the same measurementon a large number of similar systems, each of which started offin the same way, one would find that the result of themeasurement would be A in a certain number of cases, B in adifferent number, and so on. One could predict theapproximate number of times that the result would be A or B,but one could not predict the specific result of an individualmeasurement. Quantum mechanics therefore introduces anunavoidable element of unpredictability or randomness intoscience. Einstein objected to this very strongly, despite theimportant role he had played in the development of these ideas.
Einstein was awarded the Nobel Prize for his contribution toquantum theory. Nevertheless, Einstein never accepted that theuniverse was governed by chance; his feelings were summedup in his famous statement “God does not play dice.” Mostother scientists, however, were willing to accept quantummechanics because it agreed perfectly with experiment. Indeed,it has been an outstandingly successful theory and underliesnearly all of modern science and technology. It governs thebehavior of transistors and integrated circuits, which are theessential components of electronic devices such as televisionsand computers, and is also the basis of modern chemistry andbiology. The only areas of physical science into which quantummechanics has not yet been properly incorporated are gravityand the large-scale structure of the universe.
Although light is made up of waves, Planck’s quantumhypothesis tells us that in some ways it behaves as if it werecomposed of particles: it can be emitted or absorbed only inpackets, or quanta. Equally, Heisenberg’s uncertainty principleimplies that particles behave in some respects like waves: theydo not have a definite position but are “smeared out” with acertain probability distribution. The theory of quantummechanics is based on an entirely new type of mathematicsthat no longer describes the real world in terms of particlesand waves; it is only the observations of the world that maybe described in thoseterms. There is thus a duality between waves and particles inquantum mechanics: for some purposes it is helpful to think ofparticles as waves and for other purposes it is better to thinkof waves as particles. An important consequence of this is thatone can observe what is called interference between two sets ofwaves or particles. That is to say, the crests of one set ofwaves may coincide with the troughs of the other set. The twosets of waves then cancel each other out rather than addingup to a stronger wave as one might expect (Fig. 4.1). Afamiliar example of interference in the case of light is the colorsthat are often seen in soap bubbles. These are caused byreflection of light from the two sides of the thin film of waterforming the bubble. White light consists of light waves of alldifferent wavelengths, or colors, For certain wavelengths thecrests of the waves reflected from one side of the soap filmcoincide with the troughs reflected from the other side. Thecolors corresponding to these wavelengths are absent from thereflected light, which therefore appears to be colored.
Interference can also occur for particles, because of the dualityintroduced by quantum mechanics. A famous example is theso-called two-slit experiment (Fig. 4.2). Consider a partition withtwo narrow parallel slits in it. On one side of the partition oneplaces a source of fight of a particular color (that is, of aparticular wavelength). Most of the light will hit the partition,but a small amount will go through the slits. Now suppose oneplaces a screen on the far side of the partition from the light.
Any point on the screen will receive waves from the two slits.
However, in general, the distance the light has to travel fromthe source to the screen via the two slits will be different. Thiswill mean that the waves from the slits will not be in phasewith each other when they arrive at the screen: in some placesthe waves will cancel each other out, and in others they willreinforce each other. The result is a characteristic pattern oflight and dark fringes.
The remarkable thing is that one gets exactly the same kindof fringes if one replaces the source of light by a source ofparticles such as electrons with a definite speed (this meansthat the corresponding waves have a definite length). It seemsthe more peculiar because if one only has one slit, one doesnot get any fringes, just a uniform distribution of electronsacross the screen. One might therefore think that openinganother slit would just increase the number of electrons hittingeach point of the screen, but, because of interference, it actuallydecreases it in some places. If electrons are sent through theslits one at a time, one would expect each to pass through oneslit or the other, and so behave just as if the slit it passedthrough were the only one there - giving a uniform distributionon the screen. In reality, however, even when the electrons aresent one at a time, the fringes still appear. Each electron,therefore, must be passing through both slits at the same time!
The phenomenon of interference between particles has beencrucial to our understanding of the structure of atoms, thebasic units of chemistry and biology and the building blocks outof which we, and everything around us, are made. At thebeginning of this century it was thought that atoms were ratherlike the planets orbiting the sun, with electrons (particles ofnegative electricity) orbiting around a central nucleus, whichcarried positive electricity. The attraction between the positiveand negative electricity was supposed to keep the electrons intheir orbits in the same way that the gravitational attractionbetween the sun and the planets keeps the planets in theirorbits. The trouble with this was that the laws of mechanicsand electricity, before quantum mechanics, predicted that theelectrons would lose energy and so spiral inward until theycollided with the nucleus. This would mean that the atom, andindeed all matter, should rapidly collapse to a state of very highdensity. A partial solution to this problem was found by theDanish scientist Niels Bohr in 1913. He suggested that maybethe electrons were not able to orbit at just any distance fromthe central nucleus but only at certain specified distances. Ifone also supposed that only one or two electrons could orbit atany one of these distances, this would solve the problem of thecollapse of the atom, because the electrons could not spiral inany farther than to fill up the orbits with e least distances andenergies.
This model explained quite well the structure of the simplestatom, hydrogen, which has only one electron orbiting aroundthe nucleus. But it was not clear how one ought to extend itto more complicated atoms. Moreover, the idea of a limited setof allowed orbits seemed very arbitrary. The new theory ofquantum mechanics resolved this difficulty. It revealed that anelectron orbiting around the nucleus could be thought of as awave, with a wavelength that depended on its velocity. Forcertain orbits, the length of the orbit would correspond to awhole number (as opposed to a fractional number) ofwavelengths of the electron. For these orbits the wave crestwould be in the same position each time round, so the waveswould add up: these orbits would correspond to Bohr’s allowedorbits. However, for orbits whose lengths were not a wholenumber of wavelengths, each wave crest would eventually becanceled out by a trough as the electrons went round; theseorbits would not be allowed.
A nice way of visualizing the wave/particle duality is theso-called sum over histories introduced by the Americanscientist Richard Feynman. In this approach the particle is notsupposed to have a single history or path in space-time, as itwould in a classical, nonquantum theory. Instead it is supposedto go from A to B by every possible path. With each paththere are associated a couple of numbers: one represents thesize of a wave and the other represents the position in thecycle (i.e., whether it is at a crest or a trough). The probabilityof going from A to B is found by adding up the waves for allthe paths. In general, if one compares a set of neighboringpaths, the phases or positions in the cycle will differ greatly.
This means that the waves associated with these paths willalmost exactly cancel each other out. However, for some sets ofneighboring paths the phase will not vary much between paths.
The waves for these paths will not cancel out Such pathscorrespond to Bohr’s allowed orbits.
With these ideas, in concrete mathematical form, it wasrelatively straightforward to calculate the allowed orbits in morecomplicated atoms and even in molecules, which are made upof a number of atoms held together by electrons in orbits thatgo round more than one nucleus. Since the structure ofmolecules and their reactions with each other underlie all ofchemistry and biology, quantum mechanics allows us in principleto predict nearly everything we see around us, within the limitsset by the uncertainty principle. (In practice, however, thecalculations required for systems containing more than a fewelectrons are so complicated that we cannot do them.)Einstein’s general theory of relativity seems to govern thelarge-scale structure of the universe. It is what is called aclassical theory; that is, it does not take account of theuncertainty principle of quantum mechanics, as it should forconsistency with other theories. The reason that this does notlead to any discrepancy with observation is that all thegravitational fields that we normally experience are very weak.
How-ever, the singularity theorems discussed earlier indicate thatthe gravitational field should get very strong in at least twosituations, black holes and the big bang. In such strong fieldsthe effects of quantum mechanics should be important. Thus, ina sense, classical general relativity, by predicting points ofinfinite density, predicts its own downfall, just as classical (thatis, nonquantum) mechanics predicted its downfall by suggestingthat atoms should collapse to infinite density. We do not yethave a complete consistent theory that unifies general relativityand quantum mechanics, but we do know a number of thefeatures it should have. The consequences that these wouldhave for black holes and the big bang will be described in laterchapters. For the moment, however, we shall turn to the recentattempts to bring together our understanding of the otherforces of nature into a single, unified quantum theory.