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abcs - of - quantum - mechanics - by - rydnik, Notas de estudo de Física

abcs - of - quantum - mechanics - by - rydnik

Tipologia: Notas de estudo

2014

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Baixe abcs - of - quantum - mechanics - by - rydnik e outras Notas de estudo em PDF para Física, somente na Docsity! E FROM BOHR'S THEORY 10 QUANTUM MECHANICS A Remarkable Article A Little about Ordmary Waves Getting Acquainted with Mauer Waves Why Can't We See de Broglie Waves? The Wave IS Found Two-Faced Parucles Pilot Waves Together or Separately? A VISit to the'<Shootmg Range Waves of Probabrltty Probability Enters Into PhySICS Cauuous Predrcnons Waves of Particles and Particles of Waves On the Way to the Wave Law Measurmg Instruments Take over The Uncertainty Relation What IS to Blame, the Instrument or the Electron? An Attempt with Rather Faulty Tool, Another Marvel The Uncertainty Relation Once Agam Matter Waves Again The Wave Function Waves and Quanta are United ATOMS, MOLECULES, CRYSTALS Clouds, In Place of Orbits Monotony In Diversity Another Marvel - But as Yet Unexplamed The Atomic Architect at Work Crazy Atoms Atoms and Chemistry The Birth of a Spectrum Fat Lines and Double Lmcs Atoms Get Married Solid Bodies are Really Soltd Skeletons and Mulnstorey Structures of Crystals Insulators Can Conduct Current How does Current Move m a Metal ') Those Wonderful 'Seml.Thmg'i' Useful 'Om' Generous and Greedy Atoms 8 120 124 125 128 130 132 136 140 142 147 149 155 158 162 164 165 65 66 67 68 71 75 76 78 80 83 84 87 90 91 95 98 101 103 106 109 112 114 116 THE INTERIOR OF THE ATOMIC NUCLEUS On the Threshold The First Step The Second Step The Search for the Mysterious Meson The Strongest Forces of All Once Again about the Stability of Nuclei Tunnels III Nuclei ? Does the Nucleus Consist of Shells? Where do Gamma Rays Come fr~m The Nucleus as a Liquid Drop The Liquid-Drop Nucleus Splits The Secrets of Nuclear FISSion H M' Nuclei Can There Be? ow Nu~f:u~ as Shells and Liquid Drop Together: Partrcles Fly out of the Nucleus that Were Never There The Electron Has an Accomplice Electrons are Born in Nuclei The Hungry Nucleus FROM ATOMIC NUCLEI TO ELEMENTARY PARTICLES The Discovery of a New World The InVISIble Drvtdmg Lme A Bit More about the Theory of Relanviry The First Dlfficultle~ An Unexpected Discovery A Stili More Unexpected Discovery The Birth of a 'Hole' The Outlmes of Empnness Complete Emptmess? Emptiness Depends Oil Bodies I Matter and Fields There IS No Ernptmess ' What the Whales Rest on Particles Change Their GUise The Two-Faced PI·Meson A Clue to Meson Exchange The Secret of Interaction The Kingdom of Vmuahucs The Virtual Becomes Real In the Search for New Particles Sorting the Booty 171 173 176 180 181 185 187 191 193 196 198 200 203 204 207 209 212 215 219 221 224 226 228 231 ::!35 237 240 242 245 246 249 251 254 256 259 ::!64 265 268 270 9 Antiparticles Come mto Action Parucles Drsmtegrate . PhYSICiSts Classify Interactions The Mystery of the K-Mesons Is the Left Any Different from the Right? A way out IS Found I Worlds and Anttworlds . What Goes on Inside Particles? The Mystenous Resonances The Curtain Rises Resonances Get Citizenship 1 nplets, Octets- Quarks. . Old Ideas Hold One Back . The Reverse of the Obvio~s The Ubiquitous Quantum . 274 276 279 281 283 287 290 292 295 297 299 302 306 308 310 312 FROM QUANTUM MECHANICS TO Indeterminable Determmacies . . The Biography of Quantum Mechanics . Quantum Mechanics Gets Its Second Wmd 315 320 324 From classical mechanics to quantum mechanics In Lieu of an Introduction Atomic energy. Radioactive isotopes. Semiconductors. Elementary particles. Masers. Lasers. All quite familiar terms, yet the oldest IS hardly twenty-five years of age. They are all children of twentieth-century physics. In this age, knowledge is advancing at a fantastic rate, and every new step opens up fresh vistas. The old sciences are going through a second youth. PhySICS has pushed out ahead of all others and is pioneering into the unknown. As the front broadens, the attack slows up only to make renewed thrusts forward. To get at the secrets of nature, physics has had to find powerful instruments, to devise precise and convincing experiments. At the physics headquarters are hundreds and thousands of theoreticians mapping out the offensive and studying the trophies captured in the experiments. This is no struggle in the dark. The field of battle is lighted up with powerful physical theories. The strongest searchlights of present-day physics are the theory of relativity and quantum mechanics. Quantum mechamcs carne in with the twentieth century. Date of birth: December 17, 1900. It was II F The Outlines of the New World But didn't people know anything about this atom before the twentieth century? In a way they did, that is, they had guessed and conjectured. The inquisiuve human mmd had speculated upon these thmgs and had long rmagmed what became a real thing only many centuries later. In ancient times, long before the first travellers laid therr paths of discovery, man had guessed that there were people and animals and land beyond the little area m which he hved. In the same way, people felt that there existed a world of the ultrasmall long before it was actually discovered. One did not need to go far In search 12 of this new world, for it was fight at hand, lying around him in all things. In olden times, thinkers had meditated on the way nature had produced the world around us out of something quite formless. How was it, they queried, that it came to be inhabited by its great diversity of things. MIght it not be that nature worked like a hutlder that makes large houses out of small stones? Then what are these stones? Enormous mountains arc weathered away by the water, the wind, and mysterious volcanic forces. The rocks that come away are In time broken down into pieces. Hundreds and thousands of years pass, and these are pulvenzed IOta dust. Is there no limit to this dividing and subdivldmg of matter? Are there particles so small that even nature is no longer able 10 break them up? The answer was YES. So said the ancient phIlosophers EPICUruS, Democritus and others. These particles were given the name 'atom'. Their chief property was that no further division IS possible. The word 'atom' 10 Greek means 'nondivisible'. What did an atom look like" In those times, this questton remained unanswerable. Atoms might be In the form of solid impenetrable spheres, yet they might not be. Then again: How many different vaneties are there? Maybe a thousand, yet perhaps only one. Some philosophers (the Greek Empedocles, for one) beheved that there were probably four. They beheved that the entire universe consisted of four elements- water, air, earth, and fire. In turn, these elements were thought to consist of atoms. One might now think that With mformation as meagre as this there could be no talk of any progress. True, yet the first steps of sctence are usually m IJ on this day that the German physicist Max Planck reported to a meeting of the Berlin Academy of Sciences Physical Society on his attempt to overcome one of the difficulties of the theory of thermal radiation Dlfnculhes are a common thmg in science. Every day scientists come up against them. But Planck's encounter had a very special significance, for it foreshadowed the development of physics for many years to come. An enormous tree of new knowledge has grown out of the seminal ideas expressed by Planck, which served as a starung point for amazing discoveries far beyond the unagmauon of the wildest science- fiction writers. Out of Planck's concepts grew quantum mechanics, which opened up an entirely new world - the world of the ultrasmall, of atoms, atomic nuclei and elementary parucles, Quantum mechanics brought with It not only new knowledge. It gave a radically different interpretation to the phenomena of the world. For the first time. SCience gave full recogruuon to the accidental And perhaps physrcrsts are not to blame for being taken aback. Though It was only the eternal deterrmmsm which they themselves had concocted that gave way, phYSICiStS seemed to thmk that It was determinism as such that was crumbling, that the universe was governed by absolute anarchy. and that thmgs no longer obeyed exact laws. It took quue some lime before phySICS found Its way out of the deep CflSIS. The Temple Collapses Cunosrty killed the cat The saying IS probably applicable to theories as well Even If today the theory appears quite correct and capable of explainmg all the facts A theory puts m Its appearance at a certain stage 10 the development of SCience, when the latter has made a study of a Wide range of phenomena. The aim of the theory IS to give an explanation from some one POlOt of view But the very same theory proves insufficient and even erroneous when fresh facts are discovered that do not tit 1010 Its narrow framework. Classical mechanics was entirely satisfactory as long as phySICSwas confined to mechanics But the nmeteenth century saw phySICSattack a new broad front thermal processes, which gave nse to therrnodynarmcs , light, which gave rise to optics: electnc and magnetic phenomena. which served as a startmg POlOt for electrodynamics. For a time. phySICS rern.nned III a 2-70 17 rather contented state. All new discovenes continued to fit neatly Into the exisung moulds. However. as the edifice of classical physics grew upwards, ItS enormous front gave signs of fatigue, Sinister cracks appeared. and finally the entire structure began to crumble under the bombardment of new facts One of these most fundamental facts was the remar- kable constancy of the velocity of light. The most careful and objective experiments demonstrated that the behaviour of light IS radically different from what had been observed 10 all other known areas. To fit the behaviour of light into the framework of classical phySICS,SCientists had to devise a medium called the ether, which, by the rules of classical phySICS.would possess SImply fantastic properties. We shall come back to thrs ether later on and examine It In more detail. But the new ether could not save the old phySICS. Another stumbling block to classical phySICS was the thermal radrauon of heated bodies Then, finally, the discovery of radioactivity. ThIS had the most shattenng effect on classical pb ysics dunng the last years of ItS undivided rule, for the mysterious processes of radioacuvny not only smashed atomic nuclei, but exploded the very baSISof phySICS- those pnncrples that had appeared so obvious from the standpoint of common sense. Out of these cracks 10 the structure of classical mechanics grew the theory of relativity and the quantum theory. How the New Theory was Named Quantum mechanics was born at the turn of the cenlury. But why this name? Actually, the term but feebly reflects the contents of the thrngs which the new phySICS dealt WIth. 18 Probably not a single branch of phySICShas escaped a certain vagueness 10 rerrmnology. There arc many reasons for this. but they are primarily of a histoncal nature. First of all. why mechanics? There was nothing mech£JOlcal 10 the new theory. and as we shall see later on, there couldn't be The word 'mechanics' is Justified only 10 that It IS used 10 a general sense, like we speak of the 'mechanics of a watch' meanmg the prinCiple of operatIOn. The conceptual range of quantum mechamcs IS better covered by the broad definttlon of phySICS Itself. Secondly. why quantum" Quantum 10 Latin means 'discrete portion or 'quantlty', Further on we shall see that the new science does actually deal With 'discreteness' In the properties of the surrounding world. That IS onc of ItS basic pnnclples. On the other hand, as we shall sec, thrs discreteness IS not at all general. and IS not found everywherc or at all times. What IS more, It IS only one Side of the medal A no less peculiar aspect IS the duality of the properties of matter. The dual nature of matter lies 10 the fact that one and the same entity (object) comhlOcs the properties of particles and waves The new science was relined to wave mechaOlcs' But here again we have only half of It - there IS no menuon of quanta We conclude that none of the names of the new phySical theory was satisfactory But couldn't somethlOg be thought up more 10 keeping With the actual contents of the subject'! The Introduction or new terms 10 science ISa labonous and thankless job New terms come 10 slowly and l'h.lng~,,1111more slowly. PhYSICistSundcrstand the new 19,. meamng that these terms carry and so It IS for us to learn them. Physicists Build Models Imagine the monon of a ball along a rope that you are whirling round your head. It IS obviously quite simple because you can see everything with your own eyes. That IS exactly how classical physics developed - out of the observations of objects and phenomena that surround us. Roll a ball along a smooth honzontal table It continues to move after the action of the hand has ceased, that IS, after the force has ceased to operate. This and similar observations gave nse to the law of mertia that was enunciated by Newton as the first baSIClaw of mechanics. A ball will not begin to move until pushed by the hand or hit by another ball A ball moving over a smooth table and a ball at rest have one thmg m common: they are not acted upon by any forces, On the rope, however, the ball IS all the time acted upon by a force that deflects It from the reculmear path inherent an free monon. That same ball at rest on the table will, under the action of the force of one's hand. begin to move and will acquire speed (the greater, the bigger the force). ThIS observation gave nse to Newton's second law. But now the investigator - Newton again - leaves the everyday world and looks to the heavens to seek a clue to the 'harmony of the celestial spheres' which had stumped the ancient philosophers. What makes the planets move round the sun 10 the way they do and not otherwise? The word "harmony' suggests a system of order, the operation of some law governing the monon of 20 the heavenly bodies. The matter is not one of 'spheres' naturally. But there must be a law govermng the laotion of the planets, and our earth too. about the sun and the motion of the satellites about their planets. One might recall the ball moving along a swmgmg rope. The motion of the planets about the sun IS indeed very much like the uniform motion of the ball, though It IS slower and there IS no rope In short, If in one case a force is operauve, It is reasonable to suppose that It IS operative to another case too. There is of course no way to perceive directly the acuon of the force governIng planetary motions. But the force IS there. And Newton discovered It We know that It is the force of the reciprocal attractIOnof bodies. Newton's genius hes in the fact that he perceived what is common between the monon of a ball and the orbttal motton of a planet. The Important thmg for us, however, is that the ball and rope was probably one of the f rst phySical models. One gains an understandmg of such a grandiose phenomenon of nature as planetary motion through the study of things on a much smaller scale - on the assumptiOn of course that both are governed by similar laws The question arises as to whether this IS Justifiable everywhere and at all times. Is it nght to extend the laws of one phenomenon to another one which IS much larger or much smaller? In Newton's urne the answer was Simple since observation corroborates the development of some large-scale phenomenon that has been calculated on the baSISof some small-scale one, or vice versa, everythmg holds true. Roughly the same answer can be heard today as well. True, the approach IS somewhat different. Newton 21 --------~---- �-----------.W----------- story a bu - when all these models will have to be jettisoned and replaced by still more unusual ones that will be even harder to grasp. That IS how science develops. Therein hes the greatness of the physicists of this century: they were able to reach their goal through a maze of abstractions and models far removed from everyday things, they succeeded 10 constructmg a far- reach 109 theory of the new world of the ultrasmall. What IS more, on this baSIS. physicists achieved some of the greatest thmgs In the entire history of CIVilization They discovered the secret of nuclear energy. the JIO", that had been bottled up for so long. The atomic power industry and electronrcs would not be here today without the existence of quantum mechanics. phySICS in that the world of the ultrasmall is of prime Importance. The ancient Idea of the great and the small meeting finds Its confirmatIOn In outer space Enormous stars and minute atoms not only converge but exist as an Integral unit. It IS almost imposstble to wnte popularly about science without resorting to some kind of Visual represenlalions And so with quantum mechanics we shall try to find analogies, If not models, In nature. However, such analogies are in no way exact or profound. They simply help us to get a general grasp of things For Instance. as we shall see. the phrase 'electrons revolve around an atomic nucleus' hardly has more meaning to us than the words 'snow IS somethmg white. rather hke salt and falls from the sky' have for the inhabitants of tropical Afnca. The motion of an electron In an atom and the essence of the electron as such ISImmeasurably more complicated than what we know about them today and the way we picture them And not only today, tomorrow and a thousand years hence! Indeed, the development of quantum mechanics IS added proof of the hmitless drverstty. the .nexhausubrhty of the properttes of the electron And everything else as well We today still have rather fragmentary knowledge of the world about us. We are only begmnlng to penetrate into the earth's crust. IOtOthe oceans, the atmosphere We have only Just started to understand the hfe of the fields, the forests, the mountains, the fivers and the deserts If that IS so. how can we expect to know as much about the world of atoms, atorruc nuclei and elementary particles. which are sull more difficult to observe. ~7 Difficult but Interesting The unusual nature of quantum mecharucal notions and the fact that these concepts cannot be vrsuahzed properly make the subject difficult to grasp. True. some of the fault lies 10 quantum mechanics Itself. Not only because ItS range IS continually expanding and Its methods are constantly undergoing refinement. we know that It IS always more difficult 10 wrne about somethmg In a slate of nux and development. and particularly such raptd development, than It IS about firmly established theones. Not only tlus, but also because physicists themselves are still, to thrs day. arguing about the very meaning of quantum mechanics, about the specific aspects of the minute world that It descnbes. We have now entered the space age, where again physics IS called upon to pave the way. The physics of cosmic space differs radically from 'terrestrial' 26 b 2 There is exploration ahead 10 this science for hundreds and thousands of years. As yet we are only at the source of a mighty nver of knowledge. Even so, what amazing things are revealed to the explorer of this recently discovered world. What inspinng, truly fantastic horizons does this new science open up for technology, industry, agriculture and medrcme. Nuclear power stauons, radroactive isotopes, solar batteries, to name a few We are on the threshold of controlled thermonuclear reacuons and we are penetrating IOta outer space All these great attamments of the bnght present and the dazzhng future were born 10 our century out of a small seed thrown Sixty years ago, into the ferule soil of scientific knowledge by Max Planck and, since, carefully cultivated by a whole galaxy of bnlhant scientists. The first steps of the new theory Heat and Light It's nice, on a cold winter evening, to Sit near a hot stove and listen to the sputtenng flames inside and feel the warmth of the fire. But why warmth? Why IS It warm near a stove? Without even seeing the fire inside, one can feel the heat at some distance away. A stove emits some kind of mvisiblc rays that give the sensation of heat. These rays are called heat rays, or Infrared rays. A httle careful observing will show us that thermal radiauon IS quite a common thing In nature. Both heat and light are ermtted by a candle, a large fire, and our enormous sun Even the fantastically distant stars send heat rays to the earth. If a heated body glows, It definuely IS emittmg heat rays as well. The ermssion of hght and heat IS actually one process. That IS why SCientists gave the name thermal radiation to all errussions of a body that appear to be due (0 a heating process - both the errussron of hght and the thermal radiauon proper. Last century, physicists had already discovered the basic laws of thermal radranon They are familiar to all of us. Let us recall two laws 29 , II I __b__ Blacker than BlackFirst, the more a body IS heated, the brighter It glows. The quantity of radiation emitted per second varies drastically with change of temperature of the body. If the temperature IS Increased three times. the radration will mcrease almost one hundredfold ~econd. the colour of the ermssron changes with an Increase In temperature. Observe a piece of Iron pipe under the name of a torch. At first It IS quite dark, but then a faint Crimson unge appears, this turns red, then orange and yellow. And finally the heated metal begins to emit a white light. An .expenenced steelworker can gauge the temperature of an incandescent pipe quite accurately by the colour of lummescence. He wrll say that a famt crimson tinge means a temperature of about 500 C, yellow IS about 800 C, and bright white IS over 1,000 C Physicists are not satisfied with this rough qualitative descnpuon, they want exact figures. To a physicist. 'the day IS cold' means about as much as 'he had a big face'. What one needs is the peculiar features the nose, the lips, the forehead. ' Physicists had encountered a great diversity of bodies and conditions an which thermal radiation 15 emitted. But this diversity of condiuons did not satisfy them in the least. They wanted some kind of 'standard' body. a cntenon to be used as a baSIS for establishing the laws of radiauon of heated bodies. Then the ermssion of light by other bodies could be regarded as deviations from the 'standard' Picture a description like this: "The nose of the man was longer than the standard nose, the forehead was narrower, the jaw more extended, the eyes somewhat greener and somewhat smaller than normal." Rather strange to us, but the phySICISt would be delighted Here's why. Take a number of objects of the same colour, as close as possible. Now exarnme them and try to see how they differ 10 colour A careful examination Will show that there are differences. One has a faint tinge. another has a deep. rich colour Th~ difference IS due to the fact that a certain amount of light falling on the body IS absorbed and a certain amount IS reflected. Naturally, the relationships of these two amounts can vary over a tremendous range. To take two extreme cases, a shmy metallic surface and a piece of black velvet The metal reflects almost all the light that falls on It, while the velvet absorbs most of the light and hardly reflects any. Magicians make good use of this properly of velvet, for If an object does not reflect much light. It is practically mvtsrble On the stage, a box covered WIth black velvet on a black background goes quue unnoticed, and the magician can go through all krnds of tricks with handkerchiefs. pigeons and even himself appearing and drsappearmg. Physicists also found this property of black bodies very valuable. In the search for a standard body, they decided on the black body. A black body absorbs the most radiation and, hence, IS heated by thrs radiation to a higher temperature than all other bodies. Conversely, when a black body IS heated to a high temperature and becomes a source of light, It radiates more Intensely at the given temperature than any other bodies. Tbts. then, IS a verv convenient radiator for establishing the quantitative "laws of thermal radrauon. However, it was found that black bodies themselves emit radiation In different ways. For example, soot 3\ 30 I J radiation became known as the 'ultraviolet catastrophe'. That was at the end of last century. At that time, nobody could even Imagine that it was not simply a catastrophe for one. rather special, law. It was the collapse of the entire theory that gave birth to the law - the catastrophe of classical physics! Classical Physics at an Impasse There were physicists in those days who did not regard this radiatIOn-theory obstacle In the path of classical physics as significant. But any hindrance IS a grave matter. for everything In the theory ISinterrelated. If some point ISfalse, we cannot rely on the description it gives of other phenomena. If the theory is not able to overcome a httle barner, what hope IS there for big barriers'? Physicists made heroic attempts to surmount the difficulties of radrauon theory. Today, these attempts seem logically mconsistent. Yet what can one expect? When a theory gets Into a hot spot, It is like a cat in a burning house with one way out - into the river. The cat races from corner to corner, but It never thinks to Jump Into the water, for that would be against all the eat's instincts. Sornethmg Similar happens to scientists who are caught 'bur rung' In the house they have worked all their lives. The house which IS so dear to them and to which they are so accustomed. They try to put the fire out, but they can't conceive of running away and leaving it. However. It became clear to the more acute scrennsts that classical phySICShad reached an Impasse. And the theory of thermal radration was not the only blind alley. Those same years saw the ether theory collapse too ]. JS The Ultraviolet Catastrophe Physicists have a penchant for universal laws. As soon as It is discovered that one and the same phenomenon is described, In various aspects, by several laws. an attempt IS Immediately made to combine them Into a single general law encornpassmg all aspects at once. Such an attempt was made. with respect to the laws of thermal radiation. by the English physicists Rayleigh and Jeans. The unified law which they obtained stated that the intensity of radiauon emitted by a hot body is directly proportional to the absolute temperature and Inversely proportional to the square of the wavelength of the emitted light. ThIS law appeared to be in good agreement wrth experimental findings. But It was suddenly discovered that the agreement was good only for the long-wave portion of the vrsible spectrum, the green. yellow and red. The law broke down as the blue, violet and ultraviolet rays were approached. From the Rayleigh-Jeans law It followed that the shorter the wavelength, the greater should he the intensity of thermal radrauon, Experiment failed to contirm thrs, What IS more. a very unpleasant thmg was that as we move to shorter and shorter wavelengths the radiation intensity was supposed to increase without bound! Of course, this doesn't occur. There can never be an unbounded growth In wave Intensity. If a physical law leads to 'unboundedness', It IS doomed. Nature has large things. very large, even urumagmably large things. but there IS nothing WIthout bounds, except the uruverse Itself. This cunous situation that arose in the theory of 34 radiation became known as the 'ultraviolet catastrophe'. That was at the end of last century. At that time, nobody could even Imagine that it was not simply a catastrophe for one. rather special, law. It was the collapse of the entire theory that gave birth to the law - the catastrophe of classical physics! Classical Physics at an Impasse There were physicists in those days who did not regard this radiation-theory obstacle 10 the path of classical physics as significant. But any hindrance IS a grave matter. for everything 10 the theory IS interrelated. If some point IS false. we cannot rely on the description it gives of other phenomena. If the theory is not able to overcome a little barrrer, what hope IS there for big barriers '? Physicists made heroic attempts to surmount the difficulues of radiauon theory. Today, these attempts seem logically mconststem. Yet what can one expect? When a theory gets IOta a hot spot. It is like a cat in a burning house with one way out - IOta the river. The cat races from corner to corner, but It never thinks to Jump into the water, for that would be against all the eat's instincts. Sornethmg Similar happens to scientists who are caught 'burning' 10 the house they have worked all their lives. The house which IS so dear to them and to which they are so accustomed. They try to put the lire out, but they can't conceive of running away and leaving it. However, It became clear to the more acute screnusts that classical phySICShad reached an Impasse. And the theory of thermal radiation was not the only bltnd alley. Those same years saw the ether theory collapse too ]' .15 uninterrupted stream of light. Or take the smooth build-up of speed (and with it, energy) of a locomotive moving downhill, of a falling stone. Imagme for a moment that energy is acquired and given up in httle portions. One calls to mind the jerky movies of years ago. One pictures the candle Oaring up and dying down, the sun shimng in bursts, as it were, a nash of radiant energy, and then a lull until the next flash. The train moving down a slope 10 Jerks, the stone bumping along through the air in its plunge to earth. "Sneer nonsense!" was the answer Planck most likely got from his first suggestion that the energy of radiation (like matter itself) IS atomistic and that It ts released and acquired not continuously but III small portions, quanta, as Planck called them, from the Latin 'quantum' meaning quantity. If he had only known the quality that would eventually grow out of such quantity! For Planck's formula, quanta were vitally important. Without them, ii would have failed miserably and would have gone to the dusty archives of science along with so many others that have found no substantiation. These quanta of energy served as a firm foundation for Planck's formula. But the foundation itself rested on practically nothing since there was no place for It In classical physics. That is exactly what troubled the cautious Planck. It is no easy mailer to give up a lifetime of habit. The Elusive Quanta A quantum of light is an extremely small portion of energy. The most minute particle of dust has thousands of millions of atoms. The radiant energy 40 released by a uny glow-worm contains thousands of milhons of quanta. Now we come to the magnitude of these separate porttons of energy. Planck made the extremely important discovery that such portions differ for different types of radrauon. The shorter the wavelength of hght, that is the higher its frequency (10 other words, the 'more violet' rt is), the larger the poruon of energy. Mathematically, thts is expressed by means of the well-known Planck relation between the frequency and the energy of a quantum: E = hv Here, E IS the energy carried by the quantum: v IS the frequency of the quantum; h IS a proportionality factor which turned out to be the same for all types of energy that we know. It is known as 'Planck's constant' or the 'quantum of action'. The value of this number is Just as great to physics as Its magnitude is small: 6x 10-27 erg per second! It IS this insignificant magnitude of the quantum that makes the hght of a candle or the sun appear to us to burn with a constant glow. To illustrate, let us calculate the number of quanta radiated by a 25-wall electric light bulb per second. Takmg the emitted light to be yellow, we find by Planck's relationship 6 x 1019, which is 60 rmlhon million million portions of energy per second. All of that is radiated by a small 25-wall bulb every second! Quite obviously, the human eye IS not sensitive to such magnitudes of energy. Vet this is not so. The eye is an extremely sensitive instrument, as was convincingly demonstrated by the experiments of the Soviet physicist S. Vavilov. An observer was kept in the dark for a certain time (to increase the sensitivity of the eye) and then an 41 exceptionally weak source of light that yielded just a few quanta per second was switched on. The eye recorded them almost as separate entities! The pomt is not the magnitude of the quanta but the very high rate at which they follow one another. We have already seen that even a small lamp emits millions upon millions of millions every second. Now the human eye, like any other Instrument, operates with a time lag. It is not able to record events that proceed in rapid succession. This inertia-like property of the eye IS what makes moving pictures possible. We see the screen as a continuous sequence of events, although we know that the pictures are actually m the form of separate frames. Energy quanta emitted by sources of light follow one another much more rapidly, and so the human eye sees light as one continuous flow. Vavilov conducted his expenments in the 19305 when Planck's notion of quanta was generally recognized. Planck himself was not able to prove his discovery by direct experiment. The fact that a formula is corroborated by expenment but does not follow from theory always appears at first somewhat dubious. In this case, all the more so since the formula was obtained from reasoning that ran very much against the grain of accepted thought. That was why there was not much enthusiasm in scientific circles when Planck delivered his commumca- non at the Berlin Academy of SCiences. SCIentists are human beings. too, and they require time to digest something so out of the ordinary. Planck himself was fully aware of the boldness of his attack on classical physics and was eager to Justify It. But of course he could never irnagme the tremendous developments that revolutionized the whole of phySICSJust a few years later. 42 The first years of the twentieth century, 1901, 1902, 1903, 1904, went by with hardly any auennon pard to the theory of quanta. The number of scientific papers that appeared could be counted on one's fingers. An Unaccountable Phenomenon But then in 1905, a totally unkown member of the Swiss Patent Office, Albert Emstem, publIshed his theory of the photoelectnc effect 10 metal; 10 the German journal "Physjkalische Rundschau". At the ume that Einstein took up this study, the effect was well on In years. It had been discovered in 1872 by A. Stoletov. professor of Moscow University. Later on It was studied by the German physicists Hertz and Lenard. Stoletov had pumped the air out of a flask, put two metallic plates inside and attached them to the poles of an electnc battery. Naturally, there was no current through the airless space. But when the light of a mercury lamp was made to fall on one of the plates. current Immediately began to flow In the electnc curcuu. When the hght was turned off the current stopped. Stoletov drew the proper conclusion, that current carriers (later found to be electrons) had appeared in the flask and that they ongmated only when the plate was rllummated. It was qutte obvious that these electrons were ejected from the ilium mated metal much like molecules Jump into the arr from the surface of heated liquid. However, the words 'much like' really mean 'quite differently from'; the ejection of electrons from metal was funda- mentally different and, what 1S more, was of an unknown nature. 43 To begin with, hght IS an electromagnetic wave. It is difficult to imagine how a wave can knock electrons out of metal. There IS no collision here of energetic molecules, as a result of which one of them IS ejected from the surface of a liquid. Another Interesting Circumstance was noted. For each metal studied, there appeared to be a certain limiting wavelength of incident light. When the wavelength was exceeded, the electrons m the flask disappeared at once and the current ceased to now no matter how strong the hght was. This was altogether strange. It was clear that electrons are ejected from the metal because the hght m some way conveys energy to them. The bnghter the illurm- nation, the stronger the current. The meta) receives more energy and larger quantities of electrons can be knocked out. But no matter what the wavelength of the light. the metal should be receiving energy all the same. True, with increasing wavelength the energy diminishes and fewer electrons are ejected from the metal, but still there should be some kmd of current. Yet experiment showed no current at all. One would think the electrons ceased to accept the radiant energy. Why were electrons so particular about the energy food they were given? That was something that physicists Just could not grasp Photons Einstein regarded the photoelectric effect from a dilTerent angle. He attempted to picture the actual process of the ejection of an electron from a metal by hght. 44 In normal conditions, there IS no cloud of electrons hovenng over the metal. Which would suggest that the electrons are bound to the metal by some kmd of force. To knock them out of the metal, a little energy IS needed. In Stoletov's experiments this energy was supplied by light waves. But a light wave has a definite wavelength, something of the order of a fraction of a micron, and ItS energy is, as it were, concentrated in the minute volume occupied by an electron. Thrs means that in the photoelTect a ltght wave behaves like a tiny 'particle'. It strikes an electron and dislodges It from the metal. This must obviously be a particle of light: as Newton would say,a corpuscle, because Newton regarded light not as waves but as streams of particles. Then what would the energy be of such a particle? Calcula- tions show that It would be very small. Then why not suppose that It would be exactly equal to the quantum that Planck had conjured up five years before? So Einstein sard that hght is simply a stream of quanta of energy, all the quanta of a smgle wavelength being exactly the same, which is to say that the quanta carry Identical portions of energy. Later, these quanta of light energy were given the name photon. The explanation now was complete. A photon carrying a small poruon of energy stnkes an electron With sufficient force to knock it out of the metal. On the other hand, obviously, If the photon energy is insufficient to disrupt the electron bonds 10 the metal, the electrons will not be knocked out and there Will be no current. According to Planck's formula, the energy of a quantum IS determined by Its frequency, and the greater the wavelength of the light, the lower the frequency. Hence it is quite obvIOUS that the 45 ------_ ......._-------- bound by any forces to neighbouring ones ceases to be particular and responds to all kinds or energy packets. But if the electron should find itself in a metal, it gets moody and demands specific portions of energy again. Why this IS was explained some twenty years later. The Visiting Cards of Atoms Meanwhile, a young Danish physrcist, Niels Bohr, tried to apply the new quantum concepts to the respectable SCience of spectroscopy. By the twentieth century, hundreds or papers had appeared dealing with spectroscopy. Spectral analysis was moving ahead at quite a pace doing great service in chemistry, astronomy, metallurgy and other sciences. Credit for the discovery of spectra goes to the diversified genius of Newton. But spectral analysis made its appearance only a century ago. In 1859, the prominent German chemist Bunsen repeated Newton's old experiment by placing a glass pnsm in the pathway of the sun's rays and decomposing the ltght into a spectrum. In Bunsen's experiment. the role of the sun was played by a burning rag dipped in a salt soluuon. Newton had found that a ray of sunlight is expanded into a band of many colours. Bunsen didn't see any band' at all. When the rag had table salt [sodium chlonde) on It, the spectrum exhibited only a few narrow hnes, nothing else. One of the lines was a bright yellow. Bunsen got another well-known German SCientist, Kirchhoff, Interested In this fact. Both or them cor- rectly concluded that the role of the glass prism con- sisted only in sorting the incident rays of light into their wavelengths. The extended band of the solar spect- rum indicated that all the wavelengths or visible light 48 u were present. The yellow line, which appeared when the light source was a burning rag, indicated that the spectrum of table salt had a smgle specific wavelength. The formula of sodium chlonde IS Na'Cl. To which element (sodium or chlonne) did the yellow line belong? ThIS could be checked very simply. The sodium could be replaced by hydrogen, giving us hydrogen chloride, HCI, which, when dissolved 10 water, yields hydrochlonc acid. The rag was dipped in hydrochloric acid and placed 10 the flame of a Bunsen burner and the spectrum was taken. The yellow line had disappeared without a trace, which meant that It belonged to sodium. This was verified once again. The sodium was retarned, and the chlonne was replaced (caustic soda, NaOH). The Iarruhar line appeared in the spectrum immediately. There was no longer any doubt. No matter what the substance in which sodium appeared, It made its whereabouts known by the bright yellow spectral line, its visiting card. Later, It was found that sodium is no exception in this respect. Every chemical element has its own characteristic spectrum. As a rule, some of the spectra were much more complicated than that of sodium and consisted at times of a very large number of lines. But no matter what the compound or substance the element appeared in, Its spectrum was always distinct, like the photograph of a person. One rmght look for a person 111 a crowd by checking the identification card of each one, hke chemists do when lookmg for elements in rock specimens using chemical methods of analysis. But an easier way is to have his photograph. Which IS precisely how the search is done with the aid of spectral analysts. And the elements are found in places where 'looking over 4-70 49 idenuficauon cards' would be out of the quesuon c- on the sun, In distant stars, in the mferno of blast furnaces and In plasma. All that It needed IS the photographs of all the parttcrpants. Today there are over a hundred chemical elements, and nearly all of them have been classified according to their charactensnc spectra. Why do Bodies Emit Light? The successes of spectral analysis were colossal, but there was a fundamental flaw. The edifice of spectroscopy was erected on the [oundanon of the theory of thermal radiauon and bore all the traces of the baSICshortcoming of this theory The baSICweakness lay In Its answer to the quesuon: Why do bodies begin to emit hght when heated? How ISthis hght emitted? Obviously. by the component parts of the bodies - atoms and molecules. lncreasmg temperatures make the molecules move faster. Mutual colhsions are more violent and more frequent. and the molecules vibrate so fast that they begin to emit light That was the view of the old physics But then why do not bodies lummesce at room temperature, smce the molecules are still In monon? No explanation was then forthcoming. When, In 1898, the English SCIentist Thomson created the first model of the atom, the mystery of luminescence seemed about to be solved. In this model. atoms were clouds of posmvc charge Within which floated negative electrons In quantities sufficient to balance the charge. The electrons were attracted by the positive clouds and retarded in their monon. But according to classical physics, charged particles have to emit electromagnetic radration when they are 50 and Improve the model so that an electron 10 it could revolve about the nucleus and emit light and yet not fall onto the nucleus The year was 1912. Fresh in the memones of all phYSICIStSwas the sensation that Emstem had created with hrs photons. And only three years before, It was Einstein again who completed his theory of relauvrty - another sensation. Naturally, all these attacks on classical phys- ICS could not but stir up the young physicists and add boldness to their mode of thmkmg. Bohr conunued to mull over the problem and at last got an Idea. Why should an electron In an atom emit light connnuously? Because It IS always movmg at an accelerated rate? Let's reject that and say that an electron in an atom need not give ofT light even when m accelerated monon. And how IS this possible? The electron has to move along a specific path about the nucleus. In an orbit. and not Just any way. If the electron does not emit light. It can live In the atom as long as It likes. But there was no way m which classtcal physics could countenance such a situation What IS more, It didn't follow from any other theory. Bohr was not able to prove it And so he modestly called It a postulate. Bohr, mcidentally, was never able to prove It wnmn the framework of lus theory. The proof came some ten years later and was quite unexpected That we'll discuss later on But how many possible orbits are there rn which an electron can move without erruttmg light? Bohr's calculations show that the number IS great, very great. What's the distmgurshmg feature? The mean dIS- tance from the nucleus' there are close orbits and distant orbits. Yet It IS not a question of distance, but of the energy which the electron possesses m ItS orbit. Which IS understandable, because the closer an 54 electron IS to the nucleus. the faster it has to move to keep from falling onto the nucleus The reverse IS true of a more distant electron, which IS not so strongly attracted to the nucleus. and hence can move more slowly. The conclusion, then. IS that the pathways (orbits) of electrons differ as to electron energy. As long as an electron stays m ItS orbit, there IS no ermssion of light. Bohr at this point advanced a second postulate. Let us suppose an electron in orbit suddenly jumps to another orbit of less energy. Where has the excess energy gone? Energy cannot simply vamsh away into nothmg. Seek It outside the atom, says Bohr The energy is ejected from the atom In the form of a quantum, that same quantum of itght energy which Emstem had mtroduccd An electron that has emitted a photon takes up a different orbit and does not errut light any more The photon IS ejected dunng the rmnute fraction of time when It jumps from one orbit to the other. Meanwhile the photon IS making its way through the other atoms and finally gets out of the substance. It can enter our eye, It can be passed through a glass pnsm In a spectroscope and photographed. The energy contained In photons IS transformed many times before we sec Its actual Image as n black line on a photo- graphic plate This hne has a lot to say for Itself By measur- ing us POSition on the plate we can lind the wavelength of the photon and Its frequency. Then we take the Planck relauonshrp between frequency and energy of photons and determine the energy of the photon. Thrs energy comes out as the exact difference 111 energy between the old and new orbits 10 the atom The blackness on the plate at the Site of the spectral lme indicates the number of photons there' the more there 55 7 are, the blacker the line. The more photons, the brighter the body that has emitted them. What a simple and elegant explanation of spectra. All the atoms of a certain substance are exactly alike. Hence, the electrons all exist under the same con- ditions. And so the photons emitted during Jumps bet- ween two orbits are all the same. All transitions that electrons make between two orbits yield, in the final analysis. a smgle unique spectral line. We have already mentioned that there 3fC quite a few such old and new orbits. An electron can reside In anyone of them, in turn. Every Jump from a higher-energy orbit to one of lower energy IS accompamed by the birth of a photon. But since there IS a difference of energy between dif- ferent orbits, the photons WIll have different energies and frequencies. A photographic plate WIll then exhibit a senes of narrow spectral hnes. This IS exactly what the spectrum of gaseous hydrogen looks like. It has several tens of hnes with different wavelengths. Generally speaking. such a SImple spectrum as that of sodium consrstmg of only one line is a rarity. Spectra usually have many tens of lines and frequently even thousands of hnes. The spectral patterns of some che- mical compounds are so intricate that there doesn't seem to be any hope of disentanglmg them But there are laws to go by which make the task easy. Before Bohr's theory, physicists had racked their brains In attempts to decipher some of the complicated spectra. And when Bohr proved that the spectrum is the biography of the atom, more precisely, of the atomic electrons, the job was greatly SImplified. All one had to do was to combine the various electron orbits in an atom until he obtained the observed hnes of the spectrum. 56 w And conversely, by examimng a spectrum, one can draw all manner of conclusions about the conditions under which atomic electrons exist. This is very im- portanl. Actually, Just about all that we know about the electron shells of atoms has been acquired through a painstaking analysis of their spectra. From Where do We Reckon the Energy? Now that Bohr has explained how an atom emits light. let us ask WHY. Why do bodies begin to emit light only at high temperatures and why do they cease to emit light at room temperature? Before answering this question we shall have to digress a bit. The very convincmg picture of the atom which we have just drawn will have to be turned upside down. Not that there is something wrong with it. No! It is sunply the sequence of electron orbits that has to be reversed. We considered the close orbits to be the most ener- gene ones, whence it followed that a photon was emitted when an electron jumped to an outer orbit from the nucleus. Actually, It is just the other way around. Let us try to picture thrs business by digging a hole in the ground. Put a ball at the bottom of the hole and put another one on the ground near the hole. Which of the two balls has the greater energy? A knowledgeable person will immediately say: "The quesuon IS not clear. First, what energy are you talk- ing about, potential or kinetic? Second, from what level do you reckon the potential energy? If the level of the earth IS taken, then the potential energy of the ball on the ground may be taken as zero, then the ball in the hole will have a potential energy less than zero, that IS, negative energy. But If we reckon the 57 thousands of degrees, collisions result in big exchanges of energy, electrons Jump to new orbits, and light is emitted. Energy has been imparted, the electron IS In an outer orbit. Then what happens? The nucleus does not allow the electron to stay m the outer orbit for any length of time. It pulls It back into an Inner orbit, and as the electron Jumps Inwards, a photon is ejected. Our eye perceives the photon and we say that the body glows, or emits hght. The body IS now emitting light. Let us raise the tem- perature and see what happens. The thermal motion of the atoms becomes more energetic, collisions are more frequent and violent. The electron spends only a httle ume 10 Its innermost orbit. The atoms more and more frequently go into a state which physicists call "excited', then return to 'normal' only to leave It again almost immediately. At this point, photons are bemg generated by thou- sands and rrulhons every second. They build up ava- lanche-like as the temperature nses (recall the Stefan- Boltzmann law). But it is not only the number of photons that is Increasing. The lengths of the electron Jumps also increase The first timid Jumps to neighbouring orbits and back again give way to record leaps to distant orbits, far away from the nucleus. Jumping back from such orbits the electrons generate very strong photons. And we know that the higher the energy of a photon, the greater Its frequency and the smaller its wavelength. The emil- ted light becomes brighter and more 'violet' (recall Wien's displacement law). Bohr's theory was thus able to account at one stroke for the basic laws of the theory of thermal radiation and spectroscopy. After this great success, the quantum 60 nature of light and of atonuc processes was obvious. In Just a little while this was recognized by most scientists. The First Setbacks Yet It was still early to speak of a complete victory for Bohr's theory. The next ten years saw a tremendous development of the theory. There was a great expansion In the range of phenomena that it embraced. These included the most subtle processes of emission and absorption of light by atoms, and the detailed structure of atoms and molecules. In 1914, Kossel laid the foundations of quantum chemistry now included in every textbook on the subject. In 1916, Sommerfeld advanced a more exact theory of the ongin of atomic spectra. To this day it helps decipher cornphcated spectra. The new theory was able to account for recently discovered magneuc and electncal properties of atoms and mol- ecules. At the same time, the Bohr theory was encounter- ing more and more difficulties. It was not capable of explaining many new facts, some of which were the ones that gave It birth. The first was In the very spectra that Bohr's theory helped to explain. The trouble was that the explanation was not sufficient. We have already mentioned that spectral lines are characterized not only by wavelength but by brightness too. From Bohr's theory we could find the distance between the rungs of the energy ladder of electron orbits (that is, the wavelengths of the photons generated in electron Jumps from rung to rung, from orbit to orbit). But the theory was helpless as far as accounting for the brightness of the spectral hnes was concerned. 61 It was not clear how one could calculate the number of photons In the spectrum. It was obviously too early to speak of a victory for the Bohr theory over classical physics. Though he at first dispensed with the classics. he later had to revert to them. ThIS was In the form of the so-called cor- respondence principle. In a nutshell It was this Classical physics was able to calculate the brightness of spectra, but could not account for their ongm. Quantum rnechamcs was able to explain the essence of spectra. but could not cal- culate the brightness of the spectral lines. Bohr conclud- ed that both theones had to be used, and that they should be harnessed together In areas where they more or less coincided. But where did this occur'! According to classical physics. an electron in orbit about an atomic nucleus would come closer and closer to rt and finally fall onto It. In the process. it would emit a continuous spectrum with no smgle hnes. According to quantum mechanics. an electron in an atom radiates separate hnes or. as we say. radiates a discrete spectrum. What have the two spectra 10 com- mon? The rungs of the energy ladder of electron orbits have different heights The height IS less, the farther the orbit's from the nucleus The energy ladder in the atom is somewhat like a long ladder looked at end- wise. in perspective. so to speak: the rungs at the far end appear close together. In the case of the ladder, this is simply an optical illusion. while In the atom It IS an actual fact. But the height of the energy level corresponds to the energy of the photon or the wavelength of its spectral hne. Thus, long wavelength hnes of the spectrum, which From Bohr's theory to quantum mechanics A Remarkable Article In \924, the September issue of the English "Philo- sophical Magazine" carried an article by an unknown physicist, Louis de Broglie. The author described the principal points of his dissertation, which was devoted to the possible existence of matter waves. Waves of matter? Weren't they the commonly known sound, light and other such waves, which are qui te material and which are perceived by our sense organs or are recorded by instruments? No, it turns out that de Broglie had in mind quite different waves. The views expressed by de Broglie were so unorthodox and paradoxical that they could easily compete in originality with those put forward by Planck a quarter of a century before concerning quanta of energy. And not only as to their importance to phys- ics, but also in the way they were received by very many physicists: open incredulity. What are these matter waves, anyway? Before going into item, let us take a look at 'ordinary' waves, which had been thoroughly studied hy that time. 5-70 65 A Little about Ordinary Waves Throw a stone into a pond and watch the waves move over the surface of the water. Incidentally, sur- face waves are practically the only type of wave that can be observed directly in motion. It might appear that the water Itself moves with the waves. But this IS not so. Watch any httle boy throw stones behind his toy ship hoping In this way to move It back to the shore. The waves move under the craft, which Just bobs up and down In one place. This means that the water does not move away, but Just up and down. In big waves produced by big stones, there IS a little movement of the water, but never for any great distance. This 'carrying' property of high surface waves IS made use of in riding the surf, a sport common in Australia and the Hawaiian Islands. The sportsman stands on a large board and moves up and down with the big regular waves moving In towards the shore. He gets onto a wave and moves towards the shore at a tremen- dous speed. But the slightest false move and he will find himself In the trough of the wave Instead of on the crest. In this risky, exciting sport, the wave carries the sportsman piloting him towards the shore. Remember the term, pilot wave. We shall return to it later on. Last century, physrcrsrs learned that sound was also a wave motion. Sound waves were found to be pro- pagated In the air, In water, and In solids. What IS It that vibrates In sound waves? The particles of the medium through which the sound IS propagated. Mol- ecules of air. water, the atoms of solids. Take away the air, water. matter generally, and sound waves disappear. There IS no sound in a void. Future 66 astronauts will probably observe grandiose eruptions of volcanoes on distant airless planets all In complete si- lence. Only the ground shaking under their feet will be felt. On the moon, spacecraft wrll start up in absolute silence. There will be no roar of rocket engines as we know It here on earth. The phYSICIStSof last century likewise learned about the nature of electromagnetic waves produced by the movement of elect TIC charges. The light and radrowaves of distant stars and nebu- lae now arrivmg at the earth began their trip thousands and millions of years ago. Their pathways lay mostly through enormous and nearly empty interstellar spaces. On the moon, astronauts in complete silence will watch jets of dazzling fire eject from the bottom of their space rocket. In a vacuum, one can see and not hear. That is the most fundamental difference between electromagnetic waves and mechanical waves, mcluding sound waves. No intermediate medium IS needed for the propaganon of electromagnetic waves. On the contrary, a medium only reduces their speed. Getting Acquainted with Matter Waves Let us return to the matter waves. De Broglie main tamed that these waves are generated In the motion of any body, whether a planet, a stone, a particle of dust or an electron. Like electromagnetic waves, they are capable of propagatIOn 10 an absolute void. Hence, they are not mechanical waves. But they are produced in the motion of all bodies, including those not charged electrically. Hence, they are not electromagnetic waves. At that time phYSICiStSdid not know of any other 5' 67 kinds of waves. So matter waves were indeed some sort of new hitherto unknown waves. Utter nonsense, said the older physicists with a shrug. They were firmly convinced that all possible waves had already been discovered. This young Louis de Broglie speaks of waves of mailer; but are not mechanical and electromagnetic waves, waves of matter? Without matter there are no waves, in fact there is nothing at all! True, de Broglie didn't think up a very good name for his waves. But what could he do? New things get names before scientists have time to understand them properly. That is exactly what happened to de Broglie. Those mailer waves of his proved so intricate that physicists are still arguing about them. We shall have to take a closer look at the de Broglie waves because they are the foundation of present-day quantum mechanics. Why Can't We See de Broglie Waves? That was probably one of the first questions that physicists asked de Broglie. Well, how do we generally perceive waves? Not only by means of our sense organs, which are a rather poor instrument anyway. The human ear perceives sound waves with frequencies between 20 and 16,000 vibrations per second. These frequencies cor- respond 10 sound wavelengths in air of about 17 metres to 2 centimetres. The human eye reacts to light waves of length from 0.4 to 0.8 micron. Those are nature's 'windows' as far as learnmg about waves goes (if, of course, we leave out the surface waves of the sea). Physicists use special instruments to transform waves beyond the human range to lengths that lie within these two 'windows'. This greatly extends our possibilities of 68 pz--------------- Not much better than the earth's de Broglie wave- length. Absolutely hopeless of ever being detected. It IS a million, rnilhon, million times smaller than the atomic nucleus, which itself is far beyond the range of any microscope. Now let us take the electron. It has a mass of about 10-27 gram. If an electron begins to move in an electric field with a potential difference of one volt, It Will acquire a velocity of 6 x 101 centimetres per second. Puttmg these figures into the de Broglie relanon gives us 6.6 X 10-27 10-7 6 X 10' x 10 27 = cm This is something quite different. 10-7 em corresponds approximately to the wavelengths of x-rays. which can be detected. Thus, 10 principle, we should be able to detect a de Broglie electron wave. The Wave is Found But how? The de Broglie wave exists 10 theory and there doesn't seem to be any way of detecting It instrumentally. But a wave IS a wave and there must be some phenomenon In which it Will manifest Itself no matter what Its nature. An attempt was made to catch the de Broglie wave In a diffracnon experiment. the point being that diffracuon is so completely a wave phenomenon. Diffracnon consists In the fact that when a wave encounters some obstacle It passes round It. In doing so, the wave IS slightly deflected from its straight path and moves into the 'shadow' behmd the obstacle. The diffracuon pattern of waves from a round obstacle or a round aperture In a screen opaque to waves is 71 x -ray pal/ern Fi g. t typically a system of. alternate dark and light rings. Such a pattern is seen, for example, when one looks at a street lamp through a dusty glass. On frosty nights, the moon is surrounded by several light and dark rings: the moon light has experienced diffraction on minute ice crystals dancing 10 the air. Diffraction is a definite indication of the existence of waves. It was precisely the discovery of the diffraction of light at the start of the nineteenth century that served as a most convincing argument for the wave theory of light. But the wavelengths of light waves are hundreds and even thousands of times greater than those of the de Broglie waves of electrons. All the devices constructed for producing diffraction of light - slits, screens, dif- fraction gratmgs - were much too crude. The dimensions of the obstacles used to observe diffraction of a wave must be comparable with or less than the wavelength. 72 ------------'- Electron diffraction pattern Fig.2 What is possible with light waves, is utterly out of the question when dealing with the de Broglie waves. By 1924, it was known what objects to use in at- tempts to detect the diffraction of the de Broglie eleclron waves. Twelve years before, the German scientist Laue had noticed the diffraction of X-rays on crystals. Laue noticed a series of dark and light dots on a photo- graphic plate exposed to X-rays that had passed through a crystal. Several years later, Debye and Scherer repeated Laue's experiment on small-crystal samples of powders, and obtained diffraction rings. In these cases, diffraction was possible because the distances between the atoms in the crystals (like slits in a 'screen' opaque to X-rays) were of the same order of magnitude as the wavelength of Xvrays: 10-8 centimetre. But the lengths of the de Broghe waves lie precisely within this range! Which means that if these waves do really exist, then electrons, in passing through a crystal, 73 at an obstacle would, as a result of diffraction, move round and get behind it. No, waves and particles were two mutually exclusive entities. A thing was either a wave or a particle! And yet the de Broglie waves existed. It was not 'either or' but 'both'. Something had to be done to connect the unconnectable. And not for the single specific case of a diffracting electron. If an electron has wave properties, then so inevitably do all the objects of our world, from the smallest to the biggest. De Broglie suggested beginning this unusual synthesis with the concept of a pilot wave. Pilot Waves Let's go back to riding the surf. The rider gets on the crest of a high wave that carries him to the shore. The wave acts as a pilot. De Broglie's idea is that matter waves pilot moving particles of matter in a similar fashion. A particle, as it were, sits on a wave and moves wherever the matter wave carries it. The length of this wave, de Broglie says, may be very great. At small velocities of motion of an electron, the length of the electron wave is many thousands of times greater than the electron. As the velocity increases, the particle, as it were, pulls the wave into itself, and the wave becomes shorter. But even at high velocities of motion the length of an electron wave is still greater than the 'dimensions' of the electron itself. It doesn't exactly matter who leads whom, the electron the wave or the wave the electron. The im- portant thing is that the wave is connected with the electron intimately and for all time. The electron wave disappears only when the electron stops. At this 76 instant the denominator in the de Broglie relationship becomes zero and the wavelength, infinity. In other words, the crest and trough of the wave move so far apart that the electron wave ceases to be a wave. The de Broglie picture rs quite vivid: an electron riding its own wave. But where did the wave come from? It exists with the particle even when the latter is in motion in an absolute void. Which means that the wave is generated only by the particle itself. And how does that occur? De Broglie's hypothesis has nothing to say on that score. Well, maybe the hypothesis can explain what interaction there is between a particle and its wave, how the wave moves together with the particle, how it shares the fate of the particle in the latter's interactions with other particles and fields, for example, when particles are incident on an obstacle or on a pho- tographic plate. No, the hypothesis does not olfer any convincing explanation. In the search for a way out, de Broglie tried to ehminate the particle altogether. Why not imagine the wave itself to be the particle? In other words, picture the particle as a compact formation of its waves, a wave packet, as it was called by physicists. A pack- et was to consist of a small number of rather short waves; when two or more packets collide they ought to behave like particles - exactly like a short- wave photon when it ejects an electron from a metaL But no matter how compact the packet, no matter how much it resembles a particle, it consists of waves. This surely means that there must be phe- nomena in which it will exhibit its primordial wave nature. But nature rejected this proposal as well. It turned out that no matter how compact the wave packets 77 are, they cannot form a particle. Thrs IS funda- mentally impossible. The point is that these packets rapidly dismtegrate 10 time, even in a total vacuum In negligible intervals of ume, a packet becomes so smeared out 10 space that the formerly compact particle ISdiluted to homeopathic proportions. Yet we know that particles are definitely stable, there is not a trace of any kmd of spreading out In time. ThIS model too had to be given up The mecha- meal combmmg of two such mutually exclusive entities as waves and particles IOta a stogie Image was not a success, And It couldn't be But that came later De Broglie, however, did not want to give up hIS 'centaur' with the head of a particle and the body of a wave. Two years passed. In the summer of 1927, phYSICISts from all over the world arnved 10 Brussels at the Solvay Congress at which de Broglie's representauon on the relauonship between waves and particles was resoundingly rejected. For many years to come, a com- pletely different representation of this relauonship led the way It was presented at the congress by two young German physicists, Werner Heisenberg and Erwm Schrodmger Together or Separately? Heisenberg and Schrodrnger buned the de Broglie concepnons. but spoke to doing so that this deter- rruned the whole subsequent development of quantum mechanics. The prmcipal idea of de Broglie concerning waves associated with the motion of bodies was quickly taken up by scientists 10 a number of countries. Hardly a year passed after de Broglie's first paper 78 would appear that the electrons rmpmged on the plate utterly at random. But there IS one thing that attracts our attention. We measure the aperture In the diaphragm from which the electrons emerged and project the outline onto the target. It would seem that all the electrons should fit inside this outline, no matter how randomly they had fallen on the photographic plate. Actually, however, many of the hits are far outside the boundary hne. And here IS another interesting thmg. If we examme the target carefully. It will be noticed that the electrons do not strike the plate In random fashion at all. Even when the number of hits on the target IS small, there are blank places with not a single hit and there are closely bunched groups of hits. If a line is drawn through these places, little rings appear. True, they are not well deli ned, but they Improve as the number of electrons stnkrng the plate Increases. Let's playa trick. Take an ordinary nfle target and punch holes where the electrons hit the photographic plate. Then show the target to a real marksman and see what his reaction IS. "What a funny way to shoot. Look at all those hits In number 10, and not a single one in 9 or 8. Was that done on purpose? All 10 10, 7, 4 and I?" We don't say anything, and after a short while the chief marksman says, "Nonsense! No one could ever shoot up a target that way. no matter how he tried. And here's why. If the man rs a beginner, hIShits will he at random, more or less evenly distributed over the whole target. The target of an experienced marks- man looks quite different: a lot grouped around the bull's eye and Just a few In the outer nngs. Let's count the total number of hits In each nng of the target and construct a graph. 6-70 81 Fig. 3 "On one axis we layoff the numbers of the nngs (or the distance from the bull's eye, which IS the same thrng), on the other, the number of hits between two nngs. We get a smooth curve downwards as we move away to the edges of the target. "And now take your target. The graph m this case Will oscillate up and down from the centre to the Sides. The way It descends differs from our curve. "In the case of our expenenced marksmen the laws of chance hold true. And the curve we gel IS what IS known as the curve of random errors, or Gaussian curve. There is something random 10 your case too. But it obeys a different law, quite new to shooting ranges." Now let's return to our target. 82 Waves of Probability True enough, the wave-like curve is never encountered in shootmg. Electrons are not bullets. A bullet has too big a mass for it to exhibit wave properties. It was this distribution curve of electron tracks on a photographic plate after their reflection from the crystal that Born proposed calling the de Broglie wave. Wait a minute! What connection is there between the 'paper' wave and the real wave? The real wave moves with the electron, while ours remains on paper: However, they are related. The graph of elec- tron hits on the photographic plate is not a figment of the imagmation. It reflects the existence of a real wave associated with a moving electron. But the meaning of this wave is quite different from that which de Broglie gave It. Classical Newtonian physics says very definitely that the electron emerging from the aperture of the diaphragm should move m a straight hne until It hits the crystal. Then the electron IS reflected from an atom of the crystal just hke a billiard ball bounces off the side of a table. Finally, the electron moves from the crystal to the photographic plate and leaves a track on It. There is no human berng here with shaking hand and tired eye. There is no wind or streams of heated air coming up from the earth that could affect the aiming process. These are ideal conditions, and hence the accuracy should be Ideal - all into the bull's eye. In other words, the electrons should reproduce on the photographic plate an exact outline of the aperture in the diaphragm. If the opening IS a tiny hole, the photograph should reproduce a small dot and nothing more. 6' 83 It stated: "It is hopeless to think that all the mole- cules of a gas have the same velocity at every instant of time. On the contrary, they have different velo- cites and, what is more, the velocrues are constantly undergoing change due to collisions. However, despite the random nature of these changes in velocity, there exists, at every instant of time, some mean, stable molecular velocity under the given conditions. What is random as concerns one molecule becomes a regu- lanty when applied to a large number of molecules. Such IS the probabihty law of large numbers. And the number of molecules 10 our volumes of gas is indeed large, m fact so large that the law can be apphed WIthout the slightest hesrtation or doubt." Physicists began to calculate the behaviour of large assembhes of molecules statistically. according to the laws of probabihty theory. But in one respect they did not want to agree with the theory of proba- bility. They maintained that there was no random- ness In molecular monon, that every collision, every individual motion of a molecule could be descnbed by Newtonian laws and that If one desired to solve millions of millions of equations, he could express these motions with absolute precision and without any kind of mean values. We don't do that, of course; but m pnnciple it could be done! We describe the motion of a gas by means of probability laws, by the laws of stausucs, but underlying them all are the exact laws of Newtonian mechanics. Classical physics was Just a little too sure of Itself; there were simply no grounds for generalizing Newton's laws to the monon of individual molecules. The sub- sequent development of phySICS proved this. Mole- cules are not billiard balls. They move and collide and m doing so obey quite different laws. 86 Cautious Predictions These were new laws, laws obeyed by electrons, atoms, and molecules. The first to 'rebel' were the electrons. They did not want to fit into the framework of behaviour of classical phySICS. Instead of huung the photographic plates where they should. they "used their own free will and did what they wanted to!" shouted some SCientists shocked by the dis- obedience of electrons. Physicists weak 10 philosophy were easily led astray. Since the electron had a 'wnl of Its own', there were no laws that it obeyed. a real anarchist. And If that's the case, why do we need SCience, which seeks laws, if there are no laws? God. they reasoned, had made the electron (and hence all thmgs in the world) free to behave as It wished, exempt from all laws except one - the divme law of its existence. But science does not investigate this law, 11 grasps It by sheer faith. QUIte a simple matter, you see - from the 'free wtll" of the electron to out-and-out idealism. Matenahsts countered by saying that the new laws hold where the laws of classical physics fall ThIS was predicted by Lerun. Twenty years before the time we are now describing he said that no matter what unusual electron properties might be discovered. they would mean only one thmg - a deeper and more correct understanding of the surrounding world. Electrons refused to follow the laws of classical phYSICS.but they obeyed the laws of the new. quantum, mecharucs. What kmd of laws were they? First of all. they were probability laws. What did the light nngs on photographic plates (negatives) m the expenrnent with the diffraction of electrons sigrnfy? Simply that electrons 87 did not strike these places on the plate. Obviously, the electrons did not act of their own free WIll, but were restricted III their behaviour. Then we have the dark rings where most of the electron hits are made. But not all of the electrons Impinge here. There are certain greyish places between the darkest and the hghtest sections. A 'mean' number of electrons impinge on these portions. We see this very clearly on the distnbutron curve of hits In our shooung game. We now come to the most important thmg, An electron leaves Its source, passes through the diaphragm, IS reflected from the crystal and IS moving towards the photographic plate. Where will it hrt the plate'! Classical physics calculates the angles, distances and velocities WIth great accuracy and says "Here" Which IS usually not where It hits at all Quantum mecharncs says. "I don't know exactly, but the greatest probabiluy is that It WIll hit the dark nngs, there is less probability that It WIll hit the grey sections, and It IShardly at all hkely that It WIll rrnpmge on the hght rmgs." Kind of overcautious. And It sounds strange for a science that wants to be called 'exact' It doesn't even sound like science. How much mcer the 'abso- lutely exact' predicnons of classical physics. Yet, If one begins to think about It, what arrogance In such predicuons, what braggadocio and what Igno- rance too I Indeed. what else can we say about a SCIence which has Just begun to Investigate an infinitely complex world and which hardly knows anything about the events taking place In It, and at the same time makes such categoncal statements. 88 But it IS not always possible to detect them. The wavelengths of the de Broglie waves rapidly fall olf with Increasing mass and velocity of the particles. and lie beyond the sensitivity of our Instruments. Then only the corpuscular properties of particles remain. Recall our discussion of wave properties. Up to a cer- lam point. waves (for instance, electromagnetic waves) do not exhibit any corpuscular properties and behave as waves should: they experience interference, diffrac- tion, and so forth. But as soon as their wavelengths become small enough, they begin to act like particles and are able to knock electrons out of metal. The best example IS gamma rays, the shortest of all known electromagneuc waves. With what ease they dislodge particles of matter, exhibiting true corpuscular properties. De Broglie's discovery united the world of physical phenomena into a coherent whole, bridging the gap between two such Opposites and, what would appear to be mutually exclusive entities, as particles and waves. However, though the umty was discovered, there are no grounds for thinking that the opposites have disappeared. They lie, as it were, deep within thmgs and deter- mine the bizarre physiognomy or the rrucroworld, which IS the world we shall be talking about very much from now on. We shall learn about marvellous things that are possible in the world of the ultra- small and are very neatly described by 'waves of pro- bability'. On the Way to the Wave Law These waves describe the monon of electrons and other particles or the microworld. Now how IS 91 one to understand the word 'describe"? A thing or phenomenon may be described both qualitauvely and quanutauvely. In ordinary hfe we usually do the former When we hear "There'll be rain today", we pick up the umbrella and we don't ordmanly ask at what altitude the clouds WIll be. But SCience, especially such an exact science as physics, is rarely satisfied with such a qualitative descnption. FIgures are needed, and exact ones too. So far we have descnbed our diffraction pattern produced by electrons on a photographic plate mainly in a quahtauve fashion, as alternate dark and light nngs. We can also descnbe it quantitatively, by meas- unng the degree of darkness at different places on the plate and then plotting a curve, Just like the one we made at the shooting range. Now It would seem that we could produce a theory about this phenomenon and rest at ease. But It happens that there are other things that require an explanation. SCience can't construct theones about each one separately In fact, therein lies the very strength of modern science: It builds theones that embrace hundreds of Interlinked phenomena. The best and most powerful theones are those that are the broadest and most ernbracmg, In phySICSthe construction of new and large theo- nes often begins With the search for a single Important formula It is called the law of mollon. A farmhar case IS Newton's second law. which connects the acceleration of a body WIth the magnitude and direc- non of the force acting on the body. But we don't actually see the force and accelerations. all we observe is the translation of bodies in space and time under the acuon of forces. It IS this motion that Newton's 92 ) law permits us to find. Acceleration IS change of velocity of motion In time. And velocity IS change of pOSItIOn of a body in lime. So finally Newton's law relates a force to the actual translation of a body. And so when solving Newton's equation we find the type of monon of the body. It is expressed as a certain curve descnbed by the body in a certain time. ThIS curve IS called the trajectory. There IS another very general and broad law m phySICS that descnbes not the motion of bodies but the propagation of waves. Mathematically, It is written III the form of the so-called wave equation, or d'Alembert's equation, after the noted French mathema- tician of the 18th century who discovered It. Neither Newton's nor d'Alernbert's equation IS de- nved from any more general laws. They were not just thought up out of the blue, but were distilled, as It were, III a theoretical generahzauon of numerous expenments and observations made by the predecessors of Newton and d'Alembert. A genius IS not one who simply contnves things out of hrs head, rather IS he one who perceives some hidden force. some Important law In an intricate maze of events, one who shakes that law free from Its encumbrances, from accidentals and insignificant details, and polishes It clean as a compact formulation or (as In the exact sciences) a formula The new law IS now hke a Jewel of knowledge with elegant lines and bnlltant facets. What law was to serve as the cornerstone of the edifice of quantum mechanics? Naturally, for the new law of quantum mechanics to take the place of the laws of Newton and d'Alembert In classrcal phySICS It had to be at least as general and broad. What is more, this new law had to descnbe, all by Itself, 93 The point IS: In what way? The illumination depends on the dimensions of the object. The first condition for a clear Image is that the wavelength of light be less than the dimensions of the object. The ordinary light microscope operates with wavelengths of from 0.4 to 8.0 micron, and therefore produces clear-cut images of objects at least about two to three microns in size. But if we now take something, say of half-micron size, the image will be blurred. When the dimensions of the objects are of the same order of magnitude as the wavelength of light, we have a strong diffraction of the light. Instead of a clear Image we get a diffraction pattern, which consists of alternate dark and light bands that reproduce the outlines of the object. Now take a still smaller object. The light goes past It as If the object didn't exist at all. The electron is not a dust particle nor a bacterium; Its size (later on we shall see that the term size is hardly appropnate) IS roughly a thousand million times smaller than the length of light waves. So how do we Illuminate It? Luckily, there are gamma rays with extremely small wavelengths. So we take an electron for observation and light It up with a gamma ray, and we sec nothing. Just nothmg at all - there was an electron and now there isn't any. There aren't even any ditTraction rings. No matter how we try to produce an Image of an electron, we'll never be able to do it. The point is that an electron is not a dust particle and the gamma quantum IS not a photon of light. The minute gram of dust has weight, and a photon carries some energy and, hence, some momentum. Where does the photon get its momentum? We know that a photon can behave like a particle. This was already demonstrated by Einstein in hrs theory of the 96 photoelectric effect. Judge for yourself: in empty space a photon always has the same velocity, that of light, but its wavelength can be different. We apply to the photon the de Broglie relation: A=~ mv where the velocity v is made equal to the velocity of light c. Then we can find the mass of the photon (this is naturally the mass of a moving photon; the rest mass of a photon is strictly equal to zero): h m= Ie Now the momentum of a photon is the product of its mass by Its velocity: h p=mc=1: Now just a little more mathematics. From this formula it is readily seen that as the wavelength of the photon decreases, the momentum Increases rapidly. When a light photon hits a dust particle, it imparts to the latter its momentum and bounces ofT into the optical system of the microscope and into your eye. Our particle of dust doesn't even budge. If it was at rest, it remains that way; if it was moving, it hardly at all changes its direction of motion. The electron is something quite different. Its mass is out of all comparison with that of a particle of dust; and its momentum is small even for a very fast electron. And so we fire a gamma photon at it with a momentum almost a thousand million times greater than a photon of light. When a gamma photon collides with an electron, forget about images or diffraction rings. The electron is knocked right out of the picture, you might say. 7-70 97 We're not getting very far, and to make things worse we have to deal wrth velocity. Take an electron in flight; it is moving in some direction but we can't say with what velocity. We then Illuminate it with a ?~mlna photon and the electron changes its speed. Or, say the electron has zero velocity. It is at rest some place. But we can't locate it, because Just as we Illuminate It the electron IS knocked off in some direction. It was so nice with the old microscope. You have a dust particle or a bacterium, say, and you know all the time where it is and how fast it's moving. But try to locate an electron. We don't know its velocity, but if we try to determine It, we lose the particle altogether. Such are the tricks of the rmcroworld. The Uncertainty Relation What we have Just descnbed IS very close to actual fact A little calculation with our dust particle and electron WIll show this to be true. Take a little piece of dust one micron (10-4 cm) across consisung, say, of a substance Witha density of to grams per cubic centimetre (just a httle greater than the density of iron) and let It be moving m tbe field of a microscope With a small velocity of one micron per second. Then it WIll have a weight of 10-11 gram and a momentum of 10-15 gram per centimetre per second. We throw hght onto It having a wavelength of, say, half a micron (in the visible light spectrum, this IS green), its photons have momenta of only 10-22, which IS tens of rmlhons of times less than the momentum of our dust particle. It is clear as day that the photon Impacts on the dust particle WIll not produce any etTect whatsoever. Now take the electron. Even If its velocity is close to that of light - 1010 cmls - It WIll have a momentum 98 a medium-fast electron moving between a potential difference of one volt is of the order of 10' centimetres per second. The right-hand side of the uncertainty relation comes out to about 10. There are different ways of constructing this value out of the quantities t! x and t! Vx. For instance, suppose we want to measure the velocity of an electron with the same accuracy as we did for the dust particle, or 10- 3 Then our uncertainties will be: 6. Vx = 104 cmls (10': 10' = 10-'), and ts x = 10-' ern. The uncertainty in the position of the electron will be thousands of millions of times (!) the size of the electron. Let us try to get the accuracy in velocity measurement up to 100 per cent, which is the actual velocity. As physicists say, this would give the order of the quantity being measured. Then !'J. Vx = 10' and !'J. x = = 10-6 centimetre, which is still millions of times the size of an electron. No, there can be no compromise; the world of the ultrasmall doesn't want it. What is to Blame, the Instrument or the Electron? Classical physics never had to deal with dilemmas like this. It always considered that the position and velocity of any particle at any instant of time could be measured with absolute accuracy (at least in principle). This lies at the heart of the predictions of particle movements on the basis of their positions and velocities at some imtial point of time. Now we find that even in principle there can be no talk of absolute accuracy of measurements. Where's the trouble? Maybe in the instrument? True. no instrument is capable of measuring a quantity 101 with absolute accuracy. We rmght say that the history of the development of rneasunng techniques has been a history of constantly increasmg Instrumental accuracy. Precisron measurements In many fields of science and technology have today reached a fantastically high degree, and they are getting better all the time. But it would seem that the uncertainly relation puts a hrmt, an upper limit, on the accuracy of instruments. In this situation, Heisenberg (and other physicists after him) sard that the trouble was In the Instrument. The Instrument for the rrucroworld differs from the telescope used to study the universe. Both Instruments are needed, of course. Our sense organs, which we use to study the world about us, have their hmttations. In fact, that is what an instrument IS for: to translate the phenomena withm Its range Into 'human' terms of feeling. But whereas the telescope does not In any way alTect the motions of the astronomical bodies it observes, the microworld IS quite a different proposition. There, OUf Instrument (let us say our Ideal supermicroscope) inter- feres directly With the phenomenon under observation and alters its natural course. Moreover, It changes It to such an extent that we have no way of separating out the phenomenon m pure form. That IS what the uncertamty relanon does, It puts an upper lrrmt on the 'purity' of an observation. Other physicrsts sard: "The troubles he With the electron:' And their argument was rather convincing. The world of the ultrasmall hves according to Its own laws and, generally speaking, does not require measurements for Its existence. When we say that an electron has wave properties, what does that mean? Well, take the oscillauon frequency of a pendulum: to say that the frequency at a given Instant IS such and such IS nonsense. To determme the frequency, one 102 has to observe the oscillations of the pendulum for some time. Sirmlarly, one cannot say that the wavelength at a given pomt IS such and such The very mearung of wavelength IS that It IS a charactensuc of a long (strictly speaking, an mfimtely long) senes of waves. No matter what the nature of these waves, then length cannot depend on the position of anyone point In the wave. Let us take the de Broglie relation and wrtte It so that we have the velocity of the particle on the left: h v = n1'A We Immediately conclude that since the wavelength l- IS Independent of the posiuon of any point 10 the wave (for example, the POint at which we belteve the particle to be), ItS velocity cannot be dependent on the position of the particle. The failure of the instrument 1S due precisely to the wave properties of the electron. So who IS right? Those who accuse the Instrument of not being able to adjust itself to the rrucroworld, or those who blame the rmcroworld as being inaccessible to measurements? It appears that both are to blame, but only half and half. The truth of the matter IS that the Heisenberg relation exhibits the 'guilt' of both Instrument and electron. But that IS not all An Attempt with Rather Faulty Tools What do we require of an instrument? FIT',t of all, that it should provide us with the mforrnauon that we wish to know. An instrument, of course, has no independence whatsoever, it only obeys the human wrll, The instrument that we wish to use to Investigate the rrucroworld has two aspects. or two ends: an 103 Another Marvel Kids will Jump over fences to get Into a cherry orchard. And so the owner builds a higher fence. Now what does Johnny do? He takes a running Jump, or he gets a ladder, or climbs a tree and Jumps over, or ... well, there are a lot of ways, really. Boys today don't believe 10 fairy tales any more, but If they dealt wuh the world of the ultrasmall they certainly would have to imagine parucles getting through 'solid' walls! Let's take a closer look at this business of chmbing or jumping over fences. From school we know that the lower a body IS, the less Its potential energy. Standing on the ground, you have less potential energy than when sitting on a fence. And we know how much less: this quantity IS given by the product of the weight of our body by the difference 10 height of the centre of gravity of the body 10 the two positions: the difference IS roughly equal to the height of the fence rrunus one metre. If you find the energy some place, you can get over the fence. ThIS can be done by using your own muscles or those of your companions to boost you up. In either case, the work done goes to Increase your potential energy and you can get over the fence. The rest IS easy. Jumping down doesn't require any elTort. On the contrary, It takes some elTort to soften the descent against the force of gravity Now the potential energy on the other SIde of the fence has dimmished to what It was before the Jump upwards. If we plot our potential energy as we go over the fence, we get a nse In physics this IS called a potennal barrier In the atomic world there are fences of this nature. For example, a metal contains a multItude or almost free electrons that are relatively feebly bound to their 106 I'~-""\ I .... \ I .~ \ I t: \ .Sl '" , " I_~ , 0() I ~ Q,) \ ..... t::::: I '""''' I·" \I.:J... ,...... \ Patentwl,' ~ \ well ,/ ~ ' ....-------_... -------- Fig.4 atoms. But, despite their freedom, no one ever heard of electrons leaving the metal of their own free will The point IS that the electrons are not completely free. Though the bond IS weak, they are still attracted to the IOnS that thus appear [thrs will be discussed in more detail 10 the next chapter). The overall action of all the IOns on all the electrons In a piece of metal may be pictured 10 the form of a yard cut 01T from the outside world by a high wall with electrons runnmg about ins Ide the yard. Electrons 10 a piece of metal resemble the balls In the hole that we discussed In connection with Bohr's theory. Inside the metal, the electrons move with ease, but they can no more get out than our balls could leave the hole they were In. For this reason, the conditions under which the electrons lind themselves In a metal are termed a potential well. Stili, such electrons are not really 'chained' to the metal for all time. Under certain conditions they can Jump over the fence and get outside, as It were. For Instance, this occurs when the metal is rllurrunated with light of sufficiently short wavelength. An energetic photon 107 Fi g. 5 can knock an electron right over the potential barrier and make the electron really free. This is the conventional, classical way of getting over the potential barrier and does not really differ from kids climbing over a fence. You've probably already noticed that in the case of electrons in a metal the barrier 15 not exactly like a fence: it has a front part but no rear part. It is more like a step than a fence. For the ball in the hole, we can make a fence by digging up the ground around its edges. In the case of electrons in a metal, this can be done by applying to the piece of metal a strong electric field. Now both barriers - that of the ball 10 the hole and the electron in metal - are very much alike. However, their similarity ends here. H we solve the Newton equation for the ball in the hole, it will show that the ball will remain there for ever if sufficient energy to overcome the barrier is not imparted to it. We know that without any equations. Balls don't just climb out of holes, neither do boys get over fences without jumping. 108 Fig. 7 Here's what happened. The electron approached the boundary of the potential barrier. To catch the electron when it IS seeping through the barner, It is not even necessary to fix Its location; make sure that It IS located somewhere within the limits of the barrier. But that IS not all The instrument must find out the velocity of the electron at that instant in order to deterrnme whether Its kinetic energy mdeed becomes negative. Here the instrument is helpless. Only the Heisenberg uncertainty relanon can save the situation. Now in order to fix the location of an electron within the barner, the electron must be illummated by photons of short wavelength, because It is required to determine the posiuon of the electron with an accuracy not less than the width of the barrier Itself and this width IS small in the world of atoms. But the impact of a photon on an electron will introduce mto Its velocity an appreciable uncertainty. which will be such that the uncertainty It causes In the kmetic energy of the electron will be just enough to get over the highest po tnt of the barrrer. III In other words, there is no way of detecting a particle in a nonclassical passage under the barrier. In the very process of 'detecting' it, an energy is imparted to the particle sufficient for the latter to jump over the barrier in a perfectly lawful and classical manner. Almost like a policeman helping a criminal cover up evidence. The foregoing is typical of many things that occiJr 10 the world of the ultras mall. From the viewpoint of classical physics, quantum mechanics can assert the most fantastic things. And it is fundamentally impossible to prove the falsity of these assertions by the use of classical Instruments. Don't look for the particle under the barrier. you won't find it. The very concept of a particle inside a potential barner is Just as nonsensical in quantum mechanics as in classical physics. Yet the particle gets through the barrier! The clue to this mystery lies, in the final analysis, in the wave properties of the electron and of the other particles of the microworld. Matter Waves Again As we have already seen, these wave properties result in the particle velocities ceasing to depend on the positions of the particles. There are no trajectories in the world of the ultrasmall. But the position of a particle affects its potential energy, and the velocity, Its kinetic energy. And so, strictly speaking, it is impossible at one and the same time to measure accurately both the kinetic and potential energies of a particle. They are independent of each other at any given instant. Within the limits of applicability of these classical concepts, energies in the atomic world are given by the uncertainty relation. All this means that a particle in a potential well has 112 a certain probability of getting outside the well, all by itself. Which means also that there is a probability that the particle will remain in the well. If, say, we have a thousand electrons and ten of them get through the barrier, then the probability of the tunnel elTect is 1 per cent, and the probability of no tunnel effect is 99 per cent. These probabilities are called, respectively, the penetra- bility and the reflecting power of the potential barrier. Penetrability, or transparency, and reflecting power are familiar terms. They describe substances with respect to the passage of light waves. At the interface of two dilTerent substances, light always partially passes into the second medium, and is partly reflected. And isn't the potential barrier a boundary between two media? Simply not for electromagnetic (including light) waves, but for the de Broglie waves. This IS a very profound analogy. The laws of the tunnel effect coincide remarkably with the laws of reflec- non and transmission of light waves between the boun- daries of different substances. The fact that we chose a fence, i. e., something of a definite width, for our barrier is not accidental. If this barrier has only a front wall, like the step of a staircase, the tunnel effect disappears completely. Particles cannot construct tunnels in infinitely long (though very low) barriers. Here the prohibition of classical physics is complete. Indeed, OUf measuring instrument would be able to celebrate something like a victory: it would be possible to establish the location of a particle under the barrier (if it got there) with complete assurance, no mailer how great the uncertainty in the measurement of its position. Which means that the uncertainty relation would yield the exact speed and, hence, kinetic energy of the particle. This energy would then definitely be negative. 8-70 t 13 In the world of the ultrasmall, the elements of these conditions consist of nuclei, atoms, molecules, crystals, and many other things. We know that they all have a remarkahly stable structure. The stationary Schrod- inger equation was firstapplied preciselyto such elements. Most interesting results were obtained. We shall discuss them in the next chapter. Waves and Quanta are United Stationary problems in Quantum mechanics have yet another remarkable property. To understand it, we recall that the uncertainty relation embraces not only, say, the position and velocity of a particle, but also its total energy and the time. In the latter case, the Heisenberg relation states that the longer a measurement is made. the more accurate will be the resulting energy of the particle. The form of this relation is very similar to that given earlier: t.Ex t.t:;>h (again, in place of h it is more correct to write h/2n). Here, I'.E is the uncertainty in the energy E of the particle, and A r is the uncertainty as to the instant of time t at which the particle had the exact energy E. The sign:;> means that the product of these uncertainties cannot be less than h, Planck's constant. Now stationary means that the energy of a particle does not vary with time. In principle therefore we could measure for ever. Here, the indeterminacy of the time of measurement does not play any part. So we calmly put I'. t ~ 00. But then, by the rules of mathematics, h h t.E=-=-=O I'. t 00 116 Fig,8 which means that the uncertainty in measuring energy is equal to zero. In other words, under stationary conditions, the energy of a particle is determined with absolute exactitude. This is the remarkable circumstance that we just mentioned. In the Schrodinger equation, the magnitude of this energy is a very active participant. As long as E is positive (and this, as we recall, corresponds to the free motion of a particle), Schrodinger's equation has a nonvanishing solution for all values of E. And this means that the square of the solution (the probability) is likewise nonzero for all values of E. Translated into ordinary language, this means that a free particle has the right to have any energy and any velocity of motion (which, naturally, can never exceed that of light) and be located at any place in space. Now when E becomes negative (this, as we recall agam, corresponds to the bound state of a particle; 117 for instance, the ball in the hole, an electron in an atom), the solution of the equation changes radically. It appears that it does not vanish only for certain specific values of the energy E. These values of E are called allowed energy levels of the particle. Take a look at the figure. The prob- ability of the existence of a particle is nearly everywhere close to zero, with the exception of states In which It has allowed energy. Here, the probability is noticeably different from zero. Physicists have termed this situation the discreteness of energy levels. Now take a closer look. Doesn't this picture in some way resemble the allowed energy levels of the Bohr model of the atom? It certainly does. What IS more, it is the selfsame thing. The electron orbits of Bohr are the very same energy states In which the probability of an electron being there is substantially different from zero! Bohr simply conjectured these orbits, but he was not able to prove why they should exist. It is quantum mechanics that slipped the foundation under lus hypothesis. Quantum mechanics also substantiates Bohr's second postulate concerning the quantum nature of electron jumps 10 atoms. As can be seen from the Schrodmger equation, an electron 10 an atom can exist only 10 states of allowed energy. Which means that when transitions are made from one state to another, the energy does not change at random but in very specific quantines. It is Simply equal to the energy difference between the states of a jump or transi- tion. This energy difference ISprecisely the Planck quantum that initiated the new physics! Quantum mechanics united two bnlhant hypotheses - that of Planck on 118
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