James J. Morwick Oakwood Collegiate Institute 991 St. Clair Ave. W. Toronto, Ontario Canada, M6E 1A3
Most teachers of chemistry remember the Schrndinger wave equation as a terrifying academic weapon used by universities to nrod undergraduates toward their final exams. I t is a " mathematical monstrosity, bristling with Greek letters having sll the allurement of barbed wire. If the murky interior of the wave equation is dissected, i t is found to have a "solution," represented by the Greek letter psi ($). Unfortunately this $ term, although mercifully shorter than the wave equation, is not a number, but a function, itself ahoundina with sinusoidal ind exponential terms. It seems that the $-iunction is just 3s unuield\.:~~ rheequation which fdiered ir. Oh\,io~isly, $ (also known "nder a variety of aliases such as "wavefunction," 'eigenfunction," or "orbital") is far too complex to be considered in high school. In fact, what teacher has not, a t one time or another, written the Schrodinger wave equation on the blackboard in order to intimidate students from askine ".notentially embarrassing questions about it? This tactic gives both the teacher and the student the opportunity to beat an :xpeditious, but dignified, retreat from the spectre of qnantum mechanics. Obviously the mathematical basis for should be left to the universities to deal with; it is, after all, one of their functions. But does it necessarilvfollow that insights into nk,mic structure, obtained from quantum mechanical calculations, can only he discussed after the student has manipulated algebraic terms? Can high schools so easily wash their hands of one of the two great theories developed by the ~hvsicalsciences in the twentieth centnrv? Clearlv the
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university lev& But does it follow that the consequences of auantum theorv cannot be surveved without beina buried in khrodinger's Wave Equation
In constructing a differential equation capable of describing In electron, confined to a space near a nucleus, Erwin Schrolinger postulated that the electron was not a particle at all. He visualized the electron as a multi-dimensional "matter wave" whose distribution and frequency were determined by .he electric and magnetic fields emanating from the atomic
662 1 Journal of Chemical Education
nucleus which it surrounded ( I ) . In effect the nucleus diffracted this electron wave so that the electron was confined to a three-dimensional, (actually four-dimensional, if you included relativity space-time considerations), standing wave pattern in the vicinity of the nucleus. Obviously the electron wave had to have exactly the right energy and shape to fit seamlessly around the nucleus. If it did not its vibrations would crash into each other thereby destroying any hope that a self-sustaining pattern was being established. In short, the atom wasnothing more than a diffraction phenomenon arising from an electron wave "caught" in the field of an atomic nucleus. Schrodinger did not originate the idea of an electron as a wave, he borrowed it from de Broglie, who in turn got the idea from the fact that light was already being pictured as both an electromagnetic wave and a corpuscular photon. Schrodinger's electron was, in effect, an oscillating smeared-out, negatively charged continuum whose energy, charge density, distribution, and complex vibrational modes were described by a $-function (i.e. the "solution" to the wave equation) ( 2 , 3 ) . Unorthodox as these ideas seemed, the Austrian physicist could muster some persuasive arguments to support a wavediffraction thesis. In order for diffraction to occur the de Broglie wavelength of the electron wave had to he of roughly the same dimensions as the atom. If quantum numbers were independently chosen so that the wavelength also explained the experimentally observed spectral lines or ionization energy of the atom, it was found that both methods of assigning wavelength agreed with each other. Was this merely a coincidence? Clearly, an electron's wave-like nature really stood out when the electron's movements were confined to small spaces or between small obstacles whose dimensions were around 10-8 cm. In these circumstances, the electron, whose de Broglie wavelength is usually around this size, had a chance to diffract and to interfere. Of course, Schrodinger had to concede that there were times when an electron behaved just as a corpuscle would. For example, when an electron moved in wide open spaces, where its movements were unrestricted, its motion was remarkably
particle-like. But Schrodinger, like other scientists, tried to avoid using two contradictory models to explain the nature of subatomic nhenomenon. He thoueht i t would be s i m ~ l e r to explain "particle tracks" in terms of waves rather than the other wav around (4). . . Schrodineer's reasonine went somethine like this: in situations where an electron appeared to be "localized." it could not he described hv a simole. sinusoidal wave hecause this type of wave had no hbundapies; it extended evervwhere. In order to pinpoint the electron one had to postuiate a "complex wave," constructed by overlapping a large number of simple waves so that they formed a crest where the electron was, a i d cancelled out wh&e the electron was not (4, 5 ) .This "wave packet" idea, although ingenious, was met with scepticism by &her scientists. For one thing, "wave packets" were never ohsewed in nature with any other types of.wave phenomena, and they would tend to disperse and spread out very quickly in any case. Clearly Schrodinger was overplaying his hand hv ex~lainineawav the "cornuscular" electron. Even to this da;, aithoug< scientists cannot derive or prove his mathematical equations, these equations are still regarded as a brilliant intuitive description of subatomic behavior. Disnutes between Schrodineer and manv of his colleaaues arise, not over this, but over hi'determinatibn to banish the particle as nothing hut an expedient artifice. The Uncertainty Principle At the same time that Schrodinger was developing wave mechanics. Heisenhere was cookine UD an abstract mathematical de&iption ofihe atom wh';ch'only dealt with measurable auantities. like enereies of stationarv states and hi; numerical information, ohtransition tainable from atomic spectra, was expressed by using something called "matrices," a branch of mathematics perfect for handling and categorizing large amounts of raw data. By tinkering around with these matrices, Heisenherg discovered that nature had laid an inescapable trap. I t seemed that in the universe there existed a limit of precision to which certain conjugate properties could be measured simultaneously. This limit had escaped notice before this time because it was so small-of the same order of magnitude as Planck's constaut-as to be unimoortant for obiects even as hie as a hiological cell. At first. Heiseuhera totallv reiected the idea that we could fbrm mental p~cturcs'ofwhn; wu; happening inside the arum. We s i m. ~ h nut get entmeh hard data ehout the electrun . could to beahle toenvisage it in any useful way. Eventually however, Ilrisenl~ergsuccumbed to the pressures ot other scientkts to retreat from this hard line and to verbalize some of the difficulties inherent in getting reliable, absolute knowledge of the electron. This he did hv devising the "microscone analow" -" to clarify and communicate his thoughts. Before we go into the analoev. ".. we have to a t least he aware of the fundamental. hut related, ground-rules on which Heisenherg's matrices and uncertainty principle are based 1) Any material entity is the sum totalof all its properties. For ex-
ample, water is transparent, fluid,boils at 100°C,freezesat O'C, reacts with sodium, etc. Any liquid, which has d l the properties of water, must be- water. This postulate isso self-evident that we take it for granted. 2) The only properties we can depend on, as being real, are those which we can measure. or ahserve direetlv. Anv orooertv. which
it really exists at all. This too is a postulate. In order to throw some lieht on his third nremise (which is coming up, he patient) ~ e i l s e n b e r ~ d r e a m s ua p"gedankenexperiment" (thought experiment) in which an observer is permitted to perform a perfect experiment with ideal gadgets, nrovided that no fundamental law of nature is violated in the brocess. The fact that the gadget or the experiment cannot he set up in practice is conveniently overlooked.
In hisgedankenexperiment, Heisenherg pretends that the electron is a particle. This electron-particle is viewed through a make-believe, but perfect, microscope. However, in order to see that electron, (photons of) light must he bounced off its elusive surface. This brings up the first of many problems. Because of the dual nature of electromagnetic radiation, the light wave, used in illuminating an electron, should have a wavelength annroximatelv eaual to the dimensions of the electronP1f its'wavelengthis t i n long, it will simply slide past the electron, ignoring it. If its wavelength is too small, then light's particle nature hecomes dominant and the "photon" crashes into the electron. huffetine it far too much for anv measurement of its position to he reliable. But even if we manage to avoid that dilemma, another one awaits us. The difficulty this time is in our "perfect" microscope. Not that it is not made well. or that we do not know how to ;se it properly; we could pretend our way around that. The problem is that in order to see the electron, we have to ricochet more than one photon of light off it. This, in itself, is a formidable task because the electron is recoiling at the same time from the previous photon's hit. But even if we overlook that minor difficulty, we find that the rebounding photons, in passing through the microscope lens, are incapable of forming a sharp image. Because the photons have a wave-like side to their nature.. thev.are diffracted bv the lens into an indistinct and fuzzy pattern. This "microscone analoev" clarifies Heisenhere's third assertion: if we want to measure a system we would have to do an experiment on it which would involve measuring devices of some kind. The problem is that even the gentlest of devices would interfere with the properties of the system in an uncontrollable and unpredi&.able way. There would be no way to make sure what the measurement would have been if the experimental probe had not been there. For larger bodies, this hindrance is so unimportant as not to matter. But in suhatomic measurements the interaction between the observer and the observed has very far-reaching consequences. I t is these consequences that concern us, for they form the heart of quantum mechanics. Heisenherg bluntly argued that since we could not, even in principle, determine certain pairs of properties of the electron simultaneously (e.g., position and momentum). nerhaus we should d a v these properties down. Perhaps the;do noi exist simultaneously as physically real attributes. Perhaps the new quantum theory should concentrate its efforts o i qualities that are amenable to measurement, and that the ones that cannot be measured have no business being in the theory ( 6 ) .This sounds as if Heisenberg was retreating into his hard-line "mathematics or nothing" approach to the atom. And indeed he did-for a while. Eventually Heisenherg did mellow his views, hut I will come back to that later. What Is the Electron, Really? Although scientists reluctantly acknowledged the brilliance of Heisenberg's uncertainty principle and the microscope analogy, they remained leery. Einstein, for one, spent many vears thinking- up that would - other zedankenexperiments . circumvent the consequence of Heisenherg's uncertainty principle. In the end these efforts succeeded only in eroding his influence and in intellectually isolating him from the mainstream of scientific thinking. The electron "double-slit diffraction experiment" was one of many ingenious attempts to force the electron to betray and reveal its true nature. In this exneriment. two closelv -snaced. . atom-sized incisions are furrowed in an opaque foil, through which electrons from a sinele source can he nassed (7-9). Whena sinyleelectron passes through aslit opening in the luil. (tie.) it orudurrs a sinalr flnih on thr fluorescent detector blackening of the photographic sc'een or a single detector film. This is a quantum process uroduced bv a microcosmic speck. ~ndeed,if we piug up siit B, the eiectron particles will pass through slit A and produce a distribution Volume 55, Number 10. October 1978 1 663
the two models upon us. But any attempt to trace out a continuous trajectory or an instantaneous velocity for an electron between two "observations" is a waste of time. We will neuer haue any way of knowing what happens to the electron between readmgs.
Detector Screen
Opdque FoilIhe double-slit diffraction experiment.
icattered around A' on the detector. Clearly then, the electron nust he a tinv- cornuscle because it behaves like one. . If a e open up Iwth slits, this should not alter the picture nuch. Atit.r nll, it is almost the sameexperiment! The electron ;huuld pais through either slit A or slit H. At t'irst we notice h a t the scintillatimi on the detector seem to 11c randomly ,lased. Hut wait! Something strange starts to happen here! 2s morc m d more parrides arc sent intermittently to the lt.tc.ctm, a staristiml pattern of spots, very reminiscent of an nterference pattwn. eventually starts t t ~appear. At first we arc tvmpted to cume to the brilliant (but incorrect) conclushm hat Fillsteindid; namely that theelectnmunly l~ehaveslike I waw In cnra.(ls. Clexlv this is incorrect because it' we send .he electrons through thk two slit system one at a time (even lavs anart). . .. the same pattern arises. Ohviouslv one electron :odd not have passed through one slit leaving a "track" helind to guide a later electron to its Droper spot in the pattern. perhapsthe electron passed through both-slits at once (just ike a wave), then, after interfering with itself, condensed on ;he other side as a single particle with a tendency to he located rt certain points hut not at others. Far-fetched? Fuddleluddle, you say? Perhaps. But keep in mind that de Broglie's ~iewof the electron was similar to this. Essentially he saw the :lectron as a corpuscle, preceded by a "pilot wave" (matter Nave) which somehow gave the electron a choice as to which iirections it could possibly move. But, just as a water wave :auld not tell a surfer exactly where she had to go, so a matter Nave could not oredict exactlv where the electron had to surface. The irony in this situation is that by analyzing these natter waves. we could predict where the electron would not ,e with far more assurance than we can predict where it would 3e. If the electron did go through both slits a t once, surely an 4ectronic sensor at each slit would confirm this for us. To our :hagrin it does not! It reveals that each electron travels ,hrough one slit only. In fad, scientific experiments have never rucceeded in locating a fraction of an elktron. I t seems that :lectrons are always detected in discrete hits. Not only that, lut we also turn un another oiece of nuzzlineinformation: the ~ave-likeinterference pattern disappears as soon as we set ~p detection systems at the slits, no matter how subtle and mohstrusive we think such systems are. The results of this ex~erimentare bewildering. - But they do ;each us two things. There is no one satisfactory model of the electron; in fact, 3icturing the electron only as; "particle" or only as a "wave" s a futile exercise hound to lead to internal contradictions. It isalmost ns if the real electrun is mocking our pathetic atcmlrti to cateoori7e i t . If wesrt upnn experiment looking for Naves, the electron humors us by behaving as a wave. If we are ooking for particles, then the electron obligingly behaves as I particle. Secondly, the act of "observing" an electron forces one of
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664 / Journal of Chemical Education
Put the Electron Into Words For Me The mathematics (whether i t he matrices or wave eqnations) hehind quantum mechanics is universally agreed upon as being a clear-cut account of what is goingon-insidethe atom. No one, so far as I am aware, disagrees or argues with the eauations behind auantum theorv. That is the heautv of mathematics-it capitalizes o n the-relationships hetween experimental variables without emotion or predisposition. The controversy begins as soon as someone tries to translate the mathematics into words. There is an old Hindu saying (to which many theoreticians would subscribe) that as soon as you use words to explain what you mean, vou have already missed the truth. As soon as we try to use language to "explain" the world of the electron, we find, to our consternation and chagrin, that the right words do not exist. The reason is quite simple: ordinary language is anchored to the common experiences of everyday life. But the sub-atomic world behaves so entirely different from what we are used to, that ordinary language becomes dangerously overextended when it is extrapolated to this bizarre evanescent world. . So what do we do? Some of the high priests of mathematical symbolism say there is nothing we can do. Any attempts to build a mental picture of the atom are, a t best, naive and hound to he inaccurate. In any process hy which we form a MODEL, we are forced to abridge the truth. In conjuring up a model, we single out a feature which we deem to he particularly important and we discard the others. But in neglecting many revealing aspects of a system in favor of one feature, we sacrifice accuracy. We have to. There is no room for iconolatry in science (10, 11). Most scientists, although aware of the shortcomings of laneuaee. .. realize that one cannot afford to he as uncompromijing and as wtranged as all that. They prefer to luhr~cate their knowledge 01 atomic structure, especially in communi. cating it to students, with tentative, imperfect, images in order to "explain" what might lie behind the behavior of atoms. If the el&tron seems tobe doine stranee thines that is because .--. ~ ~ i t is following a set of rules outside our sphere of experience. Clearlv. the nrohlem is not the electron: the ~ r o h l e mis us. We -~~~~ cannot force the electron to conform t o o;r ideas of how it should behave. We must unhinge our thinking enough to re; alize that we have to take nature as it comes.'if the electron does not comport itself completely like a "particle" or like a "wave," that is too had. ~ u i t h a t ' the i w a i i t is!
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Does $ Mean Anything? When describing an electron in an atom, the Schrodinger equation tells us all there is to tell about the electron. Ohviously, it does not enlighten us about the trajectory, color, or texture of the electron because such questions are impertinent in this situation. In fact, these questions betray a certain stubbornness in insisting that the electron must he a "partireveal moreahout the questioner cle." Clearly, such than they do about the electron. Does the +-function have any physical meaning that all scientists recognize? Obviously not. The hard line iconoclasts of mathematical abstraction would still argue that it is impossible to visualize atomic processes and that wave equations are merelv mathematical contrivances which contain no clues about what is "really" happening. Schrodinger might (if he were still alive) conceive of psi as containing vital information of the smeared-out, negatively charged cloud we call the "electron." ~ e ~ r o ~and l iEinstein, e on the other hand, might see the
electron as being an elusive, capricious particle, that has escaped true understanding only because the "hidden variables," that govern its behavior, have yet to he disclosed. But there is one model of that is widely, but not universally, accepted. It was first articulated by Max Born in Gottingen and refined by Niels Bohr in Copenhagen. This "Copenhagen Interpretation" (as it is now called) plays down the +-function, which often has negative and imaginary values, as having no physical connotation. On the other hand, $2, being always positive (or zero) and real, can be assigned a symbolic significance as a representation of certain probabilit,ies. The Copenhagen Interpretatiun does not regard the electron itself as either oscillating or moving as a wave. What does have a wave-like nature is the "probability field" which accompanies the electron. The amplitude or intensity of this "probability wave" is given by P2.In fact, some scientists like David Bohm have even contemplated that the probability field may exert a new kind of indefinable force on the electron proper, which would have the tendency of tugging the electron into regions where V is largest (7). T h e Copenhagen Model, although it does not like the idea of "indefinable forces," does view @as defining very accurately our chance of finding an electron within a certain volunle element, if we look there an infinite number of times. In short, +2 plays an actuarial role, broadly hinting, but never guaranteeing, where the electron might be found. Due to our limitations in sensory experience (and therefore of ordinary language) we cannot grasp exactly what the electron really is. The best we can do is to rely on two mutually exclusive, but complementary, models. Questions about the origin,causes, or movement of the probability wave that guides the electron are met with a frustrating silence. No one knows. The uncertainty principle, discussed earlier, supports the dual nature of the electron. I t says that if one wants to view the electron as a corpuscle, that's fine. Or if one wishes to visualize it as a wave, that's O.K. too. But it warns us that it cannot he both a wave and a particle a t the same time. In fact, it goes further: in any experinlent to define the electron's wave and particle properties, the uncertainty principle lays out precisely to what accuracy each measurement can he carried. Beyond that limit, nature can only he defined statistically (i.e. in terms of psi functions).
+
Mathemalics-A Sixth Sense Mathematics, no longer a mere tool for snlving problems hut fundamentalvehicle for our thinking about naturp, has almo heeome one of our sens Hendrik B. G. Casim When Heisenberg came out with an abstract interpretatk of the atom based on matrices, Max Born was supposed to ha. exclaimed "As usual mathematics has been wiser than visu imagination!" (12).Mathematical descriptions in science h a ~ become so complicated that their relation to our prejudire of what the world should be, have almost vanished. lronicall this trend probably began with Einstein, grew with quantul mechanics, and extends to today's descriptions of sub-sul atomic particles. For example, the "quark" is endowed wit vague, incomprehensible new properties, spun and woven or from the abstruse symbols of higher mathematics. To hide tt embarrassing fact that the human mind cannot conceive < these properties, they are given whimsical names like "charm "color" and "strangeness." Imckily for us, the electron has not yet achieved thi splendid degree of isolation. It can he assigned imnginabl properties like "shape" or "spin," which, although not. quit authentic, are not bad approximations. In fact, these ar pruximations are essential in discussing the electron. It is nr a virtue to quibble about what these semantic terms meal especially a t the introductory level. The real crime occurs nc in using these terms, but in forgetting that they are a mod1 and not the real thing. Literature Cited ~
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11) Healhcofp. Niels HughdoVandrey,"Noh~lPriw Winnrn in Phvsicr,"Ronksfor I. brarier Press. Freeport. N.Y.. 1971. p. 222. 121 Cernuw,G., "Thiny Yeas 'That S h m k Phyrie>," Duuhledny & Cu.. Csrdrn City. N.Y
1966.p.a.
(3) ISAhn,. A , "The R i x ofTho New physic.^," h v e r Puhlicathr he.. NPI Yo& 19s: p. 714. 141 Schrudin#rr, K S r i Amur 169.52 (Sept. 11J5nl. I.r5) Ue B w ~ i i e L., , "Matter and Liyhl."Dwrr I'uhlirdiunr Inc.. Reprint Edition from \1 W. Nortnn &Co. 1nc.N.Y.. 1939, pp.244. 27". 16) Hernstein.l.,"A Omprcheusible Wurld: On M,xlern Scienceand its O~i~inr,"Randur Houre.N.Y..1967.p. 126. (7) ~ n j p p e rW. , H., T h e Quanrum ~ h y s i r i r f And r A" introdi~rrinn'1'0'rhrir ~hysies. Oxfwd University Prrn. N.Y.. 1970. p. I:X (8) Margenau. H., "Nalure of Phyrienl Reality," McGraw-Hill. N.Y. 1950, v, 325. 191 ~ o i r m n n n , he Stranee S L ~,,f~ha C ltaat~tam: :ird ~ d . ~'rliran . R O O ~ S~ t d Harm~mdrvuith,Rnsland. 191R.p. l4d. (101 Hriwni3erg. W.."Acmrr The F r m t i ~ r a . "Hixlm K I h w . N.Y.. 1974, pp. in. 120. I1IJ B~ihrn,U.."qaantom 'Theory." P~entireHall. linglewood Cliff%.N..I.. 1951.p. 138. ILII Mendelsnhn. K., "The World of Wnllhrr Nernst: The Riw and Pall ofGerman Srien? Ifflid -1941,"Univc~rity~ u fP i t t r h u r ~ hPrcrr. 1973.p. 139.
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