On Introducing the Uncertainty Principle G. M. Muha and D. W. Muha' Rutgers, The State Uniwrsity of New Jersey. New B~nswlck,NJ 08903 Given its imoortance and central role in modern ohvsical theory, a discu;sion of Heisenherg's Uncertainty ~ r i k i ~isl e often included in introductow chemistry courses. The apparent simplicity of the mathematical relationship expounded usually leads to its ready acceptance, if not understanding, by most students. ~ o < s readily o forthcoming in many instances, however, is an appreciation of the epistemoloeical sienificance of the relationshin. Ir is to this oarticular dcficultithat this paper is addresseh. When applied to the consideration of the motion of a particle (mass = m) in the nonrelativistic domain, the Heisenherg principal can be viewed in its essentials as simply a statement that the particle's position ( x ) and its velocity (u = dxldt) or, eauivalentlv, its momentum (P = mu) are not independent v&iables. Rather, the position-and momentum (or velocity) variables are statistically correlated (1)in that the products of the standard deviations ("uncertainties") in ~ (ApP = their expectation values, (Ax)2 = (xZ) ( x ) and (p2) (p)%,is given by ApAx 2 h/2r. Heisenberg's y-ray microscope experiment (2) points to the source of the correlation: the measurement of one of these two variables necessarily introduces an uncontrollable and imprecisely specified chanee in the other. Yet, because of its strong mathematical flavor, the essence and significance of this argument is often lost on beginning students. An alternative introduction to the topic, less rigorous hut certainly more intuitively appealing, can be had by focusing attention on the analogy between an individual's visual perception of an object's position and motion and the quantum mechanical description thereof. Thus, in a carefully structured analysis, Bohm (3)argues that our verceution . . of an obiect in a definite wsition seems to necessarily deny the simultaneous perception of any motion. Converselv. ".the visual oerce~tion . . of an ohiect's motion necessarily seems to imply a certain"fuzziness" in the depictation of its oosition. The featuies of Bohm's argument are illustrated in the figure. In the left Dart of the figure, the position of the hall d a t i v e to the ruler is certainly well defined. On first viewing an observer might argue that not only is the position exactly known, but likewise so is the ball's velocity (and hence momentum), for the ball appears stationary. Thus u = p = 0. But is such an assertion tenable? Could not the picture have been taken with a high shutter speed so as to "stop the action"? Then. rather than a well-defined velocitv and momentum, knoiledge of these quantities is completkly uncertain. The uoint is that from the left ohotoeranh alone. i.e.. the result of asingle experiment in the ~ e i & n d e sense r ~ (2);
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Bohm's argument (see text).
only the position of the hall can be specified with any degree of certainty. The converse situation is displayed on the right side of the figure. From the blurred image shown, an observer might correctlv infer that the hall was in motion durine-the oeriod . of exposure of the photographic film. Further, from a knowledge of the shutter speed and the ruler metric, the (average) velocity of the ball during "experiment" could he defined with reasonable precision. However, the definition of the ball's position is necessarily less precise for the blurred region extends over a considerable interval. Note that any ittempt to improve the precision of the position measurement by shortening the exposure time is self-defeating, for in the limit of a vewhigh shutter speed, the equivalent of the photo on the left is obtained. That is, the velocity becomes comoletelv uncertain. &chis the essence of Bohm's argument. Its value in develouine of the intricacies of the auan. an intuitive annreciation .. titative description of pusition and motion should beappar. ent. Yet now that it should not be inferred that the limita-
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' Present address: Georgetown University, Washington. DC 20057.
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tion inherent in visual perception "explains" the origin of Heisenherg's Principle. Rather the Uncertainty Principle serves as a quantitative statement of acondition imposed by the wave-particle duality property of matter. This duality concept is, of course, quite alien to our visual and intuitive perception of matter. We return to this point below. The reader may recognize the "experiments" described in the figure in another context, that of a dialectic dating hack to a t yeast the fifth century B.C.and known as ~eno's-Paradox of the Arrow (4). This paradox was conceived t o argue against the concept of the continuity of motion, and i t is this parallel with quantum theory we consider next. A rigorous discussion based on the mathematical theory of infinite sets is available (5),but its exposition would ordinarily be inap~ r o n r i a t ein an introductorv course. Rather we focus attention on a particularly simpliked example, that of one-dimensional motion-a golf - ball in free fall (nealectina .air resistance). I t is a basic tenet of classical mechanics that both the position and the linear momentum of the center of mass of a moving obiect can be defined at every point alone the trajectory. T h a t is, there exists some matbematicallycontinuous function, for convenience here defined in parametric form, such that the nosition is eiven bv x = f(t) and the time derivative of wlkch, r ( t ) = Idxldt, defines thd instantaneous uelocitv (nrooortional to the momentum of the ohiect). For exampie,'*in 'the case of the dropping ball expehmdnt of present concern, x = ('l~)gt2and p = mr(t) = mgt whereg is the gravitational acceleration constant. Thus for any given instant of time, the insertion of a value o f t in these classical mechanical relations provides a precise and simultaneous specification of both the position and the momentum of the falling ball, in direct contradiction of the limitation imposed by the quantum mechanical Uncertainty Principle. The source of the contradiction is to he found in the analysis of the concepts denoted as italicized words in the orecedine"oaramaoh. Consider first Zeno's critiaue of the . logical implications of the notion of an instant of time. Thus Salmon (4) remarks that a t any instant of time ". . . [Zeno] claims the arrow [ball] is where i t is, occupying a space equal to itself. Durina this instant i t cannot move, for that would require the i n s h t to have parts, and an i n s t k t is by definition a minimal and indivisible element of time." In a somewhat simplistic translation, and substituting Zeno's arrow with our golf hall, seemingly if the position of the hall is known exictlv. its velocitvmust be zero. Certainlv not the Uncertainty p;inciple resilt (which would require the velocity to be indeterminate) hut the paradox does focus attention on the troublesome concept of the instantaneous uelocity, f'(t). We might expect that when formulated in terms of the modern concept of the differential calculus, the error of Zeno's argument would become apparent. Yet the proper definition of the derivative of a continuous function, i.e., ('(0,has been the source of controversy among mathematiciansand philosophers for the past three centuries (6).Even today the-concept is apparently still undergoing evolution (7). Of present concern is the limiting value of the ratio of the quotient AxlAt as At tends toward zero. Here in our particular example, At is the length of time that the camera shutter is open giving rise to the blurred length, Ax, in the photograph. Then. viewed from the mathematician's oersnective and in the currently favored Weierstrass c-6 forkl&ion (6, 77, the velocitv of the ball. u = f'(t), is said to be defined if there existsapo&tive numb&such'thatthe quantity1A d ~-t < r where 6 is a arbitrary positive number and 0 < lAt1 < 6. This abbreviated definitionis not given here for use in the following for, as has been noted (7), i t is ". . . much harder t o understand than the concept being defined.'' Rather, the point to be made is that the mathematician's definitions of continuity and differentiability are "static" concepts-logi&
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cal concepts-used to describe the "dynamic" concepts of velocity and position (6).Thus the definition given of velocity is in terms of an existence theorem, i.e., i t defines the loeicallv consistent conditions under which f'(t) exists. Fur( 6 and ther the definition is given in terms of two 6) which are in no obvious wav connected t o the . ohvsics . of the experiment. Certainly mathematicians are free to introduce a priori anv conditions deemed necessarv to render their a d v s i s lo&ally consistent. However, t6e matter presently ti be resolved is whether such an analvsis a oosteriori orovides a correct physical description of thk phyiical phenomena considered. The analvsis involves the conceot of continuous functions.2 Is mot& continuous? The answer is no, a t least to the extent that modem quantum mechanics presents a correct and complete description of the physical universe (8).In the microscopic or quantum domain, Bohm (3)notes that the limiting value of the quotient AxlAt does not exist if At becomes too small. Further he remarks that even in the macroscopic domain this limit need not exist, for example in the case of a dust particle undereoine" Brownian motion. A descriotion in this instance in terms of a continuous trajectory is not possible; a t best onlv assertions about statisticallv averaeed auantities are appropriate. Likewise, in the auantum domain onlv statistical averages can he specified as concerns the simultaneous measurement of (com~lementary) auantities such as Dosition and momentum. 1n-particula;, hkcause of the intrinsic wave properties of matter, what we "really see" in any gross observation is simply the manifestation of a wave packet, i.e., a superposition of deBroglie waves which destructively interfere except in a small reeion of soace. " . .the reeion in which the "oarticle" is located. As is well known (9), a wave packet necessarily reoresents a ranee in values of both oosition and moment i m , neither of &ese quantities beingprecisely defined except a t the expense in the certainty of the other, a fact stated in quantitative form as the Uncertainty Principle. Further since in quantum theory the concept of a continuous particle trajectory does not exist, neither do derivatives of position nor momentum. The best that can he obtained t o paiallel the classical concepts of position and velocity is the time evolution of the corresponding quantum mechanical ooerators (10).T o oursue this matter further reouires the &e of a d v h c i d leiel concepts and hence is not considered here. Rather we simply note that the integration of the timeevolution equation leads directly to Newton's first and second laws of motion. Hence the seeminelv snecifica- .orecise . tion of position and momentum in classical mechanics (noted above) is in fact onlv a statement of average statistical behavior, a result first &en by Ehrenfest (9). ' The averaaina is of course over a time interval and hence one might inquire whether shortening the exposure time while simultaneously improving the resolution in the photograph might provide further details concerning intermediate stages in the averaged result. Certainly i t is not necessary to photograph the entire hall for present purposes. A record of the motion of a spot or mark on the ball's surface would suffice. Thus the ohotomanh might he taken throueh a microscope, a suggestion which will probably bring to mind Heisenberg's well-known critique (2) of such an experimental arrangement and the attendent limitations in the measurement. In particular, resolution can he improved only by
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Actually, for present purposes we have simplified the discussion somewhat in that we identify the continuity of a function as the necessary and sufficientcondition for differentiability.However, continuous curves, including some definedby motion, but not possessing derivatives, are well known in mathematical analysis. Boyer (ref. 6, p 285) cites examples. Our simplification,however, does not invalidate the argument as presented.
decreasing the wavelength of the light used. But then there is a concomitant increase in the photon's energy and hence in the uncertainty in the particle's position because of the scattering interaction involved in the observation process. Details of the argument are available in many textbooks and need not be considered here. We simply note that for the measurement of very small distances, the rigid ruler shown in the photograph would necessarily have to be replaced by some form of interferometric method (12). In passing we remark that although Heisenberg's discussion of the measurement process is often quoted, less well noted is Brillouin's interesting essay (12) concerning practical aspects of the same matter. Brillouin calculates the expenditure of energy required to generate the very small wavelengths required in the measurement of small distances and concludes (12):"the cost of such an experiment willsoon become incredibly high, and represent more than any laboratory, any institution, or even any big nation can afford to pay." He then speculates whether such a form of practical limitation might properly be included in the theory. The inverse relationship between the cost of particle accelerators used in high energy physics and the cross-sections measured is witness to the significance of Brillouin's observations. Finally we comment on a challenge that, in our experience, students often raise concerning the application of the Uncertainty Principle to the dropping ball experiment shown in the figure. Although i t can be stated in several forms, in essence the argument is as follows: suppose the golf hall is rigidly fastened so that it is "known that i t does not move" during the exposure of the photographic film. Then necessarily Ap = 0 since u = 0. But also Ax = 0 because of the rigid attachment. Hence the Uncertainty Principle is violated since if Ap = 0, Ax should he infinitely large, not zero. But how is i t known that the ball does not move? Surely this cannot simply he accepted as defined. Rather, to dem-
onstrate the internal consistency in the argument, such a result must he forthcomine" from the ex~eriment.Yet. as is well known ( 2 ) ,distances or features smaller than the waveleneth of the lieht used in the ohservation (sav . " 5000 A) cannot be resolved. If the resolution limitation due to the grain structure of the photographic film is ignored, then the implication is that Ax = 5000 A, not zero. Thus for a golf ball cm/s which is to he interpreted of mass 50 g, Au = 4 4 as implying that the velocity is known t o he zero to within an uncertainty o f f 2 X 10P5cmls. Certainly for practical ourposes u = 0, hut it is equally clear that when thidetails of the measurement are considered, Heisenherg's relation is not violated. As Schlegel has noted (13), the magnitude of Planck's constant is so small that when macrosopic bodies are t o he considered, the limitations imposed by the Uncertainty Principle can be neglected. The resolution of other seeming paradoxes (8, 13) can serve to introduce other aspects of the theory of measurements. Literature Cited (11 Bohm. D. "Quantum Theary"; Prontiee-Hall: New York. 1951; Chap 10. (2) Heisenberg. W. "The Physical Principle of Quantum Theory"; Dowr: New York.
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(3) Ref. I.Chap6. (4) Salmon. W. C. "Zeno'sParsdorn": Bohbs-Menill: Indianawlis, 1970: p 10. (5) Ruaacll, B. G. "Our Knowledge of the External W o r l d Norton: New York. 1929: p 182. (61 Royer. C. B. "The Hiatory of the Caleulus and itr Coneeptval Development"; Dwer: New York, 1958: Chap VII. eepeciaUypp 294-298. (7) Davis, M.: Harsh. B. Sci. Amer. I972 (June),78. ( 6 But see Primas, H. "Chemiatry, Quantum Mechanics and Reductionism" 2nd ed.; SprinprVerlsg: Berlin. 1 9 8 2 : ~16; Merrnin. N. D.,Physics Today 1985 (April) 38: UIh"0, J. I" ,oreat Currents of Mathemetical Thought": LF Lionnsia, F.. Ed.; Dover: New Y o r k Val 11, p 42. (9) Kramers, H.A. "Quantum Mechanics"; Dover. New York. 1964; pp 17-Ti. (10) Atkins. P. W., "Molecular Quantum Mechanin" 2nd d:Oxford:New York, 1983: p 96.
(11) Ref l , p l 9 5 . (12) Brillouin, L. "Science and Information Theory" 2nd ed.; Aeadcmie: N m York. 1962: Chap 22. 113) Schlegel. R. Arne,. Scientist 1948. 36. 396: Perm A. A m w J. Phya. 1980, 48. 931: ma8.R . H . A ~ C , J. . P ~ Y 1981,49.925. ~.
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