WAVE-PARTICLE DUALITY: LIGHT

Transcription

1 MISN WAVE-PARTICLE DUALITY: LIGHT by E. H. Carlson WAVE-PARTICLE DUALITY: LIGHT PM r PM 1. The Problem Posed by Light a. Overview b. Classial Partiles Classial Waves d. The Nature of Light Both Aspets in One Experiment a. The Single Slit Apparatus b. Partile Model Preditions Wave Model Preditions d. Both Aspets Observed Simultaneously The Quantum Field Theory of Light a. Light Is Not Partiles, Not Waves b. Quantum Field Theory s Desription Classial EM Wave Theory Field Theory Appliations a. Strange One- and Two-slit Preditions b. Strange Spherial Wave Preditions Designing and Using an Astronomial Telesope Aknowledgments Glossary Projet PHYSNET Physis Bldg. Mihigan State University East Lansing, MI 1

2 ID Sheet: MISN Title: Wave-Partile Duality: Light Author: E. H. Carlson, Mihigan State University Version: 2/1/2000 Evaluation: Stage 1 Length: 1 hr; 20 pages Input Skills: 1. Voabulary: partile trajetory, Newton s seond law (MISN-0-15) or (MISN-0-409); photon energy, momentum, wavelength, frequeny (MISN-0-212); two-soure interferene, diffration (MISN ) or (MISN-0-432); Snell s law of Refration, polarization of light (MISN-0-222); eletrostati fore (MISN-0-115) or (MISN-0-419); Maxwell s Equations (MISN-0-132) or (MISN-0-429). 2. Apply the photon energy-momentum-frequeny relations (MISN ). Output Skills (Knowledge): K1. Voabulary: quantum field theory, wave-partile dualism. K2. Contrast the quantum field theory desription of light with that of the lassial partile and wave models. Desribe the preditive power of eah of these. Output Skills (Rule Appliation): R1. Give reasons why any given light phenomenon an or annot be satisfatorily desribed as eah of these: a. lassial partile b. lassial EM wave. quantum field THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The goal of our projet is to assist a network of eduators and sientists in transferring physis from one person to another. We support manusript proessing and distribution, along with ommuniation and information systems. We also work with employers to identify basi sientifi skills as well as physis topis that are needed in siene and tehnology. A number of our publiations are aimed at assisting users in aquiring suh skills. Our publiations are designed: (i) to be updated quikly in response to field tests and new sientifi developments; (ii) to be used in both lassroom and professional settings; (iii) to show the prerequisite dependenies existing among the various hunks of physis knowledge and skill, as a guide both to mental organization and to use of the materials; and (iv) to be adapted quikly to speifi user needs ranging from single-skill instrution to omplete ustom textbooks. New authors, reviewers and field testers are welome. PROJECT STAFF Andrew Shnepp Eugene Kales Peter Signell Webmaster Graphis Projet Diretor ADVISORY COMMITTEE D. Alan Bromley Yale University E. Leonard Jossem The Ohio State University A. A. Strassenburg S. U. N. Y., Stony Brook Views expressed in a module are those of the module author(s) and are not neessarily those of other projet partiipants. 2001, Peter Signell for Projet PHYSNET, Physis-Astronomy Bldg., Mih. State Univ., E. Lansing, MI 48824; (517) For our liberal use poliies see: 3 4

3 MISN WAVE-PARTICLE DUALITY: LIGHT by E. H. Carlson 1. The Problem Posed by Light 1a. Overview. In lassial physis, espeially mehanis and eletriity and magnetism, two distint and very useful onepts have been invented and used repeatedly, that of partiles and that of waves. However, to explain phenomena on the atomi level of size, the distintion between waves and partiles beomes blurred, and an important modifiation, involving probability, must be made. Beause the separate onepts of partile and of wave are so natural and satisfying for phenomena on the human sale it is somewhat diffiult to explain why and how they must be modified and ombined in order to desribe the behavior of very small objets. Our most omplete theory of light is a quantum field theory having both wavelike and partile-like properties that is, inorporating wave-partile dualism. We will larify the relationships between lassial partiles, lassial eletromagneti waves and quantum field theory by asking what eah of them predits for a single slit experiment. 1b. Classial Partiles. A partile is haraterized by having infinitesimal size and a definite loation in spae at eah instant of time. It may have mass and an eletrial harge, it ertainly has momentum and arries kineti energy. As time inreases, the partile moves along a definite trajetory, whih is determined entirely by loal onditions at the point where the partile is. For example, in Fig. 1 we have a partile with harge q moving near a harged plate. At eah instant of time, the partile experienes a fore F = q E determined by the eletri field at the loation of the partile. The subsequent motion of the partile in the MISN next infinitesimal interval of time is governed by: a = F /m = (q/m) E. If there are no fores ating on it, the partile travels in a straight line at onstant speed. 1. Classial Waves. The wave onept is abstrated from many different phenomena that are only geometrially similar, suh as waves on the surfae of water, sound in gas, liquid or solid, eletromagneti waves, displaement of a taut string or a strethed membrane (drumhead, flapping sail) et. In eah ase there is some field quantity varying smoothly in 1, 2 or 3 dimensions. (For water waves, vibrating strings and membranes it is the displaement of partiles from their equilibrium position; for sound in air it an be density or pressure variations; for eletromagneti waves there are six varying fields, the three omponents of E and the three omponents of B.) We will not need to onsider the detailed eletromagneti wave piture for light that an be derived from Maxwell s differential equations of the eletromagneti field, and whih orretly predits the light s veloity, its polarization (i.e., the fat that the E vetor is perpendiular to the diretion of propagation of the light wave) and the wave s behavior at surfae boundaries with onduting and dieletri materials. Rather, we will onsider the most harateristi property of any wave, that it undergoes diffration at an obstale. The momentum and energy arried by a wave are not onentrated at a point, but are spread out smoothly over a finite volume. This is in ontrast to the very small sizes of atomi partiles, and this ontrast is what makes it hard to imagine an entity whih is simultaneously a wave and a partile. In fat, we will see that light is omposed neither of lassial waves nor of lassial partiles, but is a new kind of entity that behaves in ertain ways like a partile and in other ways like a wave. A probabilisti desription ties these aspets together Figure 1. Loal influenes on the trajetory of a harged partile near a harged plate. 5 6

4 MISN MISN soure shutter slit detetor array R D Figure 2. Single slit diffration; experimental apparatus. 1d. The Nature of Light. Is light: A wave? Evidene: diffration and interferene experiments. An eletromagneti wave? L Evidene: Maxwell s equations predit suh behavior as its speed in vauum, Snell s law of refration, polarization, generation of long waves by simple eletromagneti iruits, et. A partile? Evidene: Compton effet, photoeletri effet, photohemial reations, et. The seemingly ontraditory wave and partile natures of light are reoniled by quantum field theory. Its basi statement is: The intensity (at a given point in spae and time) of an eletromagneti wave of frequeny ν gives the probability, and only the probability, that energy in the amount hν may be transferred between the wave and an external objet at that point in spae. The wave, partile and quantum field theory ideas an be ontrasted, ompared and explained by using eah to predit the results of a Fraunhöfer single slit diffration experiment. no. of events/bin bins Figure 3. Partile Model, omputer generated single-slit bar histogram; 41 detetors, 2305 partiles. open the shutter for a ertain time interval, then examine the distribution of light energy among the various detetors. We will assume that R L so the rays of light arrive at the slit as a parallel bundle. This allows the Fraunhöfer ondition to be fulfilled so that the mathematial desription is simpler. The detetion of light takes plae in a two dimensional array of individual, finite-sized detetors, eah with a detetion threshold E 0 ; that is, a minimum amount of energy E 0 must reah the detetor in some time t in order that the detetor registers the presene of light. Examples of suh arrays inlude the eye s retina, with rods and ones; a photographi plate, with individual silver halide rystals as detetors (size typially 1 µm); a stak of photomultiplier tubes; an array of thermoouples, sensitive to the heat energy from the light. Any of these ould be used in the experiment. Now let the shutter open for a time interval T t, and look at the energy aumulated by eah detetor, as predited by eah of the models for light. 2. Both Aspets in One Experiment 2a. The Single Slit Apparatus. Light s seemingly ontraditory wave and partile aspets an be made to appear simultaneously in the single-slit experiment of Fig. 2. The general idea of this experiment is to Figure 4. As in Fig. 3, but a muh larger number of partiles. 7 8

5 MISN b. Partile Model Preditions. We assume that all the light partiles arry the same energy E p, whih is greater than the detetor threshold E 0. To eah detetor we attah a ounter, alled a bin, whih is advaned one unit by the detetor when it detets a partile. In Fig. 3 the results of a omputer alulation simulating the results of the experiment are shown. (Randomly spaed straight line paths of the partiles from the soure to the detetors were assumed). Shown on the graph is the number of partiles deteted by eah of 41 detetors in a row aross the array. A total of 2305 partiles arrived, but only 16 detetors show any partiles (an average of 144 eah); the rest are in shadow. (The end detetors of the 16 were partly in and partly out of the shadow and, espeially on the left, show fewer ounts). The random spaing between trajetories aounts for the flutuations in the numbers of partiles arriving at different detetors. Simple statistial theory predits that the average flutuation, n, in number from one detetor to another is given by n < n > where < n > is the average number of partiles reahing any one detetor in the time interval, t, that the shutter is open. As the total number of partiles reahing the detetors beomes very large, this flutuation beomes omparatively unnotieable, and so a strip of width D appears uniformly illuminated, with a sharp edged shadow, as shown in Fig. 4. This is the harateristi illumination pattern given by partile theory ( ray, or geometrial optis theory), and in fat it represents rather well everyday (every sunny day at least) observations made under ertain irumstanes. (What are they?) 2. Wave Model Preditions. Continuing the above experiment (but with real light) let us replae the detetor array with one whose individual detetors are muh smaller and loser together. We will notie that the boundary between shadow and light is not ompletely sharp, as predited by the partile model, but has a width s Lλ where λ is the wavelength of the light. This feature is explained by a wave model of light; but instead of pursuing it further, we will move on to a mathematially simpler ase by shrinking D until it is muh smaller than the width of the light-shadow boundary. Then the wave model predits a Fraunhöfer diffration pattern in whih the light of intensity I smoothly varies along the detetor array aording to the equation I(y) = I 0 (sin 2 β)/β 2, β = πdy/λl, where y measures distane along the detetor array from the enter of the pattern. MISN Figure 5. Fraunhöfer intensity pattern for 41 detetors, 1405 energy units. Now we must look at the proess of wave detetion. For partiles, our detetors ounted. But wave theory assumes that the energy is distributed smoothly in spae and time and one ould give a ontinuous readout of the energy olleted versus time. However, real detetors must have finite size and finite time resolution, and so we assume, again, that the detetor must absorb energy until it ollets an amount E 0. It then advanes its bin ounter by one unit and is reset to begin the aumulation of another amount E 0 of energy. For an array of 41 detetors whih has registered the arrival of 1405 units of energy, wave theory predits that we get the Fraunhöfer pattern of Fig. 5. But if the array ontains many more detetors, and the threshold E 0 is very small, a smooth urve results (Fig. 6). 2d. Both Aspets Observed Simultaneously. Both the wave and the partile aspets of light will show up simultaneously in our single slit experiment if: 1. D Lλ, so the diffration pattern is present. 2. the intensity is low enough so we an have one photon at a time. We also require that the detetors be sensitive enough that their threshold E 0 E p = hν. That is, they an detet single quanta of light. With all bins empty, we open the shutter. The detetors begin to detet photons at random intervals of time. As the ounts build up, however, we see neither the sharp-edged pattern appropriate for lassial partiles (Fig. 3) nor the smooth diffration pattern of a wave theory (Fig. 5) but Figure 6. As in Fig. 5, but more detetors, small threshold. 9 10

6 } MISN MISN Figure 7. A non-partile, non-wave pattern, observable under appropriate irumstanes. rather a noisy Fraunhöfer pattern suh as Fig. 7 shows after 1405 have been deteted. 3. The Quantum Field Theory of Light 3a. Light Is Not Partiles, Not Waves. Light annot be ategorized as either lassial partiles or lassial waves. If it were lassifiable as a lassial partile it would not produe a diffration pattern, but light diffrats. If it were a lassial wave, its energy and momentum would have a ontinuous distribution in spae and time, but light often ats as if it onsists of loalized photons having disrete energy and momenta. Thus light annot be said to be intrinsially one or the other. 3b. Quantum Field Theory s Desription. The priniple features of the quantum field theory desription of light may be stated thusly: 1. Eletromagneti field omponents, E and B, still obey Maxwell s equations. When rossed and sinusoidally flutuating, they still form an eletromagneti wave traveling at speed. However, the E and B values no longer give exat fores on harged partiles, hene no longer predit exat energy and momentum transfer rates to those partiles. 2. An eletromagneti wave an only transfer energy and momentum to and from harged partiles in inrements fixed by the wave s frequeny: E = hν; p = hν/. 3. The atual times and plaes of these energy and momentum transfers annot be predited exatly, but the probabilities that they will our in speified time and spae intervals an be preisely alulated. These preise probabilities are linearly proportional to the wave s eletromagneti field intensities. The first statement delineates wavelike properties of light; the seond, partile-like properties. The third statement shows how we reonile those slit jaws A B } } C } sreen /2 Figure 8.. properties by inluding the observed random nature (unpreditableness) of the energy and momentum transfers. It allows us to (orretly) predit partile-like properties for small numbers of events and for short time intervals, and wavelike properties for large numbers of events where probabilisti distributions and atual distributions merge. Here are some more details on the ways eah lassial model fails for light: 1. Light is not a lassial wave. For example, onsider the photoeletri effet using a dust athode. When light of low intensity is shown on fine metalli dust, photo-eletrons start oming off immediately, whereas wave theory predits that a long time must elapse before a dust partile an ollet enough energy over its area from a wave to emit a photoeletron Light is not a lassial partile. That is, the photon annot be thought of as having a straight-line trajetory of infinitesimal width. Experiments designed to establish whih part of the slit opening a given photon rossed (for example by having a seond, narrower slit) will ause a further diffration or other disturbane of the photon and no definite path an be established. It is most orret to think of photons only at their instant of reation or destrution, and to onsider light to be a probability wave in between these times, although in the geometrial limit (resolution of the path is less than Lλ) an approximate trajetory an be assigned to the light wave, as though it were a partile. 3. Classial EM Wave Theory. Figure 8 shows a slit with shiny metal jaws. A light wave is inident on the slit, its E vetors perpendiular to the page. 1 See The Photoeletri Effet (MISN-0-213)

7 MISN In region A the wave is propagating as in free spae. In region B, where the wave is near the onduting metal, the time varying E field auses a urrent to flow in the metal. This urrent produes a B(t) field whih produes (Faraday s law) another E(t) field. All these E(t) and B(t) fields add to yield final E and B(t) fields that satisfy Maxwell s equations for a wave different from the inident one. This is the diffrated wave. In addition, the eletrial resistane of the metal auses some of the light energy to be absorbed at the slit jaws. In region C, the urrents produe the B(t) [and thus also the E(t)] fields of a refleted wave; some light is also absorbed. The above desription uses only the eletromagneti wave theory. All we need to add to the above is that energy is absorbed in terms of photons, eah of energy hν and transferring momentum hν/ to the slit. (The refleted photons transfer momentum 2hν/ to the slit). These photons are absorbed at random plaes and times on the metal in proportion to the intensity of the light ( E 2 ) and to a transition probability that depends on the eletrial properties of the metal (suh as its resistane). Consider a little volume of light at region B in time interval t. There is a probability P a that a photon will be absorbed by the slit jaws. Then there is a probability P b = 1 P a that no photon will be absorbed, in whih ase, the light in region B propagates down to the sreen where new probabilities of the light being refleted, transmitted and absorbed an be alulated, depending on E 2 there and on the eletrial properties of the sreen. 4. Field Theory Appliations 4a. Strange One- and Two-slit Preditions. A single photon must go through both slits of a double slit diffration apparatus (see Fig. 9). With low intensity light (say an average of 1 photon/se) a distintive double slit diffration pattern builds up as many photons are deteted. If one slit is losed, the pattern beomes that of a single slit. It is diffiult to explain this on a partile model; why would the use or non-use of one slit affet a partile through the other one? (The low intensity is speified to avoid possible ooperative effets between several photons and the slits, e.g. interations between photons of one slit and those of the other.) 4b. Strange Spherial Wave Preditions. Although a lassial eletromagneti wave may extend over a large volume of spae, the pho- MISN soure shutter slits detetors R Figure 9. Apparatus for a 2-slit experiment. ton is reated or destroyed in a small, loalized region. Consider a gas disharge tube in the enter of the lab, and have photomultiplier detetors on the opposite walls (Fig. 10). Assume that, on the average, one atom per seond of the soure emits a photon, onsidered a spherial eletromagneti wave pulse that expands in all diretions with speed. After a time T = r/, either detetor 1 or detetor 2, or neither, will detet a photon ontaining all the energy of the emitted eletromagneti wave. In no ase will detetor 1 and detetor 2 both detet parts of the same spherial wave pulse if the intensity is suffiiently low. 4. Designing and Using an Astronomial Telesope. We an use the design and uses of an astronomial telesope to illustrate when it is appropriate to use the wave and when the partile approximations for light, and when it is neessary to use the more exat quantum field theory: PM r PM W Figure 10. Photomultipliers (PM) deteting a flash of light. L 13 14

8 MISN MISN PS-1 1. Size of lenses and mirrors, their plaement, image size, magnifiation, et. Use light rays, geometrial optis, laws of refletion and refration (lassial partile theory). 2. Resolving power of the telesope, diffration pattern of a point soure (star) proessed through a irular aperture (lens). Use wave theory. 3. Analysis of the light for information about distane and speed of the soure, media through whih the light passed (interstellar spae); Doppler shift, et. Use eletromagneti wave theory. 4. For very faint soures, the telesope may detet only a few hundred photons appearing at random points in a diffration pattern. Use quantum field theory to make a statistial analysis of the pattern. Aknowledgments We thank Professor James Linneman for helpful suggestions. Preparation of this module was supported in part by the National Siene Foundation, Division of Siene Eduation Development and Researh, through Grant #SED to Mihigan State University. Glossary quantum field theory: a theory that reoniles the wave and partile duality of light by stating that the intensity of an eletromagneti wave at a given point in spae is related to the probability of an energy and momentum transfer by a photon. wave-partile duality: a model that enorporates both wave-like and partile-like properties. PROBLEM SUPPLEMENT Note: these problems also our on this module s Model Exam. 1. For eah appliation given below, give reasons it an or annot be satisfatorily desribed as eah of these: (a) a lassial partile; (b) a lassial EM wave; and () a quantum field. The Phenomena are: A. design of binoulars B. arhitetural analysis of light in and around a building C. study of bee navigation (sky polarization effets) D. analysis of light from very distant galaxies using a photomultiplier tube E. study of formation of vitamin D by suntanning F. using a light mirosope to study bateria 1µm long G. installation of a magi eye door opener H. building a pin hole amera 2. An atom emitting light in the visible typially takes about 10 8 seonds to do it. How long is the resulting wave paket in spae? Brief Answers: 1. A. a, geometrial optis B. a, geometrial optis C. b, polarized EM wave D., photon statistis E., quantum of light energy onverted to bond energy F. b, size of diffration effets G., photon, photo-eletri effet H. a and b 2. About 3 meters

9 MISN ME-1 MODEL EXAM 1. See Output Skills K1-K2 on this module s ID Sheet. One or more of these skills, or none, may be on the atual exam. 2. For eah appliation given below, give reasons it an or annot be satisfatorily desribed as eah of these: (a) a lassial partile; (b) a lassial EM wave; and () a quantum field. The Phenomena are: A. design of binoulars B. arhitetural analysis of light in and around a building C. study of bee navigation (sky polarization effets) D. analysis of light from very distant galaxies using a photomultiplier tube E. study of formation of vitamin D by suntanning F. using a light mirosope to study bateria 1µm long G. installation of a magi eye door opener H. building a pin hole amera 3. An atom emitting light in the visible typially takes about 10 8 seonds to do it. How long is the resulting wave paket in spae? Brief Answers: 1. See this module s text. 2. See problem 1, this module s Problem Supplement. 3. See problem 2 this module s Problem Supplement

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