Wave-Particle Duality: de Broglie Waves and Uncertainty

Transcription

1 Gauge Institute Journal Vol. No 4, November 6 Wave-Partile Duality: de Broglie Waves and Unertainty vik@adn.om November 6 Abstrat In 195, de Broglie ypotesized tat any material partile as an assoiated wave wit / support tat Hypotesis. p. Eletron diffration seems to But ten, te eletron at rest will ave infinite wavelengt, and infinite wave pase veloity. Tis says tat for a material partile, te de Broglie relation does not old. Failed attempts to save te postulate, kept te flawed relation, and modified te waves into train waves, pilot waves, probability waves, to name a few. We keep te de Broglie waves unanged, and modify te relation. First, we observe tat te Plank energy E defines virtual eletromagneti waves. used by de Broglie Consequently, for any partile, te virtual eletromagneti wavelengt is / / m, and / m /. Refinement of de Broglie argument, indiates tat / m may be Heisenberg s unertainty in te partile loation. x, de Broglie s later analysis supports tis interpretation, and we offer an explanation to partile diffration as a onsequene of Heisenberg s unertainty. We apply / m / to obtain te dispersion relations for te de Broglie virtual waves. 1

2 Gauge Institute Journal Vol. No 4, November 6 Finally, we observe tat reinstating te matter wave, String teory avoids Unertainty. 1. de-broglie waves de-broglie [1] assoiated wit a partile at rest an internal rest frequeny defined by If te partile as speed, denote m. (1) 1 1 ( / ) Ten, de Broglie assumed tat te partile onstitutes a plane wave propagated along te x-axis wit frequeny defined by Tat is, m. () m Te variation of te pase of te wave over te time interval is l t t k l t t t( ), (3)

3 Gauge Institute Journal Vol. No 4, November 6 were, it is understood, tat stands for te wavelengt of te soalled de Broglie matter wave. On te oter and, by te lok retardation formula, te observed internal frequeny of te partile is, and te observed variation in te time interval of te partile is i / ( t) /. i t of te internal pase To ave te partile remain inorporated in its wave, de Broglie set i. (4) Tat is, / / (11/ ) m (1 [1 / ]) m (5) m Te term is measured in eletron diffration, but equating it to m leads to an infinite pase veloity of te wave assoiated wit an eletron at rest. Indeed, for an eletron at rest, we ave, 3

4 Gauge Institute Journal Vol. No 4, November 6 Now, by (1),.. m / Terefore, te de Broglie wave pase veloity for an eletron at rest is. (6) ( m / ) Tis non-pysial result points to an error in de Broglie relation (5). For a material partile, te relation (5) does not old. Were did de Broglie go wrong? We proeed to observe tat de Broglie s use of te poton energy, E implies eletromagneti waves.. E mandates virtual Eletromagneti waves. By assuming te equation m de Broglie assumed two equations: () Plank s equation for te energy of te poton, E, (7) and Einstein s equation for te energy of a partile E m. (8) A poton is eletromagneti radiation energy, wit wavelengt If equation () is used /. (9) / m. (1) Te eletromagneti radiation problem of te 19 s, tat was resolved wit Plank s equation, 4

5 Gauge Institute Journal Vol. No 4, November 6 E boiled down in later generations to a partile equation, and te eletromagneti waves were forgotten. But invoking Plank s equation means assuming eletromagneti radiation. Eletromagneti radiation as speed, wavelengt, and frequeny. tat satisfy te relation (9), wi gives te dispersion relation de Broglie assumption of / k / m, leads to an infinite pase veloity of te virtual wave assoiated wit an eletron at rest beause by (1) / m. In order to resolve te eletromagneti radiation spetrum problem, Plank s ypotesized tat eletromagneti radiation energy exists in disrete pakets of E. Einstein empasized te partile nature of te eletromagneti energy, and te Compton effet, and te Bor atom amplified te partile nature of ligt even furter. But te poton equation (7) desribes preisely, and exlusively eletromagneti radiation. A non-eletromagneti wave, as de Broglie tougt of is wave, annot satisfy te Plank equation tat araterizes uniquely eletromagneti radiation energy. If Plank s equation is assumed, te wave assoiated wit te partile must be eletromagneti. Not a matter wave, Not a pilot wave. 5

6 Gauge Institute Journal Vol. No 4, November 6 Consequently, te wavelengt of te virtual eletromagneti wave an be only. Tis resolves te problem of infinite pase veloity of te virtual wave assoiated wit an eletron at rest. de Broglie analysis leads to te term diffration, but Wat is it equal to? m tat measures eletron does not equal te virtual wavelengt m. m Can we replae te inorret relation (5) wit a pysially believable relation? 3. Te de Broglie Relation, and Unertainty Sine m, (11) te de Broglie term tat is measured in partile diffration is m If te partile is a poton, ten. (1), and te de Broglie term equals to te wavelengt, wi measures te unertainty in te x loation of te poton, x. If te partile is an eletron, poton diffration suggests interpreting te de Broglie term of eletron diffration as te unertainty in te partile x loation, x. 6

7 Gauge Institute Journal Vol. No 4, November 6 We sow tat a modified de Broglie analysis points to te relation were We ave x p, x denotes te unertainty in te loation of te partile. (1 [1 / ]) (1 1/ ) / t t / / (13) Equation (13) says tat te partile remains inorporated in its unertainty zone beause As in de Broglie analysis, is te variation in te time interval i. (14) ( t) / i l t of te internal pase of te partile, sine by te lok retardation, Now, i /. 7

8 Gauge Institute Journal Vol. No 4, November 6 t( ) / t l (15) / is te variation of te pase of te virtual eletromagneti wave over te time interval. t If te partile is a poton, ten te wavenumber is, k /, and te unertainty in te partile s x loation is te wavelengt. If te partile is an eletron, ten / funtions as an effetive wavenumber, and (16) funtions as an effetive wavelengt. Poton diffration suggests tat te unertainty in te partile s x loation, equals te effetive wavelengt (16). Tat is, x (17) It seems tat te de Broglie term m (18) m p 8

9 Gauge Institute Journal Vol. No 4, November 6 tat equals determines te unertainty in te loation of te partile. We are inlined to replae te de Broglie relation wit te relation were, m x, (19) m x denotes te unertainty in te loation of te partile. If te partile is a poton, ten If te partile is an eletron, ten x () x (1) is stritly greater tan te virtual eletromagneti wavelengt. Relation (19) is onsistent wit Heisenberg s unertainty relations. Clearly, In partiular, if ten, ( x )( p ). (), p, and x. (3) 9

10 Gauge Institute Journal Vol. No 4, November 6 We proeed to establis relation (19). 4. de Broglie later analysis and Unertainty For Srodinger s wave funtion ( x, t) distribution 1 1 ( x) exp x x x for a partile [], te Gaussian (4) measures te probability to find te partile at x, witin unertainty interval of size x. It is well known [3] tat te mean value of x is and te variation of x is x ( ), x x x x dx x x x x x ( x) dx x. (5) x x By [4, p.], te Fourier Transform of te Srodinger wave funtion, is a wave funtion for te momentum p. By [4, p. 9], sine ( x) is Gaussian, ten ( p) is Gaussian, and 1 1 ( p) exp p p p x (6) p (7) In [4, pp ], de Broglie seeks te range of te Heisenberg unertainty intervals. de Broglie argues tat in pratie p x. (8) 1

11 Gauge Institute Journal Vol. No 4, November 6 and in fat In de Broglie words [4, pp ]: p x. (9) In order to preisely define te Heisenberg unertainties, it is neessary to define te unertainty A of any quantity A, as te interval of A values su tat te probability of finding a value outside A is less tan some small quantity. For two anonially onjugate quantities A, and B, one ten finds tat ( ) A B were ( ) depends on te oie of. te funtion is ( ) infinite for, so tat ten ( A)( B) from wi it follows tat sarp edged interval) B is infinite if A, is finite (ase of a However, in pratie it suffies to oose very small, but nonzero, and ten te produt ( A )( B ) an in te most favorable ases beome as small as someting of te order, but not smaller. Pratially, we tus ave in order of magnitude. ( A )( B ) Te examples tat de Broglie uses to illustrate te above, satisfy (9). In partiular, for we ave p m 11

12 Gauge Institute Journal Vol. No 4, November 6 x. m we see tat de Broglie later analysis points to unertainty as te essene of is relation, and supports replaing is relation / wit a modified unertainty relation x m m 5. Eletrons and Unertainty In te Davisson-Germer experiment [5], eletrons wit energy of 5-35 eletron-volts were sot at a Nikel rystal. Te eletrons tat refleted elastially from te rystal, at an angle, and lost only a little of teir energy, were allowed into a detetor. Te detetor ould be moved along a trak, so tat refletion would ome from various rystal planes. Te eletrons were refleted at angles 9. Te eletrons distribution wit respet to te angle, peaked at 54Volts, at an angle of 5. 1

13 Gauge Institute Journal Vol. No 4, November 6 As in Bragg diffration, te partiles are refleted at an angle, from two planes of atoms in a rystal tat are spaed at distane d. Te pat differene between suessive refletions is d sin Sine te partiles are distributed normally over te unertainty interval x / m, minimal number of partiles are likely to be at te edges of te interval, and maximal number of partiles are likely to be at te middle of te interval. 13

14 Gauge Institute Journal Vol. No 4, November 6 Terefore, maximal number of partiles will aggregate in a diretion so tat d sin. n x (3) Sine te Fourier transform of a Gaussian urve is Gaussian, te distribution of te refleted partiles in te diretion, remains Gaussian. 6. X-rays and Unertainty X-rays were produed first by Rontgen in 1895, as te result of deelerating eletrons in a metal plate target. X-rays were not defleted by eletri and magneti fields, and were determined to arry no arge. Tus, tey were suspeted to be eletromagneti radiation. Beause of teir energy, teir frequeny would be ig, teir wave lengt sort, and diffration would require gratings wit sort distane. Te planes of atoms in a rystal are spaed at su sort distane, and an serve as diffration gratings. Te suggestion to use rystals, was made by Max Von Laue in 191. Te experiments by Friedri and Knipping yielded unertainties of 1 (.1.5)1 meter, tat were interpreted as wavelengts, in Bragg diffration of waves. But te unertainty need not be wavelengt, Bragg diffration applies also to partiles wit x d sin, and te diffration experiments did not prove tat X-rays are potons. X-rays did prove to be eletromagneti radiation in two oter experiments: 14

15 Gauge Institute Journal Vol. No 4, November 6 Te first is te X-ray spetrum tat is ut off at a given voltage V. Te utoff may be explained as a reversed potoeletri effet in wi an eletron wit energy ev omes to a omplete stop, and its kineti energy ev onverts in full to an X-ray poton of frequeny and wavelengt ev /, / ev Te onfirmation tat in X ray diffration / ev d sin (31) proves tat te X-ray partile is a poton of eletromagneti energy. Te seond experiment tat proves tat X-rays are potons is te Compton effet in wi an X-ray partile ollides wit an eletron. 15

16 Gauge Institute Journal Vol. No 4, November 6 If we assume tat te X-ray partile is a poton, tat imparts 1 energy E to an eletron, and beomes a poton, we ave v v 1 p os Tus, Sine v sin psin. ( ) ( ) os ( p) E ( m ). 1 1 e m E, 1 e we obtain (1os ) m 1 1 e. (3) Te onfirmation of tis formula in te Compton experiment, validates te assumption tat te X-ray partile is a poton. Tat is, a partile of eletromagneti energy. Sine X-rays are potons, te unertainty equals teir wavelengt. X- rays wavelengt is determined in rystal diffration. 7. Dispersion Relations For a relativisti partile, te energy relation is equivalent to E ( m ) ( m ) ( p) (33) m m (1 / ) ( p) 4 4 or p m. (34) 16

17 Gauge Institute Journal Vol. No 4, November 6 Denote Substituting in (34) E m / k., and p /( / ), we obtain te dispersion relation / ( / ) / k. For a non-relativisti partile, te energy equation is equivalent to 1, m E V 1 m E p V. (35) Substituting in (35) E, p /( / ), we obtain te dispersion relation 1 V m m k V. 8. Strings and Unertainty de Broglie s assoiation of virtual waves wit a partile lead to te seeking of partile diffration. But tere are no matter waves, te relation breaks down at, and de Broglie later analysis promotes unertainty over is virtual waves. 17

18 Gauge Institute Journal Vol. No 4, November 6 Te Strings postulate reinstates te matter wave by aving te partile itself vibrate in its unertainty region. In tat region, te energy and te momentum of te partile are unknown, and te trajetory of te partile annot be determined. Te string postulate annot be onfirmed, or denied beause no experiment may apply to te unertainty region. In partiular, vibrations in a region were pysis annot be quantified, need not lead to any pysial results, watever te number of unseen spae dimensions may be. String teory aims to resolve singularities su as 1/ x, for x. But in quantum pysis tis problem is already resolved by unertainty: Away from te unertainty region x, we ompute 1/ x. In te unertainty region, energy or momentum annot be determined, te equation may not old, and we do not apply it. Tere is no way to replae te unertainty inerent in te quantum teory wit ertainty, anymore tan tere is a way to replae te quantum ypotesis wit wave teory. In fat, we may sum up te foundations of te quantum teory as te ombination of te Plank- Einstein quantum ypotesis, and te de Broglie-Heisenberg unertainty relations. Unertainty is a fundamental property of te quantum teory. Tus, it sould not be surprising tat te result of avoiding unertainty, and reonstruting quantum teory witout it, leads nowere. Te failure of tis exerise in avoiding unertainty, reaffirms ow fundamental te Heisenberg Unertainty relations are. 18

19 Gauge Institute Journal Vol. No 4, November 6 As Feynman s observed in [6], te string postulate may not be te only way to resolve singularities, and tere are oter possibilities for resolving te singularity. For instane, one su possibility is tat te equation is wrong at te singularity. Referenes 1. de Broglie, Louis, Te Beginnings of Wave Meanis in Wave meanis te first fifty years edited by Prie, William; Cissik, Seymour; Ravensdale, Tom. Halsted Press (Wiley) p Bass, Jean, Probability, PseudoProbability, Mean Values in Wave- Partile Duality, edited by Frano Selleri, Plenum199, p Rousseau, M., and Matieu, J. P., Problems in Optis, Pergamon Press, 1973, p de Broglie, Louis, Heisenberg s Unertainties and te Probabilisti Interpretation of Wave Meanis, Kluwer, Flint, H. T., Wave Meanis Metuen, 1953, p Feynman, Riard, in Superstrings, a teory of everyting edited by P.C.W. Davis, and J. Brown, Cambridge Univ. Press, 1988, pp