Investigation of the de Broglie-instein veloity equation s universality in the ontext of the Davisson-Germer experiment Yusuf Z. UMUL Canaya University, letroni and Communiation Dept., Öğretmenler Cad., No:14, Yüzünü Yıl, Balgat, 653, Anara Turiye yziya@anaya.edu.tr PACS numbers (8): 3.65.-w, 3.65.Ca, 3.3.+p, 3.65.Pm 1
Abstrat: A general formula is derived for the wave-length of a quantum partile from the energy and momentum equations of the speial theory of relativity. Sine this equation is also valid for the non-relativisti speeds, it is used without any approximation. The formula is applied to the results of the Davisson-Germer s eletron diffration experiment and it is found that the orret wave-length is evaluated by this formula. Then it is shown numerially that the de Broglie-instein veloity equation is also valid for the nonrelativisti ases sine the phase veloity is found to be greater than the speed of light. 1. Introdution Quantum physis has its roots on the invention of Plan who showed that the radiation was disrete by solving the problem of bla body [1]. He introdued a onstant of ation whih put forward the dualisti nature of light or the eletromagneti radiation. instein used the ideas of Plan and suessfully explained the photoeletri effet []. The experimental evidene of the photon was obtained by Compton, who showed the frequeny shift of light beause of the ollision of photons with eletrons [3-5]. These advanes influened a Frenh physiist, named de Broglie. He put forward the idea of the dualisti nature of elementary partiles lie eletrons [6, 7]. The experimental approval of this idea did not delay. Davisson and Germer onstruted an experiment in whih they demonstrated the Bragg diffration of eletrons by the rystal of Niel [8-1]. In this experiment it is shown that the wave-length of the eletrons gives the same value whih is evaluated theoretially.
The aim of this paper is to show that the wave-length and group veloity of a quantum partile is orretly evaluated by using the non-relativisti equations, but the phase veloity and the frequeny is not orret. We will prove this proposal by using the relativisti equation of energy [11, 1], whih also gives the exat values for the non-relativisti ases. The experiment of Davisson and Germer will be onsidered for the omputational he of the proposal. We use the term of the quantum partile for the elementary partiles whih have the dualisti nature.. Theory First of all the relativisti equations of a quantum partile will be obtained for the wave-length, phase veloity, group veloity and the frequeny. Sine the non-relativisti equations are the approximations of these relativisti equations for small veloities, the obtained formulas are also valid for the non-relativisti ases. As a seond step, we will derive the non-relativisti equations for the quantities, mentioned above. Then the small veloity approximations of the relativisti equations will be found and ompared with the diret derivations..1. Relativisti equations of a quantum partile The energy of a partile an be given by the equation of = + (1) in terms of the ineti ( ) and rest ( ) energies. Plan showed that the total energy is equal to = hυ () 3
where h and υ are the Plan s onstant and the frequeny, respetively [1]. The relativisti equation of energy an be introdued as m = (3) g v 1 for m is the rest mass [11]. and v g are the speed of light and the group veloity of the matter wave whih represents the partile [6, 7]. The rest energy is also equal to =. (4) m The frequeny of a partile an be obtained as + υ = (5) h where the ineti energy is equal to m = m. (6) vg 1 The group veloity an be found as v = (7) g + + from qs. (1), (3) and (4). The phase veloity (v p ) an be evaluated by using the de Broglie- instein veloity equation, whih an be introdued by vg v p =. v p an be written as 4
v p ( + ) = (8) + aording to q. (7). The wave-length ( λ ) of the quantum partile an be obtained from the relation of v p λ = (9) υ whih yields h λ =. (1) +.. Non-relativisti equations The non-relativisti energy equation an be given by 1 = mv g (11) whih was also used for the onstrution of the Shrödinger equation [13, 14]. The frequeny of the partile is found to be mv g υ =. (1) h The group veloity an diretly be written as v g = (13) m 5
from q. (11). Sine the phase veloity is equal to the half of the group veloity for the nonrelativisti ase, it an be given by v p =. (14) m As a result the wave-length an be written as h λ =. (15) m quations (1)-(15) are well nown and used in the literature [15]..3. Approximations of the relativisti equations The relativisti equations, found in Setion.1, will be approximated for v g << or equivalently <<. The energy equation an be written as 1 g + mv m (16) where the first term represents the ineti energy. The frequeny an be given by υ (17) h for the non-relativisti ase. The group and phase veloities an be obtained as vg (18) m 6
and v p m (19) respetively. It is important to note that qs. (18) and (19) satisfy the de Broglie-instein veloity equation. As a result the wave-length gives h λ. () m It is apparent that wave-length and the group veloity have the same values with qs. (13) and (15), but the frequeny and the phase veloity differs. The reason of this deviation is the usage of q. (11) instead of q. (16). Sine the rest mass an not be negleted near the ineti energy for the non-relativisti speeds beause << as mentioned before. 3. Computational analysis In this setion we will tae into aount the eletron diffration experiment of Davisson and Germer [8-1] in order to ompare the results, obtained in the last setion. They sent a beam of eletrons to the rystal of Niel. They observed that the diffration fringes our by the interferene of the sattered eletrons. The wave-length of the eletrons an be evaluated by using the Bragg formula of λ = d sinθ (1) where d is the distane between the Bragg planes. θ is the angle of inidene of the eletrons. The value of d for the niel is equal to,91 A. The angle of inidene was taen as 65. 7
Aording to these values, the wavelength is found to be 1,65 A. One A is equal to 1-1 m. A voltage of 54eV was applied to the eletrons. The harge of an eletron is equal to 1,6x1-19 J/eV. The ineti energy of the eletrons an be evaluated from the formula of = ev () whih gives 18 = 8,658x1 J. The rest energy of the eletron an be found from q. (4) as 14 = 8,187x1 J. (3) The Plan s onstant is 6,66x1-34 J.se. The theoretial value of the wave-length an be evaluated as λ,66899 (4) r = 1 A by using q. (1). The value of is taen as,9979x1 8 m/se. If the wavelength is evaluated by using the non-relativisti formula of q.(15), it will be found as λ,669. (5) nr = 1 A It was also shown in q. () that q. (1) redues to q. (15) for the non-relativisti speeds. Now we will evaluate the frequeny of the eletron beam. The values are found to be υ = 1,357x1 Hz (6) r and 16 υ = 1,3 x1 Hz (7) nr 8
aording to the formulas of qs. (5) and (1), respetively. The differene ours beause of the absene of the rest energy of the eletron in q. (11). But de Broglie had defined a rest frequeny whih an be given by υ = h (8) for a quantum partile [6, 7]. This frequeny inreases by the formula of υ υ = (9) v g 1 as the veloity of the partile inreases. quation (11) an also be valid for a partile whih has zero rest energy aording to q. (9). We propose that the rest energy must also be ontained in the non-relativisti formula of q. (11). 4. Conlusion In this study we onsidered the non-relativisti and relativisti energy equations in order to evaluate the wave-length and frequeny of the eletrons in the Davisson-Germer experiment. The non-relativisti ase is examined in the literature widely. But the inlusion of the relativisti equations brings the fat that the non-relativisti energy equation of q. (11) must be onsidered with the rest energy in order to obtain right results for the frequeny and the phase veloity whih also requires the universality of the de Broglie-instein veloity equation. Referenes 1. M. Plan. Ann. Phys. (Leipzig), 39, 553 (191). 9
. A. instein. Ann. Phys. (Leipzig), 3, 13 (195). 3. A. H. Compton. Phys. Rev., 1, 483 (193). 4. A. H. Compton and J. C. Hubbard. Phys. Rev., 3, 439 (194). 5. A. H. Compton. Phys. Rev. Supp., 1, 74 (199). 6. L. de Broglie. Compt. Ren., 177, 57 (193). 7. L. de Broglie. Phil. Mag., 47, 446 (194). 8. C. Davisson and L. H. Germer. Phys. Rev., 3, 75 (197). 9. C. J. Davisson and L. H. Germer. Pro. Natl. Aad. Si. U.S.A., 14, 317 (198). 1. C. J. Davisson and L. H. Germer. Pro. Natl. Aad. Si. U.S.A., 14, 619 (198). 11. A. instein. Ann. Phys. (Leipzig), 17, 891 (195). 1. A. instein. Ann. Phys. (Leipzig), 18, 639 (195). 13.. Shrödinger. Ann. Phys. (Leipzig), 79, 361 (196). 14.. Shrödinger. Phys. Rev., 8, 149 (196). 15. S. Gasiorowiz. Quantum physis. John Wiley, New Yor. 3. 1