[Yousif*, 4.(12): December, 2015] ISSN: (I2OR), Publication Impact Factor: 3.785

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1 (IOR), Publiction Impct Fctor: IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY HEISENBERG FORM OF UNCERTAINTY RELATIONS Mohmmed Yousif*, Mohmmed Ali Bsheir, Emdldeen Abdlrhim * Sudn University of Science nd Technology (SUST), Mth deprtment, college of science, Khrtoum, Sudn. Mth deprtment, college of science, AL Neelein University, Khrtoum, Sudn. Sudn University of Science nd Technology (SUST), Mth deprtment, college of science, Khrtoum, Sudn. ABSTRACT This pper, dels with the uncertinty reltion for photons. In [Phys.Rev.Let.18, 1441 (1)], nd [1] the uncertinty reltion ws obtined s shrp inequlity by using the energy distribution on spce. The reltion we obtin here is n lterntive to the one given in [Phys.Rev.Let.18, 1441 (1)] by the use of the position of the center of the energy opertor. The fct tht the components of the center is non commuttive ffected the right hnd side of the Heisenberg inequlity. But this resolved by the increse of the photon energy. Furthermore we study the uncertinty of Heisenberg with respect to ngulr momentum nd Foureir. We end the pper by giving some exmples. KEYWORDS: uncertinty reltion for photons, quntum mechnics of photons, Foureir theory. INTRODUCTION As in the stndrd Heisenberg form of uncertinty reltion for photons is somewht difficult identity its position opertor. However, it is pprently, photon cn be influenced by the spred of momentum nd the extension in spce tht represent the fmous Heisenberg phrse Je genuer der Ort bestimmt ist, desto ungenuer ist der Impuls beknnt und umgekehrt. The photon uncertinty reltions is minly divided into two defined notions: the photon wve function s mentioned in momentum spce nd then the energy density of the quntized electromgnetic field. Thus, when the second momentum is pplied, it is generted the following form[1] r p 4h (1) This work is imed to construct definition for the uncertinty of the photons position so tht cn be nlogous to the stndrd definition, which done by Iwo Bilynicki-Birul, Zofi Bilynick-Birul Heisenberg uncertinty reltions for photons (1) [1]. This to illustrte the importnce of R s the center of energy to the first momentum. Therefore, this method is going to serve in mking uncertinty reltion s close s the originl form of Heisenberg, R P > d h () Where d is the number of dimensions. A chrcteristic feture of the uncertinty reltion for photons is tht the lefthnd side in this inequlity in two nd in three dimensions is never equl to dh/, but it tends to this limit with the increse of the verge photon momentum. Only in the infinite-momentum frme is the uncertinty reltion for photons the sme s for nonreltivistic mssive prticles. However, in one dimension, the inequlity () is sturted so tht in this cse there is no difference between photons nd mssive nonreltivistic prticles. We lso prove the following shrp inequlity Interntionl Journl of Engineering Sciences & Reserch Technology [777]

2 (IOR), Publiction Impct Fctor: R. R P. P 3 h (3) In nonreltivistic quntum mechnics, the inequlities obeyed by the two mesures of uncertinty, R P nd R. R P. P, re completely equivlent. They hve equl lower bounds nd they re both sturted by Gussin functions. This equivlence does not hold for photons. Nevertheless, the two inequlities re intimtely relted. We shll first prove (3) nd then use the informtion bout the photon sttes tht sturte this inequlity to elucidte the intricte properties of the inequlity (). Study hs been done by Schwinger, he cme up with the rough estimte tht the lower bound of R P is of the order of h. This work is endevored to derive relted uncertinty reltion for photons bems. Mking use of Coherent sttes of the electromgnetic field to find description of such bems in the limit of lrge number of photons so s to prove the shrp inequlity s follows[1], [3] R P 3 h nd we find the mode functions of the coherent sttes tht sturte this inequlity. (4) The nonexistence of the locl photon density in configurtion spce is due to the fct tht in quntum electrodynmics the opertor of the totl number of photons N involves not single but double integrl: N = 1 4π hc d3 r d 3 r : [ D (r, t). D (r, t) ε r r + B (r, t). B (r, t) μ r r ] = 1 π hc d3 r d 3 r : [ F (r, t). F (r, t) r r ] (5) We use systemticlly the Riemnn-Silberstein vector (the RS vector) F (r, t) = D (r, t) ε + i B (r, t) μ (6) Which will llow us to write mny formuls in compct form. The norml ordering removes the (infinite) contribution from the vcuum stte[1]. In contrst to the totl-number opertor, the totl-energy opertor of the electromgnetic field H (the Hmiltonin) is n integrl of locl density Where H = d 3 r ε (r, t) (7) ε (r, t) =: F (r, t). F (r, t): (8) The center of the energy opertor cn be introduced in ny reltivistic theory. All we need for this construction is the set of genertors of the Poincr e group. The Poincr e group is the group of Minkowski spcetime isometries. It is ten dimensionl noncompct Lie group. The belin group of trnsltions is norml subgroup, while the Lorentz group is lso subgroup, the stbilizer of the origin. The Poincr e group itself is the miniml subgroup of the ffine group which includes ll trnsltions nd Lorentz trnsltions. More precisely, it is semi direct product of the trnsltions nd Lorentz group. We define the opertor R s follows[1], [15]: R = 1 H N + N 1 H = 1 N 1 H H (9) Interntionl Journl of Engineering Sciences & Reserch Technology [778]

3 (IOR), Publiction Impct Fctor: Where N is the first moment of the energy distribution, N = d 3 r rε (r, t) (1) Exmple (1): Prove tht R = 1 N + N 1 H H = 1 N 1 H H Proof The symmetriztion in (9) is necessry to obtin Hermitin opertor. The inverse of the Hmiltonin is well defined, provided we exclude the vcuum stte. The spectrum of the Hmiltonin is nonnegtive, therefore the positive squre root is unique. The significnce of N is further underscored by its being the genertor of Lorentz trnsformtions. Since the opertors H nd N do not commute (the energy chnges under Lorentz trnsformtions), the equivlence of the two forms of R in (9) is not obvious nd to prove the equlity of the two forms of R in (9) we will first prove the following lemm if [H, C ] = then [ H, C ] = (11) In the proof, we use the fct tht the eigenvectors of the Hmiltonin form bsis. Acting on n rbitrry stte in this bsis E (excluding the vcuum), we hve ( H, E) = [ H, C ] E = [H, C ] E = (1) Since the fctor ( H, E) does not vnish, it cn be dropped nd the vlidity of the lemm is estblished [1]. Next, we use the commuttion reltions between the Hmiltonin nd the genertor of the Lorentz trnsformtions [H, N ] = iħp (13) to obtin [H, [ 1 N 1 H H, H ]] 1 = [ 1 H [H, N ] 1 H, 1 H ] = ħ [P i H, 1 H ] = (14) Finlly, using the lemm, we my replce Ĥ by Ĥ in the first term nd expnd the resulting double commuttor: = [ Ĥ [ 1 N 1 H H, H ]] 1 = 1 H N + N 1 H 1 N 1 H H The vnishing of the difference of two expressions for R ppering in (9) mens tht they re equl. THE GENERALIZED UNCERTAINTY RELATION Erlier we lerned bout the fmous Hiesenberg uncertinty principle which reltes the uncertinly in position to tht of momentum vi from formul (1): (15) x p ħ (16) We now generlize this reltion to ny two rbitrry opertors A nd B. First, we recll tht in given stte ψ, the men or expecttion vlue of n opertor ο is found to be[6]: ο = ψ ο ψ (17) Now let s consider the stndrd devition or uncertinty for two opertors A nd B: (A) = (A A ) (18) (B) = (B B ) (19) Using ο = ψ ο ψ we cn rewrite these two equtions s: ( A) = (A A ) = ψ A A ψ () Interntionl Journl of Engineering Sciences & Reserch Technology [779]

4 (IOR), Publiction Impct Fctor: ( B) = (B B ) = ψ B B ψ (1) We now define the following kets: X =(A A ) ψ () This llows us to write: Φ =(B B ) ψ (3) ( A) = (A A ) = ψ A A ψ = X X (4) Now consider the product of these terms[6], [11]: The Schwrtz inequlity tells us tht: ( B) = (B B ) = ψ B B ψ = Φ Φ (5) ( A) ( B) = X X Φ Φ (6) X X Φ Φ X Φ = X Φ Φ X (7) Remember tht the inner product formed by ket nd br is just complex number, so X Φ = Z = ZZ For ny complex number z, we hve: In this cse we hve: ZZ = Re(z) + Im(z) Im(z) = ( z+z ) (8) X Φ = ψ (A A )(B B ) ψ (9) = ψ AB A B A B + A B ψ = ψ AB ψ ψ A B ψ ψ A B ψ + ψ A B ψ Now A, the expecttion vlue of n opertor, is just number. So we cn pull it out of ech term giving: ψ AB ψ ψ A ψ B A ψ B ψ + ψ A B ψ = ψ AB ψ A B A B + ψ A B ψ = AB A B + ψ A B ψ (3) Now the expecttion vlue of the men, which is gin just number, is simply the men bck gin, i.e. ψ A B ψ = A B = A B (31) So, finlly we hve: X Φ = ψ (A A )(B B ) ψ = AB A B + ψ A B ψ = AB A B + A B = AB A B (3) Following similr procedure, we cn show tht: Φ X = ψ (B B )(A A ) ψ = BA A B (33) Putting everything together llows us to find n uncertinty reltion for A nd B. First we hve: ( A) ( B) = X X Φ Φ X Φ = X Φ Φ X (34) Reclling tht[6] ZZ = Re(z) + Im(z) Im(z) = ( z z ), we set Z = Φ X. Then Interntionl Journl of Engineering Sciences & Reserch Technology [78]

5 (IOR), Publiction Impct Fctor: X Φ ( A) ( B) X Φ Φ X ( AB A B ) ( BA A B ) ( AB A B ) BA + A B AB BA AB BA = ( [A,B] ) (35) Tking the squre root of both sides gives us the generlized uncertinty reltion, which pplies to ny two opertors A nd B. Definition: The Uncertinty Reltion Given ny two opertors A nd B: A B [A,B] (36) Where [A, B] is the commuttor of the opertors A nd B. For the opertors X nd P, we find tht [X, P] = iħ. So [X,P] = iħ = ħ Therefore we obtin the fmous Hiesenberg uncertinty principle: X P ħ (37) (38) Also, we cn despite ll of the differences between the nonreltivistic nd reltivistic dynmics we my derive shrp Heisenberg uncertinty reltion long one direction, sy x, for ny reltivistic system. This one-dimensionl uncertinty reltion is bsed solely on the commuttion reltions between X = R x nd P = P x nd hs the stndrd form[1] X P 1 h (39) Where X = ( P ), X = X X (4) P = ( P ), P = P P (41) The one-dimensionl uncertinty reltion holds for ny reltivistic quntum system. A simple proof of (39) uses the commuttion reltions [R i, P j] = ihδ ij nd the non-negtive expecttion vlue of the opertor: ( X iλδp )( X + iλδp ) (4) Where λ is n rbitrry rel number. The condition tht this expression treted s function of λ cn hve t most one rel root gives (39). This inequlity is sturted by the quntum stte whose stte vector stisfies the condition ( X iλδp ) Ψ =. (43) The specific form of Ψ depends, of course, on the system under study. Note tht we my remove the verge vlues ħ X nd ħ P from (43) by choosing Ψ in the form[1] Ψ = exp(i P X h i X P h) Ψ (44) Interntionl Journl of Engineering Sciences & Reserch Technology [781]

6 (IOR), Publiction Impct Fctor: Since the inequlity must hold for ll vectors, replcing Ψ by Ψ mkes no difference nd the two forms of the uncertinty reltion in one dimension, nmely, X P 1 h nd (X ) (P ) 1 h (45) re completely equivlent. In nonreltivistic quntum mechnics the equivlence holds in ny number of dimensions. A sphericlly symmetric Gussin function shifted in the coordinte spce by r nd in the momentum spce by p by the unitry trnsformtion of the form (44) will utomticlly sturte the inequlity (). This equivlence, however, is no longer vlid for reltivistic systems in three dimensions[1], [1], [7]. To extend our nlysis to two nd three dimensions, we introduce the dispersion in position tht involves two or three components of the center-of-energy vector R, R = R. R (46) Where R = R R nd the dispersion in momentum, P = P. P (47) Where P = P P. Following the sme procedure s the one used in deriving (39), we obtin (). The proof is bsed this time on the expecttion vlue of the following positive opertor: ( R iλδp )( R + iλδp ) > (48) In contrst to the one-dimensionl cse, the inequlities () nd (48) re not shrp becuse there is no stte vector tht is nnihilted by ll three components of the vector opertor  = R + iλδp nd even by two components. This is due to the fct tht the commuttors [R i, R j] = ihc Ĥ 1 Ŝ ij Ĥ 1 of the components of R do not vnish. Should there exist stte vector nnihilted by Â, then this vector would lso be nnihilted by the commuttors of the components of Â. These commuttors re proportionl to the components of spin. Therefore, for ny reltivistic quntum system endowed with spin the inequlity () cnnot be sturted. THE UNCERTAINTY RELATIONS FOR ANGULAR MOMENTUM Reclling the generlized uncertinty reltion for two opertors A nd B, A B [L x,l y ] (49) we cn write down uncertinty reltions for the components of ngulr momentum using the commuttors[] [6]. For exmple, we find L x L y [L x,l y ] = ħ L z (5) Fourier theory: The fct tht momentum cn be expressed s p = kħ llows us to define momentum spce wvefunction tht is relted to the position spce wvefunction vi the Fourier trnsform. A function f(x) nd its Fourier trnsform. F(k) re relted vi the reltions: f(x) = 1 F(k)eikx dk (51) π F(k) = 1 π f(x)e ikx dk (5) These reltions cn be expressed in terms of p with position spce wvefunction ψ(x) nd momentum spce wvefunction Φ(p) s: ψ(x) = 1 (p)eipx ħ dp (53) πħ (p) = 1 πħ ψ(x)e ipx ħ dx (54) Interntionl Journl of Engineering Sciences & Reserch Technology [78]

7 (IOR), Publiction Impct Fctor: Prsevl s theorem tells us tht[6]: f(x) dx = F(k) dk (55) These reltions tell us tht Φ(p), like ψ(x), represents probbility density. The function Φ(p) gives us informtion bout the probbility of finding momentum between p b: b P( p b) = (p) dp (56) Prsevl s theorem tells us tht if the wvefunction ψ(x) is normlized, then the momentum spce wvefunction Φ(p) is lso normlized ψ(x) dx = 1 (p) dp = 1 (57) It is fct of Fourier theory nd wve mechnics tht the sptil extension of the wve described by ψ(x) nd the extension of wvelength described by the Fourier trnsform Φ(p) cnnot be mde rbitrrily smll. This observtion is described mthemticlly by the Heisenberg uncertinty principle: X P ħ (58) We cn using p = kħ, k 1. Exmple (): A prticle of mss m in one-dimensionl box is found to be in the ground stte: Find x p for this stte. Solution: Using p = iħd dx we hve: nd: pψ(x) = iħ d dx [ ψ(x) = sin (πx ) sin (πx p ψ(x) = iħ d dx ( ħπ )] = ħπ cos (πx We found in the exmple bove tht p = for this stte. cos (πx ) (59) )) = ħπ sin (πx ) (6) p = ψ (x)pψ(x)dx (61) p = ψ (x)p ψ(x)dx = sin (πx ) ħ π = ħ π sin (πx ) dx ( ) [sin (πx )] dx πx = ħ π 3 1 cos ( ) dx = ħ π x 3 = ħ π (6) p = p p = ħ π = ħπ (63) Interntionl Journl of Engineering Sciences & Reserch Technology [783]

8 (IOR), Publiction Impct Fctor: fter clcultion (π 6) 1 x = ψ (x)xψ(x)dx = = = 4 = x (sin (πx )) dx sin (πx ) x sin (πx ) dx (64) x = ψ (x)x ψ(x)dx = x (sin ( πx )) dx = ( 3 6 π) (65) x = x x = (π 6) = 6) 1π π (π 1 =.57, so x p > ħ. CONCLUTION We conclude be mking the following points: *Our results on the uncertinty reltion re bsed on mesuring the extension of the sptil domin of the photon function *We divided the first moment of the energy distribution insted of the moment of the energy distribution see [1]. *It should be noted tht by dividing the first moment of the energy distribution by the totl energy we obtined similr nlysis to tht one of the clssicl quntum mechnics. And we shown tht the clssicl Heisenberg uncertinty reltion with respect to Fourier is > ħ see exmple (). REFERENCES [1] Iwo Bilynicki-Birul.Zofi Bilynick-Birul Heisenberg uncertinty reltions for photons(1), Center for Theoreticl Physics, Polish Acdemy of Sciences, Alej Lotnik ow 3/46, -668 Wrsw, Polnd. [] I. Bilynicki-Birul nd Z. Bilynick-Birul, Quntum Electrodynmics (Pergmon, Oxford, 1975), Chp. 9; I. Bilynicki-Birul nd Z. Bilynick-Birul, Phys. Rev. D35, 383 (1987); I. Bilynicki-Birul nd Z. Bilynick-Birul, J. Opt. 13, 6414 (11). [3] U. M. Tituluer nd R. J. Gluber, Phys. Rev. 145, 141 (1966); Brin J. Smith nd M. G. Rymer, New J. Phys.9, 414 (7). [4] R. J. Gluber, Phys. Rev. 13, 59 (1963); L. Mn-del nd E. Wolf, Opticl Coherence nd Quntum Optics (Cmbridge University Press, Cmbridge, U.K., 1995). [5] V. Grc es-ch vez, D. McGloin, M. J. Pdgett, W. Dultz, H. Schmitzer, nd K. Dholki,Phys. Rev. Lett. 91, 936 (3). [6] McGRAW-HILL New York Chicgo Sn Frncisco Lisbon London Mdrid Mexico City Miln New Delhi Sn Jun Seoul Singpore Sydney Toronto, McGrw-Hill ebooks. Quntum mechnics Demystified.Chp.4,5. [7] Bilynicki-Birul I nd Bilynick-Birul Z 6 Bems of electromgnetic rdition crrying ngulr momentum: the Riemnn Silberstein vector nd the clssicl-quntum correspondence Opt. [8] Bilynicki-Birul, photon wve function Progress in Optics, edited by E. Wolf (Elsevier, Amsterdm,1996), Vol.36; see lso ArXiv: qunt-ph/58. [9] Bilynicki-Birul I nd Bilynick-Birul Z 1975 Quntum Electrodynmics (Oxford: Pergmon) [1] Generlized theory of interference, nd its pplictions, S. Pnchrtnm, Proc. Indin Acd. Sci. A 44, 47-6 (1956). See lso reference [4] below for the formultion of Pnchrtnm s phse in quntum theoreticl lnguge. [11] The Berry phse s n pproprite correspondence limit of the Ahronov-Anndn phse in simple model, J. Christin nd A. Shimony, in Quntum Coherence, edited by J. Anndn (World Scientific, Singpore, 199) pp [1] E.P.Wigner in Group Theory nd its pplictions to the Quntum Mechnics of [13] Atomic Spectr,Acdemic Press,NY(1959); A.Vglic nd G.Vetri, Optics Communictions, 51,4(1984)39. [14] J. Schwinger in Quntum theory of Angulr Momentum, ed. L. Beidenhrn nd H. vn Dm, Acdemic Press, NY.(1965) [15] K.A. Milton, Rep. Prog. Phys (6). (66) Interntionl Journl of Engineering Sciences & Reserch Technology [784]

9 (IOR), Publiction Impct Fctor: [16] McGRAW-HILL New York Chicgo Sn Frncisco Lisbon London Mdrid Mexico City Miln New Delhi Sn Jun Seoul Singpore Sydney Toronto, McGrw-Hill ebooks. Quntum mechnics Demystified. Chp.6-1. [17] V.B.Berestetskii,E.M.Lifshitz,nd L. P. Pitevskii, Quntum electrodynmics, nd ed. (Pergmon Press Ltd., NY, 198). Interntionl Journl of Engineering Sciences & Reserch Technology [785]