Dissipative Relativistic Bohmian Mechanics

Transcription

1 [arxiv ] Dissipaive Relaivisic Bohmia Mechaics Roume Tsekov Deparme of Physical Chemisry, Uiversiy of Sofia, 1164 Sofia, Bulgaria I is show ha quaum eagleme is he oly force able o maiai he fourh sae of maer, possessig fixed shape a a arbirary volume. Accordigly, a ew relaivisic Schrödiger equaio is derived ad rasformed furher o he relaivisic Bohmia mechaics via he Madelug rasformaio. Three dissipaive models are proposed as exesios of he quaum relaivisic Hamilo- Jacobi equaio. The correspodig dispersio relaios are obaied. Tradiioally, he sae of maer is recogized from is volume ad shape properies. The solid sae possesses boh fixed volume ad shape, while he liquid sae maiais a fixed volume a variable shape. The gaseous sae has boh variable volume ad shape, adapig hem o fi he coaier. Usually he plasma is cosidered as he fourh sae of maer. From is defiiio as a eural mixure of charged paricles, however, i follows ha he radiioal plasma is a gas wih boh variable volume ad shape. Furhermore, ioic liquids (e.g. RTIL) ad crysals (e.g. NaCl) could be cosidered as liquid ad solid plasmas, respecively. The liquid ad solid meals are defiiely plasmas as well. The forgoig logic shows ha he oly possibiliy of he fourh sae of maer is o possess variable volume a fixed shape. The reaso for he differe saes of maer is he forces acig bewee he paricles of maer. The hermal eergy i solids is so small as compared o he poeial ieracios ha he posiios of he paricles are firmly fixed ad oly small vibraios aroud hem are prese. Tha is why he solids maiai boh fixed volume ad shape. A higher emperaure, he paricles i liquids sill cao separae each oher bu hey ca move, allowig he liquids o flow. I gasses, he hermal eergy is so high ha he poeial ieracios bewee paricles are egligible. I ay case, he classical poeials decrease wih icrease of he disace bewee he paricles ad a very dilue maer is always a ideal gas (he Boyle s law). From his perspecive he fourh sae of maer, which does o possess a fixed volume, cao be explaied by he classical ieracio poeials. The abiliy o maiai a ow shape eve i a very dilue sae requires ieracios, which are o depressed by he disace. A prese, he oly kow ieracio, beig idepede from he disace bewee paricles, is due o quaum mechaics. I is possible o geerae quaum paricles i such a way ha he quaum sae is defied oly for he whole sysem. Thus, he quaum sae of each paricle depeds o he saes of he ohers, wihou presece of ay classical poeial ieracios. This so-called quaum eagleme exiss eve if

2 he paricles are separaed by a large disace. Therefore, he fourh sae of maer could be a eagled oe. The quaum eagleme is quaiaively described via he Bohm quaum poeial. 1 I is laer recogized ha he Bohm poeial is a iformaio poeial ad represes he Fisher iformaio force. 3 Due o he very close relaioship bewee iformaio ad eropy, 4 he iformaio forces are eropic ad, hece, hey differ from he usual poeial ieracios. The lack of volume resricios suggess ha he fourh sae could probably be prese a cosmological scale, which requires a relaivisic reame of he quaum problem. I he begiig of he previous ceury Eisei s relaiviy ad quaum mechaics have reformulaed physics. The square roo i he special relaiviy expressio 4 E m c p c for he eergy of a paricle geeraes, however, some quaizaio problems ad a usual way o ge hrough is o cosider is quadrae. Iroducig he eergy E ˆ i ad momeum ˆp i operaors from quaum mechaics io ˆ 4 E m c pˆ c yields he Klei-Gordo equaio 5 ( mc / ) 0 (1) where / c is he d Alember operaor. This parial differeial equaio describes scalar bosoic fields ad reduces paricularly o he wave equaio 0 for phoos, sice heir res mass m 0 is zero. The Klei-Gordo equaio (1) suffers, however, serious problems wih he probabilisic ierpreaio of he wave fucio. 5 The reaso for his could be a improper quaizaio of he res mass eergy, which mus be persise i space ad ime. To resolve he problem oe ca iroduce a eergy operaor, where he res mass eergy remais cosa. Subsiuig Eˆ 4 mc i io he quadrae Eˆ m c pˆ c of he relaivisic eergy yields aoher fudameal equaio for he relaivisic quaum mechaics ( m/ i ) 0 () which reduces also o he wave equaio for phoos. Oe ca easily recogize i Eq. () he relaivisic Schrödiger equaio, which ca be rewrie i he aleraive form i / m U Hˆ / mc (3)

3 hus accouig for a exeral poeial eergy U as well. The sadard Hamiloia operaor reads E m c m c bewee he full relaivisic ˆ / H m U. A simple relaioship 4 E ad orelaivisic eergy eigevalues follows from Eq. (3), which resembles he Eisei expressio. I he limi c Eq. (3) reduces aurally o he orelaivisic Schrödiger equaio, while he relaivisic eergy expads i series as E mc /mc. To demosrae ha he relaivisic Schrödiger equaio (3) overcomes he probabiliy problems of he Klei-Gordo equaio (1) le us iroduce he Madelug rasformaio of he complex wave fucio exp( is / ), where S is he real quaum phase ad is he local probabiliy desiy. Thus, Eq. (3) reduces sraighforward o he followig wo real equaios ( S/ m) S ( S )( S ) / m U Q 0 (4) where is he sadard 4-gradie operaor wih. The firs equaio is he coiuiy equaio, while he secod oe is he relaivisic quaum Hamilo-Jacobi equaio. The laer differs from he classical aalog 6 via he addiioal erm Q /m, beig he relaivisic Bohm quaum poeial. 7 As is see, he laer ca be srog eve a vaishig maer desiy as required for he fourh sae. Sice he probabiliy is coserved, 3 dx 1, he direc iegra- 3 io of he coiuiy equaio leads o he followig expressio ( Sd x ) 0. I reflecs he coservaio of eergy E 3 Sd x, which is a iegral of moio, idepede of ime. The sysem of Eq. (4) defies he relaivisic Bohmia mechaics. If oe is ieresed i ope quaum sysems, 8 he relaivisic Hamilo-Jacobi equaio ca be furher exeded o 9 S ( S )( S ) / m U Q S (5) where he ew erm o he righ-had side describes he quaum phase decay wih a collisio frequecy. If he laer is high eough, oe ca eglec he secod oliear erm ad subsiuig equaio S S U Q io he coiuiy Eq. (4) yields a relaivisic quaum elegraph-like [ ( U Q S) / m] [ ( U Q) / m] (6)

4 The las expressio follows from he ime derivaive of he coiuiy equaio, liearized o S agai. Sice he Bohm quaum poeial is a oliear fucio of he probabiliy desiy, oe ca liearize furher Eq. (6) o o obai he followig liear equaio ( / m) ( U / m) 0 (7) I he case of a free paricle ( U 0 ) he ime-space Fourier rasformaio of Eq. (7) provides he dispersio relaio i ( / m) ( k / c ) 0 (8) A low frequecy Eq. (8) simplifies o he imagiary orelaivisic soluio i( k / m) /, while a high he specrum is modulaed by he Zierbewegug frequecy mc /. I geeral, Eq. (8) is a complex relaioship bewee hree impora characerisic frequecies reflecig he collisios, Zierbewegug ad super-relaivisic propagaio wih frequecy ck. Aoher dissipaive model implies radiaive fricio, 10 where he correspodig relaivisic Hamilo-Jacobi equaio reads S ( S)( S) / m U Q S (9) 3 If he characerisic ime equals o e /6 mc, for isace, he erm o he righ-had side 0 describes emissio of phoos. Omiig agai he oliear erm i Eq. (9) provides he rae of chage of he quaum phase, S S U Q. Thus, applyig a ime derivaive wice o he coiuiy equaio (4), eglecig he oliear S -erms ad usig he las equaio yields 3 [ ( U Q) / m] (10) Oe ca liearize furher Eq. (10) for low probabiliy desiy gradies o obai 3 ( / m) ( U / m) 0 (11)

5 I he case of a free paricle ( U 0 ) he ime-space Fourier rasformaio of Eq. (11) provides he followig dispersio relaio 3 i ( / m) ( k / c ) 0 (1) 3 which simplifies a low ad high frequecy o i( k / m) / ad i ( mc / ), respecively. Comparig Eq. (1) ad Eq. (8) uveils a effecive fricio coefficie. The las dissipaive model cosidered i he prese paper implies diffusio of he quaum phase S, where he correspodig relaivisic Hamilo-Jacobi equaio reads 11 (13) S ( S )( S ) / m U Q D S The ew erm describes reacive diffusio of S wih a diffusio cosa D. Omiig agai he oliear erm i Eq. (13) provides he rae of chage of he quaum S U Q D S phase. Iroducig i io he ime derivaive of he coiuiy Eq. (4), beig liearized o S, yields [ ( U Q) / m] D ( S / m) (14) Oe ca liearize furher Eq. (14) for low probabiliy desiy gradies o obai D ( / m) ( U / m) 0 (15) I he case of a free paricle ( U 0 ) he ime-space Fourier rasformaio of Eq. (15) provides he dispersio relaio idk ( / m) ( k / c ) 0 (16)

6 3 A low ad high frequecies i simplifies o i( k / m) / D ad i D( mc k / ), respecively. Comparig Eq. (16) wih Eq. (8) uveils a effecive fricio coefficie Dk k. I he super-relaivisic case ( ck ) Eq. (16) reduces o simple diffusio, which is also he case of Eq. (1) wih a diffusio cosa D c e /6 0mc. Therefore, i he case of a complee relaivisic reame of he phase diffusio, he relaivisic quaum Hamilo-Jacobi equaio reads S ( S )( S ) / m U Q D S (17) The paper is dedicaed o David Bohm ( ). 1. D. Bohm, Phys. Rev. 85 (195) 166. D. Bohm ad B. J. Hiley, Foud. Phys. 5 (1975) M. Regiao, Phys. Rev. A 58 (1998) C. Shao, Bell Sys. Tech. J. 7 (1948) W. Greier, Relaivisic Quaum Mechaics: Wave Equaios, Spriger, Berli, L. D. Ladau ad E. M. Lifshiz, Theoreical Physics vol. : Field Theory, Nauka, Moscow, H. Nikolic, Foud. Phys. Le. 18 (005) A. B. Nassar ad S. Mire-Ares, Bohmia Mechaics, Ope Quaum Sysems ad Coiuous Measuremes, Spriger, Berli, R. Tsekov, New Adv. Phys. 3 (009) R. Tsekov, Fluc. Noise Le. 15 (016) R. Tsekov, New Adv. Phys. 8 (014) 111