Eulerian and Newtonian dynamics of quantum particles

Transcription

1 Eulean and ewonan dynas of uanu pales.a. Rasovsy Insue fo Pobles n Means, Russan Aadey of enes, Venadsogo Ave., / Mosow, 956, Russa, Tel , E-al: as@pne.u We deve e lassal euaons of ydodynas (e Eule and onnuy euaons), fo w e ödnge euaon follows as a l ase. I s sown a e sasal enseble oespondng o a uanu syse and desbed by e ödnge euaon an be onsdeed an nvsd gas a obeys e deal gas law w a uly osllang sgn-alenang epeaue. Ts sasal enseble pefos e oplex oveens onssng of soo aveage oveen and fas osllaons. I s sown a e aveage oveens of e sasal enseble ae desbed by e ödnge euaon. A odel of uanu oon wn e ls of lassal eans a oesponds o e ydodyna syse onsdeed s suggesed. PAC nube(s): 3.65., 3.65.Ta, 5..Jj, 45.5.x, 45..-d I. ITRODCTIO Te eods used o desbe e oon of pales n lassal and uanu eans ae fundaenally dffeen. I s well nown a ee ae seveal alenave foulaons of lassal eans [,], fo w, fo ou puposes, we glg only wo: e ewonan foulaon and e Halon-Jaob eoy. In e ewonan foulaon of lassal eans, e oon of a pale s desbed by ewon's seond law (o s alenave eods n e fo of, e.g., Halon o Lagange euaons): & & () w allows e alulaon of e pale ajeoy,, v ) a e pesbed nal ondons of ( v ( ), & ( ) () and, us, a ea nsan, spefes a w pon of spae e gven pale s loaed. In e Halon-Jaob eoy, e oon of an enseble of denal non-neang pales, w we all e Halon-Jaob enseble, s onsdeed ae an e oon of a sngle pale. Ts enseble s aaezed by a densy (, ), w sasfes e onnuy euaon: dv ( v) (3)

2 wee v (, ) (4) s e veloy feld of e enseble, and e funon (, ), w as e sense of aon, sasfes e lassal Halon-Jaob euaon: (5) Te ajeoy of an ndvdual pale n e Halon-Jaob eoy an be found ee usng Jaob s eoe [,] w e oplee negal of E. (5) o by solvng e syse of odnay dffeenal euaons: & (, ) (6) usng a soluon of e Halon-Jaob euaon (5). Te neonneon of bo foulaons of lassal eans s well nown and obvous: usng e Halon-Jaob euaon (3) as a soue and sepaang e poenal enegy, we an onsu ewon s euaons (), and ve vesa. sng a soluon of ewon s euaons (), we an onsu a oplee negal of e Halon-Jaob euaon (3) [,]. Fo a aeaal pespeve, euaon () desbes e aaess of e Halon-Jaob euaon (5). In uanu eans, e oon of a pale s desbed by a wave funon ψ (, ), w s e soluon of e ödnge euaon. Te lassal noon of e ajeoy of a sngle pale n uanu eans s eanngless, and s only possble o dsuss e pobably of fndng e pale a dffeen pons n spae bu no ow e pale ae o a paula pon. Aodng o Bon s pobabls nepeaon [3,4], e pobably densy of fndng e pale a a gven pon s popoonal o ψ (, ). On s bass, one an ague a, n uanu eans slaly o e lassal Halon-Jaob eoy, we do no onsde a sngle pale bu ae an enseble of denal non-neang pales, w an be alled e ödnge enseble. ueous aeps o onsu a uanu eans of ndvdual pales usng lassal oneps, nludng "pon pale", "veloy", and "lassal ajeoy", wee unsuessful [5,6]. Ts ls sould also nlude e so-alled Boan eans [7-9], w anno be onsdeed a opleely lassal foulaon of uanu eans beause e veloy feld, n w a oon of ndvdual pales s alulaed, s e soluon of e ödnge euaon. Fo s eason, Boan eans sould be alled Boan neas, wle e oespondng dynas s sll desbed by e ödnge euaon.

3 Quanu eans s assoaed w lassal eans oug e Halon-Jaob eoy [4]. If we epesen e wave funon n e fo of ψ exp( ) and sepaae e ödnge euaon no eal and agnay pas, one aves a bo e onnuy euaon (3) and e followng euaon: 3 w, foally, s e Halon-Jaob euaon fo a lassal pale ovng n a poenal feld wee ef (, ) (8) (, ) (9) s so-alled uanu poenal assoaed w e wave popees of e uanu pale. Te oon of a uanu pale n an exenal poenal feld (), a leas foally, s euvalen o e oon of a lassal pale n a poenal feld (8). Fo s eason, one an say a e ödnge enseble s a uanu Halon-Jaob enseble, w we all e Halon-Jaob-ödnge enseble. Te ödnge euaon n e fo of Es. (3) and (7) s used o jusfy e anson fo uanu o lassal eans n e l : n s l, e uanu eans beoes e Halon-Jaob eoy fo a lassal pale. One of nepeaons of uanu eans, Boan eans, s based on s analogy. sng (7) and (8), one an foally we ewon's seond law () fo a uanu pale as & & ef () Tus, e oon of uanu pale, a leas foally, an be alulaed wn e ls of lassal eans f e effeve poenal feld (8) s nown. Howeve, e dffuly of s appoa s a e uanu poenal (9) s no a pedeened funon of e oodnaes, as n lassal eans. Insead, depends on e pobably densy (e densy of e enseble) (, ), w an be found usng e soluon of e ödnge euaon. If one onsdes euaons (3), (4), and (7) as a ydodyna despon of e Halon- Jaob-odnge enseble (Madelung flud), n s ase, e uanu poenal (9) plays e ole of pessue. A Madelung flud s opessble one, bu e pessue does no depend on e densy, as does fo a lassal opessble flud; depends on e seond devaves of (7)

4 e densy w espe o e oodnaes. Tus, even su a se-lassal ydodyna odel as fundaenal dffules, bo fo e pespeve of e lassal nepeaon of uanu eans and fo e nueal (ydodyna) odeg of uanu pale oon. In s pape, we sow a ee s anoe ydodyna foulaon of uanu eans n w ee ae no su pobles and a s lose o e ydodyna despon of flows of lassal nvsd deal gases. II. VARIATIOAL PRICIPLE I CLAICAL AD QATM MECHAIC Te vaaonal pnples n pyss play an poan ole. On e one and, ey ae a foal way o deve e fundaenal laws of naue [,,], and, on e oe and, ey ae pa of a plosopal pnple a sows a aue s aanged aonally and spends a nal effo n s developen. Te euaons of oon of a lassal syse of pon pales ae deved fo e leas aon pnple []: δ () unde e ondon δ ( ) δ ( ) () wee L(, &, ) d (3) ( ), ae e onsans, L (, &, ) s e Lagange funon, and () ae e genealzed oodnaes of e syse. Fo one pon pale, ovng n a poenal feld, wle, fo e ue ajeoes of e pale, v L ( ) (4) v ( ) d 4 (5) n (6) (o, oe pesely, ends o a seady value) fo any nsan, unde ondons () and onsan,.

5 Te expesson (5) an be ewen n e fo d d v ( ) d (7) o, usng e δ -funon, as ( ( )) d v δ ( ) s dd (8) d wee s ( ) s a oon law of e pale and e negal w espe o s aen ove e ene spae (e ene onfguaon spae fo e syse of pales). Le us onsde e Halon-Jaob enseble onssng of a se of denal non-neang pales w dffeen nal ondons. Te Halon-Jaob enseble oespondng o a sngle pale an be epesened as a opessble flud (gas), wee e flow s desbed by e veloy feld (4). In s ase, we sould ansfe fo an ndvdual (Lagangan) despon, w s used fo a sngle pale, o a onnual (Eule) despon []. As a esul, one obans wee, n aodane w (4), d d ( v ) (9) v () ung (8) ove all of e pales n e Halon-Jaob enseble, one obans δ ( s ( ) ) ( ) dd () s Le us un o a onnuous dsbuon of pales n e Halon-Jaob enseble ove spae. Fo s pupose, one nodues e densy of pales ove spae (densy of enseble): Evdenly, s e nube of pales n a volue Ω. δ ( ( ) (, ) ) () s s (, ) d Ω I s onvenen o edeene e densy () usng In s ase, (, ) as a noalzaon (, ) δ s ) s ( ( ) 5

6 (, ) d Ω and an be onsdeed a pobably densy o fnd e pale of e Halon-Jaob enseble a a gven pon. E. () en aes e followng fo: (, ) ( ) dd (3) Tus, e ue oon of e Halon-Jaob enseble (and us e ue oon of ndvdual pales n s enseble) oesponds o e funons (, ) and (, ) a sasfy e ondon (3) and e ondons () and () sulaneously. We assue e funons (, ) and (, ) ae ndependen. By vayng e expesson n E. (3) w espe o (, ) and euang e vaaon o zeo, one obans e Halon-Jaob euaon (5); by vayng e expesson n E. (3) w espe o (, ) and euang e vaaon o zeo, one obans e onnuy euaon fo e Halon-Jaob enseble (3), (4). Tus, we ave a e vaaonal pnple of Halon-Jaob eoy n lassal eans. Le us now onsde uanu eans. Te ödnge euaon an be wen n e fo of (3), (4), and (7), w s foally denal o e lassal Halon-Jaob eoy fo a pale n e effeve poenal feld (8), w depends on e densy of e enseble. Genealzng e expesson (3) o uanu pales, one an we (, ) ef ( ) dd (4) Ts genealzaon s based on e foal slay of euaons (5) and (7). Te uanu poenal (9) an be wen as 4 By subsung E. (8) n E. (4) and onsdeng E. (5), afe sple ansfoaons, one obans (, ) ( ) dd ( ) 8 d d 4 wee e negal ( d ) s aen ove e sufae boundng e spae egon n w e (5) (6) Halon-Jaob-odnge enseble oves and sufae. d s e veo eleen of aea of s 6

7 We assue a, a e bounday of e egon ouped by Halon-Jaob-ödnge enseble, ( d ) Ts ondon an always be sasfed beause one an always onsde a egon of spae o be u lage an e sze of e Halon-Jaob-ödnge enseble, su a e densy of e enseble a e egon bounday s a onsan o even eual o zeo. As a onseuene, one obans (, ) ( ) dd 8 By de alulaon, s easy o e a e ndependen vaaon of E. (7) w espe o (, ) leads o e onnuy euaons (3) and (4), wle e vaaon w espe o (, ) leads o e Halon-Jaob euaon (7) w e effeve poenal enegy (8). Togee, ese euaons fo e ödnge euaon. Tus, e anson fo e lassal vaaonal pnple fo E. (3) o e uanueanal vaaonal pnple fo E. (4) s jusfed based on e foal slay of euaons (5) and (7), aloug, n uanu eans, e effeve poenal enegy (8) s vaed by vayng e densy (, ). Te expesson (5) sows a e uanu poenal wee an be wen as (7) (8) 8 (9) 4 I s neesng o noe a bo oponens (9) and (3) of e uanu poenal ene no e Halon-Jaob euaon (7) fo a uanu pale, wle only e oponen (3) of e uanu poenal appeas n e negal vaaonal pnple (7). Based on s analyss, we onlude a pesely (9) ay be alled e uanu poenal, wle e oponen appeas n e euaon (7) due o vayng e poenal (9) w espe o densy. Tus, fo e pon of vew of e vaaonal pnple (7), e oon of e uanu pale s euvalen o e oon of a lassal pale n a poenal feld of ef 8 (3) 7

8 Ts effeve poenal dffes fo e poenal (8) based on e foal slay of e ödnge euaon n e fo of (7) and e Halon-Jaob euaon (5). As we sow below, pesely e poenal (9) s esponsble fo e unusual (non-lassal) beavo of uanu pales. III. HYDRODYAMIC OF THE HAMILTO-JACOBI-CHRÖDIGER EEMBLE A. Pale oon n a uly osllang feld Le us onsde e oon of a lassal pale n an exenal (slowly angng) poenal feld (, ), on w e followng uly osllang foe sulaneously as: f (, ) f (, ) f (, ) sn (3) wee s e feueny and f (, ), f (, ) ae e ve, w depend on e oodnaes s and ae wealy dependen on e. Te feueny sasfes e ondon >> T, wee T s e aaes e of e pale oon n an exenal feld (, ) a f (, ) f (, ). Te wea dependene of e ve f (, ), f (, ) on e e eans a e aaes e of e ange s u bgge an. s nde e aon of e exenal foe (3), a pale pefos a oplex oon a onsss of e aveage oon along a soo ajeoy R ( ) ( ) and e fas osllaons w a feueny aound. I s well nown [] a, aveaged ove e osllaon, e oon of e pale s desbed by e euaon wee ef s R& & ef (33) (, ) (, ) f (, ) (, ) fs 4 Tus, e aon of e uly osllang foe (3) esuls n e eaon of an addonal poenal enegy of f (, ) f 4 w s sply e ne enegy of e osllaoy oon. p s (, ) s (34) (35) 8

9 Fo exaple, wen a aged pale oves n e feld of an eleoagne wave, a uly osllang foe (3) s aused by an ele feld: f (, ) E(, )exp( ), wee s e ele age of e pale and E (, ) s e slowly vayng aplude of e ele feld. In s ase, e addonal poenal enegy (35) aes e fo p and s alled e pondeoove poenal [3]. E (, ) 4 (36) B. Eule euaon By opang Es. (34) and (3), one an onlude a e uanu poenal (9) an be epesened, a leas foally, as an addonal poenal enegy (35) a ases as a esul of e aon of e uly osllang foe: f (, ) (37) Tus, e oponen of e uanu poenal (9) an be explaned wn e ls of lassal eans f one assues a a uly osllang foe (37) oe an an exenal poenal (, ) as on a pon pale. By defnon, s foe as a vey g feueny and, oespondngly, a lage aplude ~. Of ouse, s foe s no a ue lassal one beause depends on e densy of e Halon- Jaob-ödnge enseble. Te nepeaon of s foe s onsdeed below. Te analyss above sows a e uanu pale an be onsdeed a lassal pale ovng n an exenal feld (, ), on w an addonal uly osllang foe (37) as. Tunng o e Halon-Jaob-odnge enseble, s easy o see a e uly osllang foe pe un volue of e Halon-Jaob-ödnge enseble s f (, ) (38) In e onnual despon, one an we e euaon of oon fo e Halon-Jaob- ödnge enseble as v ( v) v (39) w s e ydodyna Eule euaon. Ts euaon sould be solved ogee w e onnuy euaon: dv ( v ) 9 (4)

10 Hee and below, e subsp efes o e ue (op) paaees of e Halon- Jaob-ödnge enseble, nludng e u osllaons, wle e paaees wou e subsp efe o e aveage oon of e enseble. Te las e on e g-and sde of E. (39) an be wen as ( p ), wee p (4) an be nepeed as e pessue n e Halon-Jaob-ödnge enseble, w s a uly osllang funon of e. Te uanu Halon-Jaob-ödnge enseble s an nvsd gas w nenal pessue (4). Howeve, s pessue s no e onvenonal pessue a ous n a lassal gas beause s sgn-alenang and eefoe anno be explaned by e lassal ne odel [4]. Foally, e elaon (4) an be onsdeed e euaon of sae of a lassal deal gas p T. In s ase, e epeaue of e Halon-Jaob-ödnge gas s deened by e expesson T ( ) (4) w s also sgn-alenang; fo s eason, anno be onsdeed as an aveage ne enegy of e ando oon of pales. Te veloy v a enes no e Eule euaon (39) and e onnuy euaon (4) s e nsananeous veloy of e pales n e enseble, wle e veloy defned by E. (4) s an aveage ove e fas osllaons of e pales' veloes. A onsan, e Eule euaon (39) as a soluon n e fo of e poenal flow of e Halon-Jaob-ödnge enseble: v (43) wee e funon (, ) s dffeen an e aon defned by e ödnge euaon (7) and sasfes e Halon-Jaob euaon (44) Te densy sasfes e onnuy euaon (4) w e veloy (43). Euaon (44) dffes fo e Halon-Jaob euaon (5) fo a lassal pale n a onans a uly osllang poenal a depends on e densy of e Halon-Jaob- ödnge enseble. Foally, euaon (44) an be onsdeed an odnay Halon-Jaob euaon fo a lassal pale ovng n a poenal feld

11 Te oponen ef (45) (46) an be alled a ue uanu poenal n onas o aveagng e pale oon ove e fas osllaons. n E. (9), w s e esul of Euaons (39) and (4) ae foally e ydodyna euaons of an nvsd deal gas w a uly osllang sgn-alenang epeaue (pessue) and an easly be solved nueally usng e onvenonal eods of lassal opuaonal flud dynas. Passng fo e Halon-Jaob eoy o ewon's despon of a uanu pale, one an foally we e ewon's law as dv ef d (47) IV. DIRECT DERIVATIO OF THE CHRÖDIGER EQATIO A. Pale n a poenal feld In e pevous seon, based on ae geneal bu non-goous easonng, we ae o euaon (39), w, ogee w e onnuy euaon (4), sould be euvalen o e ödnge euaon beause gves a dealed despon of e oon of a uanu pale (oe pesely, e Halon-Jaob-ödnge enseble); oweve, aveagng ese euaons ove e fas osllaons sould lead o e ödnge euaon. Le us sow a e oon of e Halon-Jaob-ödnge enseble, as desbed by euaons (39) and (4), s aually desbed by e ödnge euaon wen aveaged ove e fas osllaons. Ts analyss s oe sply pefoed usng e Halon-Jaob euaon (44) and e onnuy euaon (4) w e veloy n E. (43). Le us see e soluon of Es. (4), (43), (44) n e fo, (48) wee and ae slowly vayng funons w a aaes e sale and ae e uly osllang funons w feueny a sasfy e ondons T >> and ; (49)

12 Hee,... denoes aveagng ove e fas osllaons. By subsung E. (48) no Es. (4), (43) and (44), one obans ( ) ) ( (5) dv (5) Wen wng euaon (5), one assues a <<, so ) (. Ts ondon s poven below. Aveagng euaons (5) and (5) ove e fas osllaons wle onsdeng E. (49) allows e sepaaon of e fas and slow oponens. As a esul, one obans (5) dv (53) ( ) ) ( (54) ( ) dv (55) By onsdeng osllaons and sall, we an es ou onsdeaon o e ea appoxaon, n w euaons (54) and (55) an be wen as ) ( (56) dv (57) We sow below a e onveve es ) ( and, w ae elaed o e aveage flow of e Halon-Jaob-ödnge enseble, ae u less an e oe es n euaons (56) and (57). Tus, e es n Es. (56) and (57) desbng e onveve anspo, w s assoaed w e aveage flow of Halon-Jaob-ödnge enseble, an be negleed. As a esul, euaons (56) and (57) ae e fo

13 (58) Euaons (58) and (59) ave e soluons dv (59) sn (6) Le us opae e onveve e ( ) (6) w a uly osllang oponen n euaon (56) usng e soluon (6). Beause e osllaons of ese es ae sgn-alenang w zeo ean values, aes sense o opae e oo-ean suae values. We ave e esaons wee ( ) ( ) V ~ L ~ T V ~ s e aaes veloy of e ean flow of e Halon-Jaob ödnge enseble, L s e aaes spaal sale of e ean flow of e Halon- Jaob-ödnge enseble, and T L V s e aaes e of e ange of e paaees of e ean flow of Halon-Jaob-ödnge enseble. Beause we onsde vey fas osllaons sasfyng e ondon T <<, e esaon above sows a e e ( ), n ean, s u less an e uly osllang e and an be dsaded, w was done n euaon (58). laly, one an opae e oponens of Es (6) and (6). One obans and V ~ L n euaon (57) usng e soluons T L 3

14 ~ L Gven a << T less an e e e ao : n e ase onsdeed, one onludes a e e, n ean, s u and an be negleed, w was done n euaon (59). Le us esae wee ~ L ~ VLT T VL s e sale of aon fo e ean flow of e Halon-Jaob-ödnge enseble. Fo uanu pales, and us one onludes a e ondon << s sasfed n ean and a e ge e feueny n opason w pesely s sasfed. sng e soluons (6) and (6), s easy o oban T e oe (6) 8 (63) (64) 4 ubsung Es. (6) oug (64) no euaons (5) and (53) fo e ean oon of e Halon-Jaob-ödnge enseble gves 8 dv 4 (65) (66) Ts se of euaons ondes w euaons (3), (4) and (7) of e Halon-Jaob eoy, w e uanu poenal (5) and, onseuenly, w e ödnge euaon fo e wave funon ψ exp( ). We ave poven a euaons (39) and (4) ae euvalen o e ödnge euaon by easonng fo e Halon-Jaob euaon (44). Te sae esul an be obaned by onsdeng e Eule euaon (39) dely and wng e veloy of e Halon-Jaob enseble n e fo v v w, wee w s e uly osllang oponen of e veloy sasfyng e ondon w and v s e ean veloy of flow of e Halon-Jaob- ödnge enseble. 4

15 B. Caged pale n an eleoagne feld We onsdeed e pale n an exenal poenal feld and deonsaed a e oon of e Halon-Jaob-ödnge enseble, beng desbed by Es. (39) and (4), afe aveagng ove e fas osllaons, s desbed by e ödnge euaon. ow, we wll sow a s saeen s also ue fo a non-elavs spnless pale ovng n an exenal eleoagne feld. Le e Loenz foe age, F E [ vh ] a on a aged pale, wee s e pale A E ϕ, H oa ae e exenal ele and agne felds, and ϕ, A ae e sala and veo poenals of e eleoagne feld. Te naual genealzaon of E. (39) s as follows: v ( v) v E [ vh] Le us assue a e veo and sala poenals of e exenal eleoagne feld, A and ϕ, ae slowly vayng funons n a e aaes e of e angng s (67) T <<. ubsung e ele and agne felds, expessed n es of e sala and veo poenals of e eleoagne feld, no E. (67) gves Gven a v A ( v) v [ voa] ϕ (68) we ewe E. (68) n e fo ( v ) v v [ v o v ] v v A ϕ [ v ov A] (69) Euaon (69) as a soluon n e fo of wee e funon sasfes e euaon v A (7) v ϕ (7) 5

16 6 w, by onsdeng E. (7), s e Halon-Jaob euaon fo a non-elavs pale n an exenal eleoagne feld w an addonal osllang poenal (4): ϕ A (7) Coespondngly, e onnuy euaon (4) aes e fo dv A (73) As n e pevous seon, we see e soluon of euaons (7) and (73) n e fo of (48) and (49). ubsung E. (48) no Es. (7) and (73) gves ( ) A A (74) dv A A (75) Hee, as above, s assued a, on aveage, <<, so ) (. Aveagng euaons (74) and (75) ove e fas osllaons, wle onsdeng E. (49) allows e sepaaon of e fas and e slow oponens. As a esul, one obans A (76) dv A (77) ( ) A (78) ( ) dv A (79) Assung a e osllaons and ae sall, euaons (78) and (79) an be wen n ea appoxaon n e followng fo: A (8)

17 7 dv A (8) By analogy w e pevous seon, an be sown a e es n euaons (8) and (8) desbng e onveve anspo w e ean flow of e Halon-Jaob-ödnge enseble ae, on aveage, sgnfanly less an e oe es n ese euaons and an be negleed. As a esul, Es. (8) and (8) ae e fo (8) dv (83).e., ey onde w Es. (58) and (59) fo a pale n a sala poenal feld. Euaons (8) and (83) ave e soluons (6) and (6), fo w e expessons (6) - (64) follow. ubsung Es. (6) oug (64) no euaons (76) and (77) fo e ean oon of Halon-Jaob enseble leads o 4 8 A (84) dv (85) I s easy o e a euaons (84) and (85) ae euvalen o one ödnge euaon fo a aged pale n an exenal eleoagne feld: ψ ϕ ψ A (86) wee ) exp( ψ. Tus, we ave sown a e euaons of ydodyna ype (67) and (4) w a uly osllang sgn-alenang poenal (4) ae euvalen o e ödnge euaon fo a spnless pale n an exenal eleoagne feld: a vey g feuenes, an aveage flow of e Halon-Jaob-ödnge enseble s desbed by e ödnge euaon (86). C. yse of neang pales Le us onsde a syse of neang pales.

18 8 A lassal Halon-Jaob enseble fo a syse of neang pales s desbed by e Eule euaon n e onfguaon spae ),..., ( ) ( v v v (87) and onnuy euaon ) ( v (88) wee ),,..., ( s e densy of e Halon-Jaob enseble n e onfguaon spae ),..., (, ),..., ( v v V s e veloy of e pon epesenng e syse n e onfguaon spae, ),,..., ( V s e veloy feld of e Halon-Jaob enseble n e onfguaon spae, ),,..., ( s e poenal enegy of e lassal neang pales, dependng on e oodnaes of all pales, s e ass of e pale, and, wee,...,. A naual genealzaon of E. (39) o e syse of neang pales gven E. (87) s as follows: ) ( v v v (89) Euaon (89) as a soluon n e fo v (9) wee e funon ),,..., ( sasfes e Halon-Jaob euaon: (9) As befoe, we see e soluons of euaons (88), (9) and (9) n e fo of (48) and (49). ubsung E. (48) no Es. (88), (9) and (9) gves ( ) ) ( (9) (93) Aveagng euaons (9) and (93) ove e fas osllaons, wle onsdeng E. (49) and sepaang e fas and slow oponens gves

19 9 (94) (95) ( ) ) ( (96) ( ) (97) Assung a e osllaons and ae sall, we ae esed o e ea appoxaon; usng s appoxaon, Es. (96) and (97) an be wen n e fo (98) (99) Euaons (98) and (99) ave e soluons sn () () sng soluons () and (), s easy o sow a () 8 (3) 4 (4) ubsung Es. () oug (4) no euaons (94) and (95) fo e aveaged paaees of e Halon-Jaob-ödnge enseble leads o 4 8 (5) (6)

20 I s easy o see a e se of euaons (5) and (6) s euvalen o e ödnge euaon fo a syse of neang pales: ψ ψ ψ (7) wee ψ (,...,, ) exp( ) s e wave funon of e syse of neang pales as defned n e onfguaon spae. Tus, e Halon-Jaob eoy (88) and (89) fo a syse of neang pales w a uly osllang pessue (4) onans e uanu eans as a g ase a vey g feuenes. V. DYAMIC OF IDIVIDAL QATM PARTICLE I s well nown a e Eule euaon fo an deal gas an be obaned fo an analyss of e oon and so-ange neaon of e uludes of pales [4]. One an expe a euaon (39) an also be obaned fo an analyss of e oon of ndvdual uanu pales. We sow below a, wn e ls of lassal eans, one an we e euaons of oon of ndvdual pales, w edue o euaons (39) and (4) fo e Halon-Jaob enseble. Appaenly, ee ae dffeen euaons of oon of a lassal pale oespondng o euaon (39) fo e Halon-Jaob enseble. Le us onsde e foal lassal euaon of oon of ndvdual pales, leadng o euaon (39): dv a d δ ( ) (8) d V d wee V s e veloy of an ndvdual pale, a s soe funon of e, ae e onsan ve, and δ ( ) s e Da dela-funon, wee, fo any volue Ω: δ ( ) d f pon les wn e volue Ω and s oewse eual o zeo. Te ve aδ ) an be onsdeed poenal foes a ae eaed by e δ -soues ( loaed a e pons n spae. Ω

21 Te fs euaon (8) s e onvenonal ewon's seond law. Tus, e se of euaons (8) desbes e oon of a lassal pale n an exenal poenal feld n e pesene of spaally dsbued saeng enes n e fo of poenal δ -soues. Le us onsde e pase spae of e pale (, V ). Te sae of e syse a any gven nsan an be epesened by a pon n e pase spae. Le us nodue a sasal enseble onssng of a se of denal non-neang syses. Te sasal enseble s epesened by e ulude of pons n e pase spae. We noe a e sasal enseble s defned n e pase spae n onas o e Halon-Jaob enseble, w s defned n e onfguaon spae. To desbe e sasal enseble, we use e sandad appoa of sasal pyss [4]. Le us nodue e densy of e sasal enseble n e pase spae P(, V, ). If s densy s noalzed o uny, P(, V, ) an be onsdeed e pobably densy a a pale as e pedeened poson and veloy. Te densy P(, V, ) sasfes e onnuy euaon n pase spae, w, fo syse (8), as e fo P ( V) P ( ) aδ P (9) V Hee, as usual, e poson and e veloy V of e pales ae onsdeed o be ndependen. Due o e neaon of ndvdual pales w e δ -soues, e dsbuon funon as a sall-sale suue w e aaes spaal sale on e ode of e ean dsane beween adjaen δ -soues. If one assues a e aveage dsane beween adjaen δ - soues s u less an e aaes spaal sale of oon of e sasal enseble, one an aveage euaon (9) ove a sall volue d Ω w a sze u lage an e dsane beween adjaen δ -soues bu u less an e aaes spaal sales of oon of e sasal enseble. By defnon of δ -funon dω P(, V, ) δ ( ) d P(, V, ) () f e pon les wn e volue d Ω ; oewse, s negal eual o zeo. As a esul of e aveagng of euaon (9) ove e volue d Ω, one obans P a ( ) P P (,, ) P V V V V dω () dω wee e suaon s ove all δ -soues nsde e volue d Ω ;

22 P(, V, ) P(, V, ) d () dω dω s e aveaged densy of sasal enseble n pase spae ove e volue d Ω. If e δ -soues ae dsbued andoly n spae, one an we P(, V, ) np(, V, ) dω dω (3) ( dω) wee n s e densy of e δ -soues n pysal spae and ( dω) s e nube dω of δ -soues nsde e volue d Ω. Ten, euaon () aes e fo P ( ) P P ( anp(,, ) ) V V V V (4) Te densy of e Halon-Jaob enseble s defned by e expesson (, ) P(, V, ) dv (5) wle s veloy v (, ), w s nvolved n E. (39), s defned by e expesson Le us onsde a enso v VP(, V, ) dv (6) Π V V P(, V, dv (7) j j ) Te veloy of e pales n e sasal enseble an be deoposed no e ean veloy v, w ondes w e flow veloy of e Halon-Jaob enseble, and e ando veloy oponen w V v w (8) wee w (9) Ten, e enso (7) an be wen as Π j ( v v w w ) () j Fo soop ando poess w, ww j τδj () wee τ w fo any,, 3 s soe paaee a s assued o be onsan. Inegang euaon (4) w espe o V wle onsdeng a e pobably of nfne veloes s eual o zeo gves j dv ( v ) ()

23 .e., e onnuy euaon (4). Mulplyng euaon (4) by V and negang w espe o V wle onsdeng Es. (5) oug () gves v ( v) v ( an τ ) By opang euaon (3) w euaon (39), one onludes a ey onde f we ae (3) an τ (4) Tus, we ave sown, a leas foally, a, f e oon of a pale s desbed usng e lassal dynaal euaon (8) w e paaees (4), e Halon-Jaob enseble of e pales s desbed by e euaons of ydodyna ype (39) and (4). In dong so, e ean oon of e Halon-Jaob enseble s desbed by e ödnge euaon. VI. COCLDIG REMARK We ave sown a e ydodyna euaons (39) and (4) w a uly osllang sgnalenang pessue (4) ae euvalen o e ödnge euaon n e sense a a vey g feuenes, e flow of e Halon-Jaob-ödnge enseble, defned by e euaons (39) and (4), afe aveagng ove e fas osllaons, s desbed by e ödnge euaon. I follows a e ödnge euaon n s odel s an appoxae euaon and desbes only e ean oon of Halon-Jaob-odnge enseble, wle euaons (39) and (4) desbe e oon of e enseble n deal, w onans fas osllaons. In paula, Es. (39) and (4) ay onan a nube of fne pysal effes a ae absen n e ödnge euaon beause ey ae los due o aveagng ove e fas osllaons. I s neesng o nvesgae ese effes. We ave sown a e ödnge euaon oesponds o e l of vey g feuenes,. Wa s s feueny? One an assue a s feueny s a naual feueny of e pale assoaed w s es ass:. A leas fo e an non-elavs uanu pales (.e., eleons, poons, neuons), s feueny an be onsdeed vey g n opason o e feuenes of all poesses nvolvng ese pales. Te ödnge euaon, n e end, an be obaned fo e euaons of oon of lassal pales (8), w s of gea nees fo e pespeve of e poble of nepeng uanu eans [5,6]. Le us onsde a dyna nepeaon of ewon's euaon (8). 3

24 Le ee be a ulude of δ -soues dsbued n spae w a onenaon n a eae a poenal feld a δ ( ) ; e aplude of e soues s defned by E. (4) and osllaes w a g feueny. In addon, ee s a lassal poenal feld (, ) n spae. If a lassal pale oves n s spae, s oon s desbed by e euaons of lassal eans (8), wle e Halon-Jaob enseble oespondng o s syse s desbed by euaons (39) and (4). Te enseble exeues a oplex oon a onsss of e ean (obseved) oon and e fas osllaons w feueny. As sown above, e ean (obseved) paaees of su an enseble ae desbed by e ödnge euaon. Ts odel esebles e faous pnball gae n w e ball oves andoly beween sall epulsve obsales. Fo s eason, s odel an be alled a pnball odel. Obvously, e pnball odel esonaes w e soas odel of uanu oon [5,6]. We ae no gong o dsuss e naue of ese δ -soues; we noe only a ey an be elaed o e pysal vauu. Refeenes [] L.D. Landau and E.M. Lfsz, Means (Buewo-Heneann, 3d ed., 976), Vol.. [] H. Goldsen, C. P. Poole, and J. L. afo, Classal Means (Addson Wesley, 3d edon, ). [3] M. Bon, Z. Pys. 37, 863 (96). (Repned and anslaed n Quanu Teoy and Measueen, eded by J. A. Weele and W. H. Zue, (Pneon nvesy Pess, Pneon, J, 963). [4] A. Messa, Quanu Means (Dove Publaons In. ew Yo, 999). [5] R. Onès, Te Inepeaon of Quanu Means (Pneon nvesy Pess, Pneon,.J. 994). [6] G. Aulea, Foundaons and nepeaon of uanu eans (Wold enf Publsng Co. Pe. Ld, ). [7] D. Bo, Pys. Rev. 85: (95). [8] D. Bo, Pys. Rev. 85: 8 93 (95). [9] D. Dü and. Teufel, Boan Means. Te Pyss and Maeas of Quanu Teoy (pnge-velag Be Hedelbeg 9). [] L.D. Landau, E.M. Lfsz, Te Classal Teoy of Felds (Buewo-Heneann, 4 ed., 975), Vol.. 4

25 [] L.D. Landau and E.M. Lfsz, Quanu Means: on-relavs Teoy (Pegaon Pess, 3d ed., 977), Vol. 3. [] L.D. Landau and E.M. Lfsz, Flud Means (Buewo-Heneann, nd ed., 987), Vol. 6 [3] V. B. Kapev, Pys. Rev. Le. 4, 497 (979) [4] L.P. Paevs and E.M. Lfsz, Pysal Knes (Pegaon Pess, s ed., 98), Vol.. [5] E. elson, Pys. Rev. 5: (966). [6] E. elson, Quanu Fluuaons (Pneon ees n Pyss, Pneon nvesy Pess, Pneon, J, 985). 5