Thermodynamics of the Primary Eigen Gas and the Postulates of Quantum Mechanics

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1 Thermodyamis of the Primary Eige Gas ad the Postulates of Quatum Mehais V.A.I. Meo, Gujarat Uiversity Campus, Ahmedabad , Idia. Abstrat The author shows that that for eah quatum mehaial property of a mirosystem there is a orrespodig thermodyami oe i the primary eige gas approah. He further shows that the basi postulates of quatum mehais have equivalets i the primary eige gas approah provided time is aorded diretioal symmetry. The iterferee patter obtaied i Youg s double slit experimet is explaied i terms of the primary eige gas approah usig the diretioal symmetry of time. PACs umbers: w, 3.65 Ta, b, a, Ce Key ords 1 Itrodutio : Primary eige Gas, Stohasti quatum mehais, Atio- etropy equivalee, Quatizatio of spae ad time. It was earlier show that eletro a be represeted by a ofied helial wave (CH wave) formed by the ofiemet of a plae polarized eletromageti wave after it aquires half spi. The properties of a partile lie mass, eletri harge ad spi were foud to emerge from the CH wave [1],[2][3],[4]. These itrisi properties of eletro are foud to be expressed by the spae-depedet ompoet of the CH wave whih gets ompated ito the iteral oordiates while the time depedet ompoet beomes the plae wave whih represets the partile i the laboratory oordiate system. Extedig this idea further it was proposed that ay partile a be represeted by a CH wave where the ofiemet taes plae o a omposite wave whih has osillatios ot oly i the eletromageti field, but also i other appropriate fields. It was further show that the states oupied suessively i time by its iteratios with the vauum flutuatios may be tae to form a gas alled the primary eige gas. The oly differee betwee the primary eige gas ad the real gas is that while i the real gas the mirostates are oupied simultaeously, i the ase of the primary eige gas the miro-states are oupied suessively i time. But for this differee, the statistial mehais of the primary eige gas is equivalet to that of the real gas [5]. It was observed that the primary eige gas piture of a partile ad the wave piture of a partile are equivalet oes ad this equivalee is a diret result of a ew symmetry alled the i symmetry. I fat, it was proposed that quatum mehais ould be uderstood i terms of the statistial mehais of the primary eige gas where time has ot lost its diretioal symmetry [5]. e saw earlier that the probability futio represetig the state of the primary gas with eergy E i is give by i 1 h A g( Ei ) e (1) This futio a be obtaied by applyig i s operator o the plae wave represetig the quatum mehaial state with eergy E i [6] ie; 1 1 i A h A R i R[ B( Ei ) e ] g( Ei ) e (2) Note that i s operator replaes 2πiN by N wherever N ours i the futio. e saw that this equivalee emerges from the i symmetry. I other words, the plae [1]

2 wave whih is the eige state of a partile i spae-time represetatio beomes the probability futio of the primary eige gas represetatio. I this paper we shall examie the ompatibility of the basi postulates of quatum mehais to the statistial mehais of the primary eige gas. Besides, usig the eige gas approah we shall try to explai the iterferee patter obtaied i the double slit experimet usig a eletro beam. 2 Quatum Mehais as Reversible-Time Thermodyamis he we represet a partile i terms of a plae wave, we ever speify the umber of sigle waves ostitutig the wave trai or the legth of the wave trai whih ostitutes the plae wave. It is left udefied i the usual treatmet. This is beause, i quatum mehais, the eige state is supposed to be the most basi state. But i the primary eige gas approah we are attributig a ier struture to the eige state. I order to strie equivalee betwee the two approahes, we assume that there are N miro-states that ostitute a primary eige gas just as there are N wavelets that ostitute a plae wave state. e do ot ow whether N is a uiversal ostat or a arbitrary umber whih gets fatored out i ay observatio. I fat, i the iteratio betwee two partiles, N a be oveietly tae as a ostat. The justifiatio for suh a assumptio is that the variatio a be aouted by whih represets the umber of the primary eige gas states ivolved i ay iteratio. Oe eed ot tae both N ad as variables [5]. The Copehage iterpretatio (CI) treats eige state as the most basi elemet of the quatum reality. Aordig to CI ay observatio o a miro-system a tae us oly up to the level of a eige state. Besides, ay physial system a be iterpreted oly i terms of the observable reality [7]. Suh a otio did ot permit ay further aalysis of the eige state. But i terms of the eige gas approah, we observe that a eige state a be tae as equivalet to a eige gas whih is a primary gas. I our approah we do ot tae the primary eige gas as the most basi state. e assume that the primary eige gas state is ostituted by a group of plae wavelets oupied suessively. This may appear rather otrived as the wavelet is below the level of observatio. But suh a struture is proposed here as it simplifies the piture ad we are able to explai the laws of quatum mehais i terms of thermodyamis of the primary eige gas. Note that this approah does ot aept the view that there is a objetive reality to the miro-system. It goes by the approah based o Copehage iterpretatio that the miro-system exists i a state where all possibilities exist i a umaifested maer ad the system rystallizes ito reality oly i a observatio. e shall ow loo for the primary eige gas equivalets of other quatum mehaial properties. e ow from the i symmetry that the egative atio is equivalet to etropy [5]. ie; # A h S K (3) This may be expressed i terms of the eergy-mometum of the partile as # ( Et px) h E t / S K (4) But we ow that t o = NT eo = Nh/K o, where o is the temperature of the vauum flutuatios bagroud whih iterats with the partile [5]. Therefore, we have o o S # NE o K (5) Note that the rest eergy of the partile E o behaves lie the itrisi heat otet. e shall ow try to list out the reversible time equivalet of other quatum mehaial etities below. o [2]

3 Imagiary time Reversible time i) ave Futio State futio of the primary eige gas ii) Negative Atio Etropy iii) Total eergy Iteral eergy iv) quatum of time Iverse of temperature v) partile veloity Drift veloity vi) Lagragea Negative Itrisi heat otet vii) Uertaity priiple. Equatio for flutuatios viii) Zero poit eergy Thermal eergy of vauum flutuatios 3 The Basis for the use of Boltzma s Miro-aoial Distributio I the approah followed above, oe questio that may be raised is regardig the use of the Boltzma s distributio to represet the primary eige gas state. The justifiatio for this is quite straight forward. First of all we should remember that we are dealig with oe path of progressio at a time. The system i the primary eige gas represetatio oupies oly a partiular miro-state (wavelet of a plae wave) at a time. Suh a wavelet state is oupied out of very large umber possible wavelet states available for oupatio. I fat there is o limitatio to the umber of wavelet states that ould be oupied here. I that sese, the situatio is equivalet to a large umber of boxes available for oupatio with a limited umber of balls. This is the situatio where Boltzma s distributio holds good. If the umber of boxes ad the umber of balls were of the same order, the we would have bee fored to apply the Fermi-Dira distributio. The oept of the esemble will have to be give a fresh loo i the ase of the primary eige gas. I the ase of the aoial esemble of a real gas, we deal with eergy ad volume alog with their ojugate variables, temperature ad pressure. However, i the ase of the primary eige gas we tae the traslatioal mometum i plae of volume, ad veloity i plae of pressure. e ow [5] that the iteral heat of the primary eige gas is give by Q Therefore, the etropy of the system will be give by Nq NE Nþv. (6) S Q N( E vþ ) (7) Earlier we saw that suh a primary gas may be tae to be equivalet to a real gas [5]. Oe importat differee betwee volume i the ase of a real gas ad mometum i the ase of a primary gas is that volume aot be trasformed out by suitably seletig a frame of referee while mometum a be. This meas that we a always deal with the miro-aoial esemble by itroduig suitable trasformatio ad study the system more oveietly. Afterwards, we may trasform it ba by shiftig the frame of referee suitably. This is made possible by the fat that the etropy of the system remais ivariat i suh a trasformatio. I short, we do ot have to deal with the aoial esemble to wor out the thermodyamis of the system. Miro-aoial esemble will do. 4 The Primary Gas ad the Postulates of Quatum Mehais I the earlier papers we saw that a wave futio represetig a miro-system will beome a probability desity futio o beig operated by i s operator R. e also saw that suh a operatio does ot alter the physial situatio. Oly the imagiary time piture gets replaed by the reversible (real) time piture whih is same as statig that the wave ature of the miro-system gets replaed by the primary [3]

4 eige gas piture. This meas that the dyamis of the system whih is determied by the wave ature of the miro-system will have to be replaed by laws of statistial mehais. This i tur meas that the basi postulates of quatum mehais ould be uderstood i the light of the laws of statistial mehais. e shall ofie ourselves to the three basi postulates of quatum mehais for study [8]. These three postulates that apture the essee of the operator form of quatum mehais. There are further postulates, but these three are the most importat oes. They firmly establish the existee of the wave futio ad its status as a omplete desriptio of the state of a quatum partile. The replaemet of the values of the observable quatities (lassial mehais) with their orrespodig operators (quatum mehais) provides a reipe for usig the wave futios ad operators to alulate the values of the observables. Let us tae the first postulate of quatum mehais to start with [8]. Postulate I : The state of a quatum mehaial system is ompletely desribed by the wave futio. Here the subsript serves as a short had for the set of oe or more quatum umbers o whih the wave futio depeds. Let us tae the plae wave represetatio of a quatum mehaial system. Note that a plae wave is the eige futio of a mometum state (taig eergy as the fourth ompoet of the mometum) i the oordiate represetatio. This meas that the system may be oupyig ay of the plae wave states deoted by. If the wave futio is represeted as a vetor i a ofiguratio spae (Hilbert spae), the will be orthogoal to m uless = m. I this way, the state of a system a be represeted as a vetor i a -dimesioal ofiguratio spae. This assumes that the system may oupy all possible states simultaeously i a virtual way ad for suh a possibility to be aeptable, time has to be treated as imagiary. It should be ept i mid that the orthogoality of the wave futios beomes importat oly at the istat of observatio as two eige states aot be realized simultaeously. The higher the magitude of the vetor larger will be the probability of athig the system i that state. I the reversible time approah the state futio of the primary eige gas state whih is a probability desity futio plays the same role as the plae wave i the imagiary time. However, sie we are treatig as real time it may appear that the oept of simultaeous oupatio of various primary gas states may ot be possible. But agai we have to remid ourselves that while time i the primary eige gas approah may be treated as real, it is also treated as reversible. Therefore, by the proess of reverse jump i time it is possible for the system to oupy all states at the same istat [7]. I other words, there is o logial iosistey i assumig that all primary eige gas states exist simultaeously. Therefore just as i the ase of the plae wave represetatio, i the primary eige gas approah also it is possible to represet the state of a partile as a liear ombiatio of the primary eige gas states. The ext questio that we have to ofrot is that whe we tae the liear ombiatio of the primary eige gas states, whether they will also udergo iterferee just as i the ase of the plae wave states. Oe may thi that as the primary eige gas state does ot exhibit udulatory behavior it is impossible to aout for the iterferee pheomeo i this approah. But the we should ot forget that primary eige gas approah ad the plae wave approah are two ways of viewig the same pheomeo. The pheomeo itself remais the same i both approahes. Here we have to reall that i the ase of the primary eige gas approah, the quatum of time is oe period of osillatio of the plae wave. This meas that the periodi ature of the wave gets pushed ito the iteral struture of the quatum ad gets ompated ito the iteral oordiates. But whe two miro-states belogig two differet primary eige gases oupy the same spatial regio, the of ourse these iteral oordiates ome ito play ad the iterferee ours as usual. [4]

5 This meas that there is o differee i the et result whether we go by the plae wave approah or by the primary eige gas approah. The oept of the orthogoality of the eige states emerges from the fat that the probability for observig the system i the th state at a istat is give by Ψ Ψ. Here all the paths of evolutio of states our simultaeously ad therefore whe we tae the itegral m d m (8) I the eige gas approah also the equivalet relatio give below will hold good: m d m (9) here deotes the reversible (real) time. This is tae are of by the oept of forward ad reverse jump i time. The resultat piture that emerges is o differet from the approah based o the plae wave. Oe importat issue that arises out of this reversible (real) time approah is that the wave equatio whih applies to the wave ature of the partile will o more be valid i the primary eige gas approah. Let us tae the Shrodiger s equatio first ad try to uderstad the impliatio of replaig imagiary time with real time. e also ow that the Shrodiger equatio for a free partile o arryig out i s operatio will trasform ito the Diffusio equatio [9]. e shall tae omplex ojugate of the Shrodiger equatio ad apply i s operator to obtai 2 2 R{ 2m i } t 0 ie; tˆ h 2 ˆ 2m. (10) Note that sie t = NT e, where ad N are itegers while T e deotes the period of the plae wave represetig the partile. t will be obtaied by arryig out i s operatio o t whe 2iN will be trasformed to N. This meas that t = 2it. Similarly x = 2it. This is the heat equatio or diffusio equatio where (h/2m) represets the diffusivity. whih stads the probability desity a also represets the partile desity whih i tur a deote the eergy desity. I this oetio we should reall that [5] the rest eergy of a partile a be treated as its iteral heat. This meas that the Shrodiger equatio ad the diffusio equatio are two ways looig at the same reality. Needless to say, we may view a potetial as oe whih itrodues a gradiet i the vauum flutuatios field that direts the diffusio proess. I this oetio it is worthwhile to reall a alterate approah to quatum mehais alled stohasti quatum mehais has bee proposed by Nelso, Yasue et el [10][11][12][13]. This approah has met with partial suess. But this approah has ot bee able to obtai profoud isights lie the atio-etropy equivalee ad the i symmetry. The mai reaso for this shortomig may be that the stohasti approah shied away from proposig a basi struture to the partile ad was basig its bet o the oept of poit partile. The striig similarity betwee the plae wave ad the probability desity futio was ever probed i depth. The approah seems have foused too muh o the similarity betwee the Shrodiger equatio ad the diffusio equatio. But it is quite obvious that the primary gas approah may be developed further usig the methods of [5]

6 stohasti quatum mehais for uderstadig the behavior of the miro-systems i greater depth. Let us ow tae the Dira equatio ow. m 0 x, (11) where 0 i i, 1,2,3. ; Here σ represet Pauli s spi matries. O operatig with i s operator Ř, the Dira equatio trasforms to give m h 0 x, (12) Note that x = Nv T e, where T e = h/k. e observe that the wave equatio give i (11) exhibits udulatory behavior while that give i (12) does ot have suh a property sie is just a probability futio. Therefore, prima-faie it may appear that these two equatios represet differet realities. But we should eep i mid that the udulatory behavior holds good oly withi oe wave legth. If we treat oe wave legth as a sigle uit, the we will ot observe ay udulatory behavior. Istead, we would observe that the system progresses i a uiform maer. e shall examie this issue i more detail i the ext setio. e ow that we a extrat the Shrodiger equatio from the Dira equatio i the o-relativisti regio if we igore spi [9]. I view of this it is reasoable to assume that the Dira equatio i real time also represets some sort of a diffusio equatio. But we are ot familiar with its lassial aalogue. Postulate II: Observable quatities are represeted by mathematial operators. These operators are hose to be osistet with the positio-mometum ommutatio relatios. To larify the piture, let us tae the plae wave represetatio of the eige state give by Be 1 i þ x (13) The mometum operator þ = -iħ/x so that i x þ I other words, the mometum operator is defied i suh a way that beomes the eige futio ad þ its eige value. he we go over to the primary eige gas piture after itroduig i s operatio, we observe that þ will be the mometum of a primary eige gas. The probability desity futio for the primary eige gas i forward time represetig a partile travellig alog the x-axis will be [6] give by Be N e 1 ( Et þx) The orrespodig mometum operator will be h eige value equatio (14) (15) x ad we obtai the similar h x ˆ þ (16) [6]

7 Here also the average value of mometum would be obtaied by averagig the values of þ. It is obvious that i the operators i the primary eige gas approah a be obtaied by applyig i s operatio o the orrespodig operators i quatum mehais. It a be easily see that i s operatio trasforms oly x ad t leavig the form of the other operators uhaged. Let us ow examie how the ommutatio relatio appears i the primary eige gas approah. e ow that the mometum operator i quatum mehais satisfies the relatio ( x þ - þx) i (17) I the primary eige gas approah by by replaig x by x we obtai ( xþ - þx) - h (18) Postulate III : The mea value of a observable is equal to the expetatio value of its orrespodig operator. For a speifi wave futio, the expetatio value of the operator A is defied by the expressio A A d d (19) where τ deotes the geeralized volume elemet. I quatum mehais this expressio is itrodued rather arbitrarily. It emerges from the fat that is the eige futio of the operator A ad a is the orrespodig eige value. Therefore, we have A a Multiplyig both sides by ad itegratig over all spatial elemets yields A d a d (20) (21) This is possible beause a is a mere umber. If is ormalized, we obtai A (22) Sie the wave futio a be a omplex futio, the operator too may be a omplex futio. However, if postulate III is to mae sese, the eige value of a operator represetig a observable must be a real quatity beause this is somethig that a be measured i a experimet. Operators whose eige values are real are alled Hermitia operators. As a result of this property of the operators ay two eige futios are orthogoal. That is, taig the eige futios as ormalized, we have a d m (23) Despite beig a omplex futio while probability is a real umber, the above equatios holds good beause the phase part of gets removed as we are multiplyig it with its omplex ojugate whih leaves us with the square of the amplitude. But the we ow from (13) ad (15) that B 2 This will diretly lead us from (19) to [7]

8 A A d (24) d Stritly speaig (24) represets the lassial defiitio of the probability. The probability postulate of quatum mehais appears more otrived ad adho. Note that i the ase of a real gas existig i progressive time, the average value for mometum will be give by the relatio A 5 More o i Symmetry d A d (25) e saw that the i s operatio trasforms the plae wave ito the state futio of the primary eige gas. I this operatio, 2πiN gets overted to N. Let us examie what is happeig here. e ow that the basi differee betwee the primary eige gas approah ad the plae wave approah is that the former treats eah sigle wavelet as the basi uit or quatum. This does ot mea that this approah ompletely igores imagiary time aspets ad the aompayig phase hage that taes plae withi a sigle wavelet. It is just ompated ito its iteral oordiates. To put it differetly, the primary eige gas approah aouts for the phase of the wavelet as the itrisi property of the miro-state. I the plae wave approah there is o quatizatio ad therefore the imagiary ature of spae ad time gets expressed i the exteral oordiates themselves. Aother importat poit that emerges from the i symmetry is the relatioship betwee the probability for the forward evolutio ad the reverse evolutio i time. e ow from the earlier disussio that the state futio (x,t) represets the sum total of all probability for forward jumps i time to the th state at the spae-time poit (x,t) from the past states. Liewise, ( x, t) represets the probability for the reverse jumps to the past from the th state at the same spae-time poit. e ow that whe time has ot lost its diretioal symmetry, the probability for the forward jump should be equal to that for the reverse jump. I other words =. Let us see if suh a relatioship atually emerges from the primary eige gas piture. Let us start with the primary eige gas state represetig the partile travellig alog the x-axis give by ( Et px) h g e (26) To mae matters simple, we shall trasform this primary eige gas to a frame of referee with whih it is at rest give by [6] N NE o Ko ( o) B e e (27) Sie the system is i equilibrium with the vauum flutuatios, we may tae E o = Kθ o.therefore, (29) a be writte as N N ( o) B e e B (28) It a be easily show that the orrespodig expressio for (o) would be give by N N ( o) B e e B (29) This ofirms our assumptio. [8]

9 6 Collapse of the Primary Gas i a Observatio Till ow we have ot explaied what happes to the primary eige gas i a observatio. e had assumed that a primary eige gas state is defied by N mirostates (plae wavelets) oupied suessively. This meas that a partile state is spread over a spatial legth of N, where N is assumed to be a large umber ad therefore the partile state is spread over a very large regio. But i a experimetal set up, we selet a small regio to observe the partile ad this small regio would otai paths of various primary gas states. Therefore, whe we observe a partile, the orrespodig gas state eed ot belog to oe sigle primary gas state. The N states ostitutig the eige state may belog to various primary eige gas states whih have their paths passig through the regio i questio. I fat i the regio of loalizatio, the iterferee amog the miro-states belogig to various idividual paths will reate a sharp maximum. Oe should eep i mid that the iterferee pheomeo taes plae i the reversible time also; the oly differee is that the phase of the miro-states is defied i their iteral oordiates. But that is oly a aoutig jugglery ad does ot affet the atual situatio. The importat poit is that N states whih ostitute the observed gas that represet the partile is o more a primary eige gas state as the mirostates are ot oupied suessively i time. Suh a observed gas state ould be alled the ollapsed primary eige gas state. But the based o the priiple of statistial equivalee, we a always fid a primary eige gas state mathig with the ollapsed primary eige gas state. Therefore, it will be proper to say that o observatio, the system rystallizes to oe of the primary eige gas states. 7 Explaiig Iterferee patter of the Double-slit Experimet Let us ow examie the iterferee patter obtaied i the double slit experimet usig a eletro beam. e ow the iterpretatio of this pheomeo based o the wave ature of the eletro [14]. e shall briefly desribe it here (fig.2). The quatum mehaial iterpretatio is that the waves represetig the partiles spread aroud as a wave frot from the soure. The seodary waves emergig from slit A ad slit B arrive at P o the sree where they udergo iterferee. If both Slit A Sree P S (Soure) Slit B O the basis of the wave piture, eletro from the soure S goes through slits A ad B ad the seodary waves from the slits arrive at the sree ad the iterferee betwee the two waves frots reates the friges at P. Figure 2 waves reah P i phase the itesity will be maximum ad if they reah with a phase differee of π, the, the itesity will be miimum. This explais the formatio of the iterferee patter o the sree. Note that this iterferee would be formed eve if oly oe eletro is emitted by the soure at a time. This would mea that a sigle eletro iitially disembodies ito a wave frot ad passes through both slits simultaeously ad arrives at P o the sree whe suddely it throws away its wavedisguise ad appears as a partile. This iterpretatio may appear strage, but that is [9]

10 what quatum mehais has to offer. The issue of the wave partile duality is still a topi whih is ot properly uderstood i quatum mehais. I the primary eige gas approah, the eletro startig from the soure jumps forward i time to slit A ad from there it jumps forward to P alog all possible paths ad the jump ba i time to reah A ad ba agai to S. I a similar maer, we may imagie the same proess taig plae through slit B also. I fat, these forward ad the reverse jumps i time reates a wave frot emaatig from S. This will also explai the seodary waves emaatig from the two slits. I fat this iterpretatio is i tue with the Maxwell s equatio whih has solutios with waves movig forward i time ad baward i time. e ow obtai the same result as obtaied i the wave represetatio of the partile. The iterferee patter is reated due to the fat that the quatum has a iteral struture of a wavelet ad at P the wavelets emergig from slit A ad slit B arrive simultaeously ad is aught i a observatio where their iteral phases ome ito play ad reates the iterferee patter. Referees 1. V.A. I. Meo, Rajedra, V.P.N. Nampoori, vixra: (quat-ph) (2012). 2. V.A. I. Meo, Rajedra, V.P.N. Nampoori, vixra: (quat-ph) (2012). 3. V.A. I. Meo, Rajedra, V.P.N. Nampoori, vixra: (quat-ph) (2012). 4. V.A. I. Meo, Rajedra, V.P.N. Nampoori, vixra: (quat-ph) (2012). 5. V.A. I. Meo, vixra: (quat-ph) (2013) 6. V.A. I. Meo, vixra: (quat-ph) (2013) 7. D.Bohm, Cusality ad Chae i Moder Physis, Routledge & Kega Paul pl, Lodo, p (1984) 8. J. Baggot, The Meaig of Quatum Theory, Oxford Uiversity Press, Lodo, p 42-8, (1992) 9. J.C.Zambrii, Phy. Rev.A, vol.33, (1986) 10. E. Nelso, Phy. Rev. 150, 1079 (1966) 11. E.Nelso, Dyamial theories of Browia Motio, Prieto Uiversity Press, Prieto, N.J., (1967) 12. E. Nelso, Quatum Flutuatios, Prieto Uiversity, Prieto, N.J., K.Yasue, J. Math. Phys. 22, 1010 (1981) 14. R.P.Feyma, Quatum Mehais ad Path Itegrals, MGraw-Hill I.,N.Yor, p.2-17, (1965) [10]