Similarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle

Transcription

1 Similarity betwee quatum mechaics ad thermodyamics: Etropy, temperature, ad Carot cycle Sumiyoshi Abe 1,,3 ad Shiji Okuyama 1 1 Departmet of Physical Egieerig, Mie Uiversity, Mie , Japa Istitut Supérieur des Matériaux et Mécaiques Avacés, 44 F. A. Bartholdi, 7000 Le Mas, Frace 3 Ispire Istitute Ic., Alexadria, Virgiia 303, USA Abstract Similarity betwee quatum mechaics ad thermodyamics is discussed. It is foud that if the Clausius equality is imposed o the Shao etropy ad the aalogue of the quatity of heat, the the value of the Shao etropy comes to formally coicide with that of the vo Neuma etropy of the caoical desity matrix, ad pure-state quatum mechaics apparetly trasmutes ito quatum thermodyamics. The correspodig quatum Carot cycle of a simple two-state model of a particle cofied i a oe-dimesioal ifiite potetial well is studied, ad its efficiecy is show to be idetical to the classical oe. PACS umber(s): w, a 1

2 I their work [1], Beder, Brody, ad Meister have developed a iterestig discussio about a quatum-mechaical aalogue of the Carot egie. They have treated a two-state model of a sigle particle cofied i a oe-dimesioal potetial well with width L ad have cosidered reversible cycle. Usig the iteral eergy, E (L) = ψ H ψ, they defie the pressure (i.e., the force, because of the sigle dimesioality) as f = d E(L) / d L, where H is the system Hamiltoia ad ψ is a quatum state. The, the aalogue of isothermal process is defied to be f L = cost. correspodig to the classical equatio of state of the ideal gas at fixed temperature, whereas the aalogue of adiabatic process is characterized by f L 3 = cost., which is the Poisso-like law. The efficiecy of the cycle they obtaied reads η = 1 E C E H, (1) where E H (E C ) is the value of E alog the isothermal process at high (low) temperature. This discovery is remarkable, sice it shows how the work ca be extracted from a simple quatum-mechaical system through particular deformatios of the quatum state as well as the potetial width. However, our poit here is to grasp the above discussio as a hidde similarity betwee quatum mechaics ad thermodyamics at a certai level. The, it is equally iterestig to examie if it is possible to fully establish such similarity. To do so, of

3 cetral importace is to idetify the aalogues of the etropy ad temperature. Accordig to Kelvi, the efficiecy of the Carot cycle ca be used to defie the absolute thermodyamic temperature as E C / E H = T C / T H []. Thus, a aalogue of the law of equipartitio of eergy, which is violated i quatum theory, is assumed i the Carot cycle i Ref. [1]. Therefore, a questio that aturally arises the is if this temperature is cosistet with the cocept of etropy. I this paper, we revisit this itriguig problem of similarity betwee quatum mechaics ad thermodyamics, ot by requirig the aalogies of the classical equatio of state, equipartitio of eergy, the Poisso law, etc., but by imposig oly the Clausius equality as a fudametal thermodyamic relatio. The, we show that the value of the Shao etropy becomes formally equal to that of the vo Neuma etropy of the caoical desity matrix, ad pure-state quatum mechaics apparetly trasmutes ito equilibrium quatum thermodyamics [3]. However, decay of a pure state ito a mixed state itself ever appears explicitly. We shall also discuss, just for compariso with the result i Ref. [1], the correspodig Carot cycle i a simple two-state model of a particle cofied i a oe-dimesioal ifiite potetial well. Its efficiecy is foud to be i complete agreemet with the classical oe. Cosider a quatum-mechaical system i a state ψ. The mea value of eergy to be idetified with the aalogue of the iteral eergy is E = ψ H ψ, () where H is the system Hamiltoia. Uder chages of both the Hamiltoia ad the 3

4 state, E varies as δ E = ( δ ψ ) H ψ + ψ δ H ψ + ψ H ( δ ψ ). (3) This expressio has a formal aalogy with the first law of thermodyamics δ'q = δ E + δ'w, (4) if the followig correspodece relatios are oticed for the chages of the aalogues of the quatity of heat ad work: δ'q ( δ ψ ) H ψ + ψ H ( δ ψ ), (5) δ'w ψ δ H ψ, (6) respectively. To further elaborate the aalogy betwee quatum mechaics ad thermodyamics, it is ecessary to idetify the cocepts of etropy ad temperature. Regardig the etropy, oe might aively imagie that the etropy relevat to quatum thermodyamics may be the vo Neuma etropy S vn = k Tr (ρ l ρ). (7) As kow well, however, this is ot the case, sice it idetically vaishes i the case 4

5 whe the desity matrix, ρ, is of a pure state. Istead, the authors of Ref. [1] suggest the use of the Shao etropy. Usig the complete orthoormal system, { u }, which are the eigestates of the Hamiltoia, H, satisfyig H u = E u with { E } beig the eergy eigevalues, we ca expad a geeral state, ψ, as follows: ψ = a u, (8) a = 1. (9) The, the Shao etropy is give i terms of the expasio coefficiets as S S = k a l a. (10) S vn is ivariat uder uitary trasformatios of ρ, whereas S S is ot ivariat uder uitary trasformatios of ψ, i geeral. That is, S S depeds o the choice of a basis. The reaso why the eergy eigebasis is employed here is as follows. I the subsequet thermodyamic-like discussio, a crucial role is played by S S, which should ot exhibit explicit dyamical time evolutio. The expasio coefficiets, a s, i terms of the eergy eigebasis dyamically chage i time as follows: a a exp( ie t / ). Therefore, a remais uchaged, leadig to time idepedece of S S, as desired. It is also oted that the positive costat, k, appearig 5

6 here is still ot ecessarily the Boltzma costat i this stage. However, it turs out to be impossible to establish the Clausius-like equality for reversible processes δ S S = δ ' Q T, (11) withi the framework of Ref. [1]. Thus, similarity betwee quatum mechaics ad thermodyamics is ot yet complete i the quatum Carot cycle. Now, our idea is to impose Eq. (11) to defie the aalogue of temperature. To do so, first we rewrite Eq. () as follows: E = E a. (1) Sice we are iterested i the thermodyamic-like situatio, the time scale associated with the chage of the state is much larger tha that of a dyamical oe, ~ / E. I this case, the adiabatic theorem [4] applies. The chage of the state i Eq. (5), i.e., the aalogue of the quatity of heat, is give by δ ' Q = E δ a. (13) Regardig Eqs. (1) ad (13), see the later discussio above Eq. (17). O the other had, the chage of the Shao etropy reads 6

7 ( ) δ S S = k l a δ a. (14) A poit of crucial importace is that if the Clausius-like equality i Eq. (11) is imposed o Eqs. (13) ad (14), the the expasio coefficiet becomes the followig caoical form : 1 a = Z e E k T, (15) where Z = ( ) exp E / kt. Accordigly, the Shao etropy is calculated to be ( ) / Z ( ) / Z S S = k exp E / kt l exp E / kt, which precisely coicides with the value of the vo Neuma etropy i Eq. (7) of the caoical desity matrix, ρ = 1 Z e H k T (16) with Z = Tr exp( H / kt ). This implies that impositio of the Clausius equality trasmutes pure-state quatum mechaics ito geuie quatum thermodyamics, if k ad T are regarded as the Boltzma costat ad real temperature, respectively. Based o this fact, let us discuss as i Ref. [1] the Carot cycle of a simple two-state model of a sigle particle cofied i a oe-dimesioal ifiite potetial well with width L, i order to compare pure-state quatum mechaics with quatum thermodyamics. A arbitrary state, ψ, is give by superpositio of the groud ad 7

8 first excited states, u 1 ad u : ψ = a =1, u. L varies much slower compared with the dyamical time scale, ~ / E, as metioed earlier. The system Hamiltoia, H, its eigestate, u, ad the correspodig eergy eigevalue, E, all deped o chagig L. Write them as H (L), u (L), ad E (L), respectively. They satisfy the equatio, H (L) u (L) = E (L) u (L), i the adiabatic approximatio. This justifies Eqs. (1) ad (13). The expectatio value of the Hamiltoia i this state, ψ, is give by E = π m L ξ + 4π (1 ξ), (17) m L where m is the mass of the particle ad ξ a 1 [0, 1] (ad thus, a = 1 ξ ). The chage of this quatity through the variatio of L cotais two cotributios, as i Eq. (3). As repeatedly emphasized, the time scale of this variatio is assumed to be much larger tha the dyamical oe, ad so the chage of the state is represeted by the chage of ξ. Therefore, as i Eq. (13), the chage of the aalogue of the quatity of heat is δ ' Q = 3π m L δ ξ. (18) O the other had, the chage of the work i Eq. (6) i the preset case is calculated to be 8

9 δ ' W = π (4 3ξ) δ L. (19) 3 m L Also, the chage of the Shao etropy reads δ S S = k l 1 ξ 1 δ ξ. (0) From Eqs. (18) ad (0), it is see that idetificatio of the iteral eergy with temperature is ot cosistet. Here, we itroduce T through Eq. (11). Accordigly, ξ is solved as a fuctio of L ad T as follows: 1 ξ (L, T ) = 1+ exp 3π m L kt. (1) Note that 1 / < ξ (L, T ) < 1, which guaratees that the probability of fidig the system i the groud state is always larger tha that i the excited state, as desired. Now, let us cosider the Carot cycle (Fig. 1): A B C D A, where the processes A B ad C D are the isothermal processes with high- ad low-temperature, T H ad T C, respectively, whereas B C ad D A are the adiabatic processes with fixed ξ. The values of the volume (i.e., width) at A, B, C, ad D are L 1, L, L 3, ad L 4. The heat absorbed by the system is calculated from Eq. (18) as 9

10 (B) Q H = d 'Q = 3π d L 1 m L (A) L L 1 d ξ (L, T H ) d L = 3π m 1 ξ (L, T H ) 1 ξ (L, T ) 1 H L L 1 + kt H l ξ (L, T ) 1 H ξ (L, T H ). () O the other had, usig Eq. (19), we ca calculate the work durig each process as follows: (B) L W A B = d ' W = d L π 4 3ξ (L, T m L 3 H ) (A) L 1 = π m 1 L 1 1 L + kt l ξ (L 1, T H ) H ξ (L, T H ), (3) (C) L 3 W B C = d ' W = d L π 4 3ξ (L m L 3, T H ) (B) L = π m 4 3ξ (L, T ) 1 H L 1 L 3, (4) (D) L 4 W C D = d ' W = d L π 4 3ξ (L, T m L 3 C ) (C) L 3 = π m 1 L 1 3 L 4 + kt l ξ (L, T ) 3 C C ξ (L 4, T C ), (5) 10

11 (A) L 1 W D A = d ' W = d L π 4 3ξ (L m L 3 4, T C ) (D) L 4 = π m 4 3ξ (L 1 4, T C ) L 1 4 L 1. (6) Usig the equalities associated with the adiabatic processes, B C ad D A, ξ (L, T H ) = ξ (L 3, T C ), (7) ξ (L 4, T C ) = ξ (L 1, T H ), (8) that is, L T H = L 3 T C, (9) L 4 T C = L 1 T H, (30) we fid that the total work doe by the cycle is give by W = W A B + W B C + W C D + W D A = 1 T C T H Q H (31) with Q H give i Eq. (). Therefore, we obtai the efficiecy of the cycle as follows: η = 1 T C T H, (30) which precisely coicides with that of the classical Carot cycle. 11

12 I coclusio, we have studied a structural similarity betwee quatum mechaics ad thermodyamics. We have focused our attetio o the Shao etropy i the eergy eigebasis ad the aalogue of the quatity of heat. We have foud that if the Clausius equality is imposed o them, the the value of the Shao etropy becomes that of the vo Neuma etropy of the caoical desity matrix, ad pure-state quatum mechaics apparetly trasmutes ito quatum thermodyamics. We have, however, recogized that decay of a pure state ito a mixed state itself does ot explicitly appear i the discussio. To examie the result i compariso with the work i Ref. [1], we have also studied the Carot cycle of a two-state model of a particle cofied i a oe-dimesioal ifiite potetial well ad have see that its efficiecy is idetical to the classical oe. S. A. was supported i part by a Grat-i-Aid for Scietific Research from the Japa Society for the Promotio of Sciece. [1] C. M. Beder, D. C. Brody, ad B. K. Meister, J. Phys. A: Math. Ge. 33, 447 (000). [] E. Fermi, Thermodyamics (Dover, New York, 1956). [3] J. Gemmer, M. Michel, ad G. Mahler, Quatum Thermodyamics, d Ed. (Spriger-Verlag, Berli, 009) [4] M. Bor ad V. Fock, Z. Phys. 51, 165 (198). 1

13 Figure Captio FIG. 1 The Carot cycle depicted i the plae of the width (L) ad force (f). The processes, A B (C D ) ad B C (D A ), are isothermal expasio at high temperature (compressio at low temperature) ad adiabatic expasio (compressio), respectively. The cycle ca i fact be realized with the fuctioal fuctio form of ξ (L, T ) i Eq. (1). f A B D C L Figure 1 13