Quantum mechanical description of thermal equilibrium states as eigenstates of relative-phase interactions

13 November 1995

ElSEWEiR Physics Letters A 207 (1995) 320-326

PHYSICS LETTERS A

Quantum mechanical description of thermal equilibrium states as eigenstates of relative-phase interactions

Tsunehiro Kobayashi ’ InsMute of Physics, University of Tsukuba. Ibaraki-305, Japan

Received 4 September 1995; accepted for publication 19 September 1995 Communicated by P.R. Holland

Abstract

Thermal equilibrium states which satisfy the principle of equal a priori probabilities are investigated in a system composed of N oscillators, which is described by the free Hamiltonian HO = E Et, Ajax. Interactions depending on the relative phases

among the oscillators are introduced as Hf( [19]) = e,(Z?/N*) [ cl, c:, ai(B)aj(B) - N-’ cz, ]Oj)(Oj]], where fi =

EL, IVj with # = Ajax, ajj( 8) = aj exp( -iti,) and aj ]Oj) = 0. It is shown that the eigenstates of H = Ho + Hf( [ 01) satisfy the principle and we can derive density matrices for microcanonical and canonical ensembles by averaging over the phases.

PACS: 03.65-w; 05.30.-d; 05.70.-a

1. Introduction

It is well known that there is an apparent gap between quantum mechanical states and thermal equilibrium states, that is, the former are described by pure states as

P+ = c Ic”l*l#4 +~~c"w)(mI (1) ” nh

for the state jr,&) = C,, cnln) where the complete set

In) fulfill the equations H&r) = Z&In) and (nlm) = s II,“,? whereas the latter are described by mixed states

pstat = C exP(-PEn)ln>(nl z ’ n

(2)

for canonical ensembles in statistical mechanics, where /3 = (/CT) -i (k is the Boltzmann constant)

’ E-mail: [email protected].

and Z = Tr(C, exp( -p&) In)(nl) (the partition function of the canonical ensemble). The main difference between them is represented by the following two points:

( 1) The thermal equilibrium states have no inter- ference terms (decoherence problem).

(2)The relations

(c,l* m exp(-PW vn (3)

must be fulfilled in the mixed states pStat for the canonical ensembles.

The first problem is quite similar to the problem of the quantum theory of measurements, which has been studied for several decades [ I] (for a review see Ref. [ 21) , while the second one is a purely thermodynami- cal problem which is based on the principle of equal a priori probabilities. The important difference of thermal equilibria from quantum mechanical states is described by the existence of heat baths in thermal equi-

0375-9601/95/$59.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00718-O

T. Kohyashi / Physics L.etters A 207 (1995) 320-326 321

libria. We know that the states of the heat baths have

not yet been succesfully described in quantum mechanics. We should also not forget the role of the heat

baths in the first problem, i.e. the decoherence prob-

lem. The role of the heat baths is similar to that of detectors in quantum measurement processes, where the decoherence mechanism is also required [ l-31. Both

the heat baths and the detectors will play an essen-

tial role in the realization of the decoherence in both

processes. In order to describe the thermal equilib-

rium states in the framework of quantum mechanics,

the quantum mechanical description of the heat baths will be the first step. We shall write the heat baths as

thermally isolated systems which are described by mi-

crocanonical ensembles in statistical mechanics. The

basic assumption for the microcanonical ensembles is

the principle of equal a priori probabilities, that is, the states having the same energy eigenvalues must have the same probabilities. We may write the principle in

terms of the relations

/c,l* = lc,,,12, for En = E,,,, (4)

in ( 1) . These relations are quite unusual for general

quantum mechanical states. Actually the states fulfill-

ing the relations span only a subspace of the whole physical space spanned by the complete set of In). In

order to realize the projection from the whole space 31 to the subspace ‘&e,,,,ai (hereafter we call ‘?-Lit,ermat the thermal subspace), we have to introduce some new

interaction (HI). Of course, the new interaction is so

weak that the energy eigenvalues of the total Hamil- tonian H = HO + HI well approximate those of HO. Namely, we may expect that the thermal equilibria are

realized in the limit of HI --+ 0. The following point is also important in the introduction of the new interac-

tion: when we study a thermal equilibrium for a heat

bath and an object, the thermal interaction between the heat bath and the object, which is contained in HI, must not disturb the thermal equilibrium. This means that the interaction induces the transition only on the

thermal subspace. Furthermore the density matrix for the object must be described by that for the canonical ensembles given in (2) in the limit HI -+ 0. It should not be forgotten that in order to obtain the density matrix for the object the partial trace operation with respect to the heat-bath variables, which is called the internal trace operation [ 3-51 in the quantum theory

of measurements, must be introduced because most of

the quantum numbers of the heat bath are not mea- sured in the measurements. We may summarize the

conditions fulfilled by HI for describing the thermal

equilibria in quantum mechanics in terms of the following four criteria:

Criterion 1. The thermal interaction HI must project

the whole physical space of HO onto the thermal sub-

space of HO, where the principle of equal a priori prob-

abilities is fulfilled.

Criterion 2. The decoherence of the states on the

thermal subspapce ( ?fri,ermai) with respect to the eigenstates of Ho must be realized in averaging over physi-

cal quantities which become immeasurable in the limit

HI + 0 (derivation of microcanonical ensembles).

Criterion 3. For arbitrary divisionsof a thermal equi-

librium system described by HI into two subsystems (we may consider that one is the heat bath and the other the object), the interaction between the two sub-

systems contained in HI induces only the transitions

between the states on the direct-product space of their

individual thermal subspaces Xg&$’ @7-L$ztal. This means that HI preserves not only the thermal equilibrium of the whole system but also those of arbitrary

subsystems.

Criterion 4. In the same process as noted in Criterion 3 canonical ensembles for the object must be derived

and then the temperature between them must uniquely be determined by using the physical quantities of the heat bath.

In this paper we shall present an exactly solvable

model satisfying the above four criteria.

2. Model

Let us consider the system composed of N oscil-

lators (or N particles) which are described by the

Hamiltonian HO,

Ho=eeajaj, (5) j=l

where a/ and ai are, respectively, the creation and annihilation operators of the energy quantum E and follow the commutation relations [ ai, a,’ ] = 6, and

[ CL~, Uj ] = [ at, a: ] = 0. The eigenstates of HO with the eigenvalues EM = EM (M = 0, 1,2, . . .) are given as

322 T. Kobayashi / Physics Letters A 207 (1995) 320-326

(6)

where 1~) is the usual number state fulfilling the equa-

tions af/ni) = Jm]nj + l), Ui]?rj> = fi\ni - 1)

and ./Ui]ni) = ni]ni). The most general expressions of the states with the energy EM are described by the superposition

(M; Px~ilH) = c C([&l)IM [nil), P[Wl

(7)

where xpIni, stands for the sum over all the different combinations of [nil = (nl , ~22, . . . , nN) and the arbitrary complex numbers C ( [ ni] ) satisfy the normalization condition xp,nil 1 C ( [ ni] ) I2 = 1. In general the states ]M; [C ( [%] ) 1) do not satisfy Criterion 1 (the principle of equal a priori probabilities), that is, in general (C( [ni]>12 # const, V[ ni J . We can easily write the states satisfying Criterion 1 as

(8)

where the number of different combinations of [nil W( M, N) is given by

W(M, N) = (Mf N- l)!

M!(N- l)!

It is apparent that these states are the eigenstates of Ho but span only a subspace of the whole physical space of HO. It is also obvious that the states cannot lead to any decoherence between the states /M; [ ni] ) with different combinations of [nil. We can, however, im- age that there is an interesting possibility for realizing the decoherence, that is, we can introduce phases in the eigenstates such that

(M; [nil, [dil) = ]M; Fnil)exp(iknjfJj), (9) j=1

which induce a phase difference #jk = 8j - 6k between two arbitrary eigenstates, and the average over those phases will be an interesting possibility for the realization of the decoherence [ 61. Note that even after the introduction of such phases the states 1 M; [nil, [ di] ) are the eigenstates of Ho. We shall study the following scenario:

( 1) We construct the new interaction Ht( [O] ) of which the eigenstates are described by

IM, N, [el) = c IM; [nil > [oil). p[ni] &mm (10)

(2) In the limit that HI goes to zero and then the 8j-dependences in the Hamiltonian H = Ho + HI dis- appear, the integrations over the phases @j, which rep- resent the average over immeasurable phases, lead to the decoherence between the states I M; [nil).

3. Introduction of phases and derivation of decoherence

Let us study the introduction of the phases in /M; [nil). Here we introduce the following new operators,

a;(d) 3 ui(e)(fQj + 1)-t/2,

aj(e> s (8j + i)-1/2ai(e), (11)

for i = 1,. . . , N, where uj(e) = Uj eei*, u;(0) =

ui eiB, Rj = ajaj is the number operator for the jth oscillator [ 63. We can easily show that

~~(e)(n,,ej)=[“j+l,ej) V?Zj=O,1,2 ,...,

(ui(B)I?Zj,8j) = ITZj - 1,Sj) Vnj > 1, (12)

with ~j(0)lOj) = 0, where Inj,ej) = Inj) ei”jeJ. Note

also that the operators aj E aj(# = 0) and *I E

aI(e = 0) are nothing but the operators used for the description of the phase operator by Susskind and Glogower [ 71.

Now we can write the interaction depending on the relative phase between the jth and kth oscillators as

jk HI t t 0: (yI(o)Lyk(o) = ajffjt? '+jk, where +jk = 8j -

ok is the relative phase between the two oscillators [ 83. Since any constituents have no special role in thermal equilibria, we may suppose that the interaction is written in terms of totally symmetric operators Af, =

T. Kobayashi / Physics Ldters A 207 (I 995) 320-326 323

where 6s is the coupling constant and $ = c:, fij

(total number operator). Unfortunately the states IM,N,[8]) arenottheeigenstatesofHr([8]),

=EgM lM,N,[B])-~~lOj)("I(M~N'[el) ( ) ’ j=l

(14)

where aj]Oj) = 0. From the above equation we easily see that the relative-phase interaction which has the

states ) M, N, [e] ) as the eigenstates is given by

H,‘( LOI ) = HI( [WI - AHI, (15)

where

AHI = -Es$ 2 ]Oj)(Oj] j=l

(16)

is independent of the phases. Actually it is rewritten in

terms of the relative-phase operators cos ($jk - djk ) E

3[aj(e)(Yk(e) +~~(~)~j(~) + lOj)(Ojl+ IOk)(Okll as

COS($jk - 4jk) . >

(17)

We obtain the relation

HIM, N, [@I) = (E + e,MfIM N, WI) (18)

for H = HO + Hf( [e] ) and then we see that the phys-

ical space of the total Hamiltonian H is spanned by thesetofthestates]M,N,[0])forM=0,1,2,.... (Hereafter we call the states ]M. N, 1~91) the heat-bath states.) We may consider that the relative-phase interaction just plays the role of the projection operator

from the physical space of Ho onto the thermal sub-

space of Ho, which is nothing but the physical space of the total Hamiltonian H. Criterion 1 is obviously satisfied and Criterion 2 can also be shown to be ful-

filled by introducing the integrations over the phases

in the density matrix as

N 2rr Peff = n

s jM,N, [el)(M,Nv tell

j=l o

= wcL NI C IM; [Ql)(M; [&II. (19) ’ PIhI

The integrations over the phases may be understood as the average over the immeasurable quantities 0j.

The dependence of H disappears in the limit of

es --+ 0. Note that in the integrations with respect to the N phases in peg only the N - 1 relative

phases are independent [ 8 1. Because of the free-

dom in the choice of the total phase, (M, N, [ 01) =

eiM’IM,N,[dl), and as the integration over 0

(&%@/2?r)]MN, [el)(M,n, [WI = IM,N, [41) x (M, N, [ 411) does not change any physical results,

we may always use the heat-bath states I&f, N, 181)

having N phase parameters instead of the relative- phase states ]M, N, [ ~$1).

4. A simple expression of Hi([@]) and new operators on the thermal subspace

Here we shall study a new expression of the relative- phase interaction Hf ( [ fl] ) on the thermal subspace of HO. Let us introduce the following new operators,

&E J fi+N ---A@,

N (20)

where & = CE, cj(t?)/fi with fij(e) s .jeisj

x J-

lQj + 1. Note that cyi(B)Lyi(B) = Ajax = I?j,

[aj(@,~k(d)l = ajk and &j(e)]nj,ej) = (nj + I) x )nj + 1,8j) [ 6,8]. Then we easily see that

L49,Asl= 1,

de(M. N, vi) = J;i?lM - 1, N, vi),

deIM,N, [e]) = diGTIM+ l,N, [e]),

&blM,N, [e]) = MIM,N, [e]). (21)

It is important that & is not the Hermitian conjugate of & on the whole physical space of the free Hamil- tonian HO, but Eqs. (21) show that & behaves as the Hermitian conjugate of & on the thermal subspace

(7-&-i) spanned by the states 1 M, N, [e] ) with M =

324 T. Kotuyashi / Physics Letters A 207 (1995) 320-326

0,1,2 ,... . From the last equation of (2 1) we can rewrite Z$( [ 01) in the following very simple form on

7-t thermal>

$([@I) =&+49. (22)

In the following discussion we shall use da and & instead of A* and A!. Therefore the following argu-

ments are exact only on the thermal subspace ?LI,~~~I.

5. Derivation of the canonical ensemble

Let us derive the canonical ensemble. We divide the total system into two groups, bj E aj (j = 1,2,..., No) (b-group) and dk z aj_N,, for j 2

No + 1 (k = j - No = 1,. ..,N - No) (d-group).

Writing the operators & and & in terms of those of

the two groups, t30 and 2% for the b-group and ‘De

and l?e for the d-group, i.e. de = t30 + IDo and so on,

we see that the interaction Hf ( [ f?] ) having the factor

J&de = ri&r f fi*DD, + B&9 + OYl30 (23)

contains the interaction between the two groups writ-

ten as &Do + ‘&238, which induces only the transitions between the states on the direct-product space

of the two thermal subspaces ?&_r @ Xi,,,,,,,. This means that not only the thermal equilibria of the total

system but also those of the two groups are preserved

via the changes induced by the interaction. Now we

can say that Criterion 3 is satisfied. Let us consider Criterion 4 [6,81. The eigenstates

of the total Hamiltonian IM, N, [ 01) can be written down in terms of the product of the eigenstates of the

two groups as

IM, N, [~1[41)

Sh40+MLM di.qiiqq Jw(Mo’ No) IMo, No, [tl])b

MO=0 MI=0

8 ,./~IM,>N, t+l)d,

where Nr = N - No and

IMO. NO, mb

(24)

for the b_group,

[Ml; [WI7 IhI) = c, P[Wl W(MI,NI) ’

for the d-group. (25)

Note that (MO; [njl, [&I> and IMr; [mi], [+il) are, respectively, the eigenstates of the free Hamiltonians

for the b-group and the d-group with the eigenvalues

EMO and EMI. Hereafter we shall treat the b-group as the heat bath and the d-group as the object. Then the

b-group is taken to be much larger than the d-group,

i.e. No >> Nr and Ma > MI. Let us evaluate the density matrix for the object.

Since we do not measure any quantum numbers of

the heat-bath state, the effective density matrix for

the object can be derived by performing the internal

trace(Trr) for the heat-bath variables,

M

=5c aMo+M,.M W(Mo,No)W(Ml,Nl)

MO=0 M,=o W(M, N)

x IMI,NI, [4l)d &‘lvN. [4II. (26)

In the limit where the interaction with the heat bath is small enough to ignore, i.e. es < E, we may again

introduce the average over the immeasurable phases

[ +i], because HE” = E Cz, didk has no [4i]-

dependence. Then we obtain the density matrix for

the mixed states with respect to the eigenstates of Hgb

N, =TT Peff E n I i=l o

M M

= c c bfofM,.M !ggy’ MO=0 M,=o

(27)

Now we can easily see that the effective density matrix fulfills Criterion 1 for the eigenstates of Hgb with the

same energy eigenvalues EOb = E ( M - MO),

(28)

T. Kobayashi/ Physics Letters A 207 (1995) 320-326 325

for all the combinations of [ mj] satisfying x2, mj = We can determine the temperature by using the quan-

M-MO. tities of the heat bath as

Let us investigate how the temperature is intro-

duced in the present scheme. As discussed in statistical mechanics, the canonical ensemble is realized in the state having the maximum probability. The probability of finding an energy eigenstate of the object,

IM-Mo;N-No, [41) d, in p,f( [ 41) is evaluated as

P(M - MO) = W(Mo, No) W(M - MO, N - No)

W(M, N) (29)

kT=E[ln(l + NO/MO)]-'. (33)

We easily see that the temperature increases as the energy of the heat bath increases and the temperature becomes zero for the ground state (MO = 0). Then

we can write the density matrix in the large heat-bath

limit M- MO, N N No + 00 as

Then at the maximum of the probability, where the equation aP( M - MO) /aMo = 0 is satisfied, we find

the following relation, X c IMI; [mJ)(M1; [mill

, I’ I m I zh

(34)

I dW(Mo, No) W(MO> No) dM0

1 aW(Ml,N - No) = W(M,,N- No) aMI M,=M-MO

(30)

This is the well-known relation in statistical mechanics

for two systems connected only via heat transfer. Then

the temperature is introduced as the common physical

quantity between the two systems,

T_, _ Whb) as( ob) _.=-

aEhb aEOb E&E-E"b' (31)

where S( hb) and S(ob), respectively, stand for the

Boltzmann entropies of the heat bath and the object defined by S = kln W, and Ebb = EMO and EOb = EMI are their energies. The temperature must be determined only by the properties of the heat bath. It is trivial

that the temperature is independent of Ml and N1,

because the coefficients for the eigenstates (MI; [mj]) in the density matrix p,~, W( MO, No)/W( M, N), are written only in terms of the heat-bath variables MO and NO. Using the Stirling formula, we can estimate the change of the coefficients with respect to the change of the energy of the object, which is described to the

order O(M,), that is, AEOb = EAMI with AMI N

MI e M, MO, as

W(Mo -AMl,No) W(Mo,No) N (MoYN~)AM'

= exp -iAEUbIn( 1 + NO/MO) . >

(32)

where the normalization factor is evaluated as ZN, = Z,“’ , where Z1 = ( 1 - e-p’) -I is nothing but the par-

tition function for the canonical ensemble. Criterion 4

is now satisfied.

6. Summary and remarks

We may summarize the important results in the

present model as follows:

( 1) The thermal subspace ‘&e,,,,al of HO is spanned

bythestatesIM,N,[8])withM=0,1,2 ,.... (2) The relative-phase interaction q( [ 191) plays

an essential role in projecting the whole physical space

of HO onto the thermal subspace &,erma~. (3) The partial trace operation which describes the

average over immeasurable phases (di) induces the decoherence with respect to the eigenstates of Ho.

(4) The canonical ensemble is derived by the partial

trace operation with respect to all the heat-bath variables and the average over the immeasurable phases of

the object, and the temperature can uniquely be written only in terms of the physical quantities of the heat bath.

(5) On the thermal subspace ‘&,ermal we can simply write the interaction H;( [19]) in terms of the new operators de and &. They are mutually Hermitian

conjugates on 7-&hermal, but not on the whole physical space of Ho. Note that the introduction of the new operators makes the discussion of tither,& very simple.

We still have many problems to investigate. The most important one is to check the existence of the

326 T. Kobayashi / Physics Letters A 207 (1995) 320-326

relative-phase interaction in thermal equilibria. Even in the limit l s -+ 0 we know of some kinds of interactions which describe the energy transfer between the heat bath and the object in thermal equilibrium, that is, ELM will be a measurable finite number (not zero) even in the limit es + 0. We shall be able to measure precisely the effects induced by such interactions.

It is also interesting that in the present scheme we can keep the total number of oscillators N fixed in the derivation of the thermal equilibrium. When N is a fixed number the density matrices pea have some de- viation from the exact canonical ones obtained in the limit N -+ co. Such deviations might be observable in realistic physical processes such as mesoscopic phe- nomena for systems consisting of a few constituents.

What we have presented is still a toy model. We have to investigate thermal equilibria for more real-

istic systems having different energy scales EL (i =

1,2,X. . .) and l i # Ej for i # j. This problem will be discussed elsewhere.

References

[ I] 1. von Neumann, Die Mathematische Gnmdlagen der Quantenmechanik (Springer, Berlin, 1932)

[2] J.A. Wheeler and W.H. Zurek, eds. Quantum theory and measurement (Princeton Univ. Press, Princeton, 1983).

[3] T. Kobayashi, Phys. Lett. A 185 (1994) 349. [4] T. Kobayashi and K. Ohmomo, Phys. Rev. A 41 (1990) 5789. [5] T. Kobayashi, Nuovo Cimento 107B (1992) 657. [ 61 T. Kobayashi, preprint of University of Tsukuba, UTHEP-294

(1995). [ 71 L. Susskind and J. Glogower, Physics 1 ( 1964) 49. [8] T. Kobayashi, preprint UTHEP-303 (1995). University of

Tsukuba.

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Quantum mechanical description of thermal equilibrium states as eigenstates of relative-phase interactions