- Email: [email protected]
On the full quantum description of nonlinear dynamic systems in the phase-space representation
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 37 (2008) 369–386 www.elsevier.com/locate/chaos
On the full quantum description of nonlinear dynamic systems in the phase-space representation S. Maximov a
a,*
, L.S. Kuzmenkov
b
Instituto Tecnolo´gico de Morelia, PGIIE, Av. Tecnolo´gico No. 1500, Lomas de Santiaguito, C.P. 58120, Morelia, Mich., Me´xico b Moscow State University ‘‘M.V. Lomonosov’’, Vorobyevy gory, Moscow, 119899, Russia Accepted 1 September 2006
Communicated by Prof. M.S. El Naschie
Abstract An approach to the completely quantized description of nonlinear dynamic systems with chaos is considered. The method bases on the correspondence between operators and their symbols in a phase-space representation. Heisenberg equations in a wide class of phase-space representations are obtained. Nonlinear oscillations of the coupled quasiparticles-oscillator system are analyzed in the completely quantized description in terms of symbols of operators in the Wigner representation. The asymptotic solutions of the Heisenberg equations for the symbols Qt, Pt, etc. are obtained for a stable region of the phase-space, and the respective quantum observables are calculated. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The correspondence between classical nonlinear dynamics and its quantum counterpart is of interest for the class of problems denominated ‘‘quantum chaos’’. The analysis of completely quantized nonlinear dynamic systems is sometimes complicated. Therefore different kinds of approximations, such as quasiclassical approximation or Born–Oppenheimer approximation, are applied to treat these systems. Another approach to this problem consists in the analyzing the development of the density matrix in the Wigner or the Husimi representations [1]. Nevertheless, the problem of the completely quantized description of nonlinear dynamic systems remains actual. In the present paper, a possible approach to the fully quantized description of such systems is offered. The approach is based on the correspondence between operators and their symbols in a phase-space representation. This correspondence maps the dynamics of operators in the Heisenberg representation to the dynamics of symbols. Respective equations of the dynamics of symbols are obtained for a wide class of phase-space representations. A complete description of any quantum dynamic system requires determination of the propagator in the phase-space representation. For nonlinear dynamic systems with chaos this problem is very difficult due to non-integrabilities. Nevertheless, evolution of the symbols such as qt and pt, which correspond to the operators ^ qt and ^pt , does not require *
Corresponding author. Fax: +52 443 3171870. E-mail address: [email protected] (S. Maximov).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.040
370
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
calculation of the full quantum propagator. As a result, the quantum observables hqti and hpti can be calculated relatively simply. Nonlinear oscillations of the coupled quasiparticles-oscillator system are considered in the completely quantum description in terms of symbols of operators in one of the phase-space representations. The interest in the dynamic properties of this system is due to the applicability of the mathematical model to a wide range of physical phenomena. The model leads to the spin-boson Hamiltonian of the next form [2]: 1=2 b aþ ^ b 2 Þ 1 ð^aþ ^a2 þ ^aþ ^a1 Þ þ p b ¼ 1 ðP^ 2 þ r2 Q H a1 Þ; r Qð^ aþ ð1Þ 1 2 2 a2 ^ 1^ 2 2 2 b ¼ 1, ½^ak ; ^aþ ¼ dkj , ½^ak ; Q b ¼ ½^ak ; Pb ¼ 0. In the mixed quantum–classical description the dynamic where h 1, i½ Pb ; Q j properties of this system have been previously considered in the papers [2–4]. In [5] the quasiclassical approximation was applied to the quasiparticle-oscillator system to obtain asymptotic solutions. Heisenberg equations, with the Hamiltonian (1) in the quasiclassical approximation, was led to a single ordinary differential equation. The stability problem was investigated and the asymptotic solution around one of the stable points was found. b Pb , ^ In this paper the system of equations for the symbols of the operators Q, q1;2 , and ^ p1;2 is obtained. In the Wigner representation these equations are similar to the equations in the quasiclassical approximation, obtained in the paper [5]. The results of paper [5] are used to obtain the quantum solutions for this system. The respective quantum observables are calculated for a postulated initial density matrix in the Wigner representation.
2. Symbols of operators b B; b . . . in the Schro¨dinger representation and functions b . . . ; C; Let us consider a correspondence between operators A; A(x), B(x), . . . , C(x), . . . of a set of variables x=(x1, x2, . . . , xn): b $ AðxÞ; B b $ CðxÞ; . . . b $ BðxÞ; . . . ; C A b b ðxÞ as follows: This correspondence can be realized by means of the operators DðxÞ and U b b ðxÞ AÞ; AðxÞ ¼ trð U Z b ¼ dx DðxÞAðxÞ: b A
ð2Þ ð3Þ
b ðxÞ and DðxÞ b Eqs. (2) and (3) lead to the next two relations between the operators U b ðxÞ Dðx b 0 ÞÞ ¼ dðx x0 Þ; trð U Z 0 b 0 0 0 b dxha1 j DðxÞja 2 iha2 j U ðxÞja1 i ¼ dða1 a1 Þdða2 a2 Þ;
ð4Þ ð5Þ
where {jai} is a set of vectors orthogonal and complete. We call functions A(x), B(x), . . . , C(x),. . . symbols of operators b B; b . . .. b . . . ; C; A; Let the mapping (2) be such, that ð^q; ^pÞ7!ðq; pÞ, where at the initial moment of time q and p are eigenvalues of the operators ^q and ^p, i. e. ^qjqi ¼ qjqi, ^pjpi ¼ pjpi. Then, we can take (q, p) as the set of variables x, and the relation ^ ¼ ð^q; ^pÞ and the symbol x = (q, p) takes the next form: between the operator x Z b ðxÞ^ b ^ ¼ dx DðxÞx: x ¼ trð U xÞ; x ð6Þ Mapping (2) transforms commutators into Moyal brackets in the phase-space: Z Z i b b ½ A; B7!fAðxÞ; BðxÞgM ¼ dx1 dx2 Kðx; x1 ; x2 ÞAðx1 ÞBðx2 Þ; h where
b 1 Þ; Dðx b ðxÞ i ½ Dðx b 2 Þ : Kðx; x1 ; x2 Þ ¼ tr U h
b ðxÞ maps polynomial operators of ^ Let the operator U q and ^ p to the corresponding polynomials of q and p: 1 1 X X b¼ Am;n K½^qm ^pn 7!Aðq; pÞ ¼ Am;n qm pn : A m;n¼0
m;n¼0
ð7Þ
ð8Þ
ð9Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
371
Eq. (9) is a simple generalization of the well known ^q^p-, ^ p^ q- and Weyl-quantizations [6], in which polynomial operators b differ from each other only by means of the ordering K½^ pn of the operators ^ A qm ^ qy^ p: X X m n m n cP P½^q ^p ; cP ¼ 1; cP 2 R: ð10Þ K½^q ^p ¼ P
P
P is a permutation of the operators ^q and ^p in q^m ^pn : mþn X P½^qm ^pn ¼ ^qa1 ^p1a1 ^qa2 ^p1a2 ^qamþn ^p1amþn ; ak ¼ m;
ð11Þ
k¼1
where each ak takes the values 1 or 0. Excluding coefficients Amn from the formula (9), we find Z Z 1 X 1 omþn AðxÞ m n b ¼ dx b ^ p dðxÞ K½^ q ¼ dx DðxÞAðxÞ; A m!n! oqm opn m;n¼0 where b DðxÞ ¼
1 X ð1Þmþn ðmÞ d ðqÞdðnÞ ðpÞK½^qm ^pn : m!n! m;n¼0
ð12Þ
Defining explicitly the ordering (10) we completely determine the type of quantization. For example, any linear quantization leads to the correspondence of the next form [6]: b pa $ L2 a AðxÞ; ^qa A b qa $ L2 a AðxÞ; b $ L1 a AðxÞ; A^ b $ L1 a AðxÞ; A^ ^pa A ð13Þ p p q q a
where Lip are the first order differential operators s X a ab o ab o ab b b AðxÞ; aab q þ b p þ f þ g Lip AðxÞ ¼ i i i i oqb opb b¼1
ð14Þ
ab ab ab i ^b ¼ dab $ fpa ; qb gM ¼ dab . Let us pa ; q where the coefficients aab i ; bi ; fi ; g i are those to satisfy the correspondence h ½^ operate differently. We generalize the Eqs. (13) and (14) by imposing the next condition for the form of the operator b DðxÞ (12):
b i b o DðxÞ ½ DðxÞ; ^q ¼ ; h op b i b o DðxÞ ½ DðxÞ; ^p ¼ h oq
ð15Þ ð16Þ
b we obtain on the one hand b b ðxÞ. From the Eq. (15), for an arbitrary operator A, and search the operators DðxÞ and U i b b i ½ DðxÞ; b b ^q ¼ tr DðxÞ ½ A; ^q ¼ fA; qgM : tr A h h On the other hand we have ! b o DðxÞ o b b oAðxÞ b ¼ tr A DðxÞ ¼ : tr A op op op Similar manipulations we can fulfill for the Eq. (16). Thus the Eqs. (15) and (16) lead to the following relations for every symbol A(x): i b oA ½ A; ^q $ fA; qgM ¼ ; h op
i b oA ½ A; ^p $ fA; pgM ¼ : h oq
These relations are more general than the relations (13) and (14). b Operator DðxÞ in the Eq. (12) can be considered as the result of quantization of the classical microscopic distribution function d(x x(t)): 1 X ð1Þmþn ðmÞ d ðqÞdðnÞ ðpÞqm ðtÞpn ðtÞ dðx xðtÞÞ ¼ m!n! m;n¼0 pn of the operators ^ qm ^ q and ^ p. On in which the product of canonical variables qm(t)pn(t) is replaced by the ordering K½^ 0 ^ the other hand, Eq. (4) permits the next clear interpretation: function d(x x ) is the symbol of the operator DðxÞ. In
372
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
b other words, DðxÞ is the operator corresponding to the classical microscopic distribution function; for every phaseb space representation of quantum mechanics, symbol of the operator DðxÞ coincides with the classical microscopic distribution function. b is similar to the classical statistical average of a function A(x) with the distriQuantum average of the operator A bution function f ðx; tÞ: Z Z b ¼ dxtr q ^ ^ðtÞ A ^ðtÞDðxÞ hAi ¼ tr q AðxÞ ¼ dxf ðx; tÞAðxÞ; ð17Þ where ^ ^ðtÞDðxÞ f ðx; tÞ ¼ tr q :
ð18Þ
Eq. (17) shows the necessity of considering symbols of the type (18). Therefore, alongside with the symbols A(x), we shall consider dual symbols AðxÞ which are given by the next equations: Z ^ ðxÞ A ^ b ; A b ¼ dxDðxÞAðxÞ; AðxÞ ¼ tr U ^ ðxÞ and DðxÞ ^ ^ ðxÞ and DðxÞ b where the operators U are related with the operators U through the formulae ^ ðxÞ ¼ 2ph DðxÞ; b U
^ b ðxÞ: DðxÞ ¼ ð2phÞ1 U
b b 3. Structure of the operators DðxÞ and UðxÞ qmk ^ pnk . Such Commuting operators ^q and ^p, we expand the permutation (11) in series in terms of the type ði hÞk ^ terms appear after commuting k pares of operators ^q and ^ p. As a result we obtain K½^qm ^pn ¼
minfm;ng X
ðihÞk C km;n ^qmk ^pnk ;
ð19Þ
k¼0
P where C km;n 2 R. From the equation P cP ¼ 1 we obtain the initial condition for C km;n : C 0m;n ¼ 1. Substituting (19) into b (12) we come to the next formula for the operator DðxÞ: minfm;ng 1 X X ð1Þmþn ðmÞ qmk ^ pnk : d ðqÞdðnÞ ðpÞ ðihÞk C km;n ^ m!n! k¼0 m;n¼0 Pminfm;ng P we obtain Rearranging the summations 1 m;n¼0 and k¼0
b DðxÞ ¼
b DðxÞ ¼ ¼
1 X 1 X ð1Þmþn ðmÞ d ðqÞdðnÞ ðpÞðihÞk C km;n ^ qmk ^ pnk mn k¼0 m;n¼k 1 k X X ðihoq op Þk k¼0
m;n¼0
ð1Þmþn Ck dðmÞ ðqÞdðnÞ ðpÞ^ qm ^ pn : ðm þ kÞ!ðn þ kÞ! mþk;nþk
ð20Þ
b Coefficients C km;n completely determine the operator DðxÞ. To find these coefficients we substitute (20) into (15) and (16). The left-hand side of the Eq. (16) gives 1 1 X X ð1Þmþn C kmþkþ1;nþk ðm þ 1Þ ðmÞ i b ½ DðxÞ; ^p ¼ oq d ðqÞdðnÞ ðpÞ^ ðihoq op Þk qm ^ pn : h ðm þ k þ 1Þ!ðn þ kÞ! k¼0 m;n¼0
Substituting this result into the Eq. (16) we obtain the next equation: oq
1 1 X X ð1Þmþn C kmþkþ1;nþk ðm þ 1Þ ðmÞ pn ðihoq op Þk qm ^ d ðqÞdðnÞ ðpÞ^ ðm þ k þ 1Þ!ðn þ kÞ! k¼0 m;n¼0
¼ oq
1 k X X ðihoq op Þk k¼0
m;n¼0
ð1Þmþn Ck pn : dðmÞ ðqÞdðnÞ ðpÞ^ qm ^ ðm þ kÞ!ðn þ kÞ! mþk;nþk
On the other hand, substituting the formula (20) into the Eq. (15), we find
ð21Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
op
373
1 1 X X ð1Þmþn C kmþk;nþkþ1 ðn þ 1Þ ðmÞ d ðqÞdðnÞ ðpÞ^ pn ðihoq op Þk qm ^ ðm þ kÞ!ðn þ k þ 1Þ! k¼0 m;n¼0
¼ op
1 k X X ðihoq op Þk k¼0
m;n¼0
ð1Þmþn pn : dðmÞ ðqÞdðnÞ ðpÞ^ qm ^ Ck ðm þ kÞ!ðn þ kÞ! mþk;nþk
ð22Þ
Eqs. (21) and (22) result in the following system of recurrent equations for the coefficients C km;n with the respective initial condition 8 > Ck ¼ mþk C kmþk1;nþk ; > m < mþk;nþk C kmþk;nþk1 ; C kmþk;nþk ¼ nþk ð23Þ n > > : C 0 ¼ 1: m;n
From the first equation of the system (23) we obtain C kmþk;nþk ¼
ðm þ kÞ! k C k;nþk : k!m!
Substituting this result into the second equation of the system (23) we finally find C kk;nþk ¼
ðn þ kÞ! k ðn þ kÞ!k! C k;k ¼ fk ; n!k! n!
where C kk;k fk ðk!Þ2 :
The initial condition in the system (23) results in f0 ¼ C 00;0 ¼ 1. Finally the solution of the system of Eqs. (23) has the next form: C kmþk;nþk ¼
ðm þ kÞ!ðn þ kÞ! fk : m!n!
b Substituting this result into the Eq. (20) we find the operator DðxÞ: 1 X b b ^q^p ðxÞ; where Lx ¼ fk ðihoq op Þk : DðxÞ ¼ Lx D
ð24Þ
k¼0
b ^q^p ðxÞ corresponds to the ^q^p-quantization of quantum mechanics: D Z 1 X ð1Þmþn ðmÞ du dv iuðq^qÞ ivðp^pÞ b ^q^p ðxÞ ¼ e e ¼ jqihqjpihpj: d ðqÞdðnÞ ðpÞ^qm ^pn ¼ D 2 m!n! ð2pÞ m;n¼0
ð25Þ
^ If exists the inverse operator L1 x for the differential operator Lx , then the operator U ðxÞ has the form Z b ðxÞ ¼ dx0 Dðx; x0 Þ U b ^q^p ðx0 Þ; U b ^q^p ðxÞ can be obtained from (25) where D(x, x 0 ) satisfies the equations Lx Dðx0 ; xÞ ¼ Lx0 Dðx0 ; xÞ ¼ dðx x0 Þ. Operator U b ^q^p ðxÞ ¼ 2phjpihpjqihqj U to satisfy the Eqs. (4) and (5). P1 k UðzÞ , where U(z) is an analytic function of z that satisfies Let us suppose that converges the series k¼0 fk z ¼ e b b ðxÞ obtain the form U(0) = 0 (because f0 = 1). Then, the operators DðxÞ and U b ^q^p ðxÞ; b DðxÞ ¼ eUðihoq op Þ D Uði h o o Þb q p b ðxÞ ¼ e U ^q^p ðxÞ: U
ð26Þ ð27Þ
b Substituting the operator DðxÞ from the Eq. (26) into (3) and integrating by parts over the variables q and p, Z Z Z Uðih~ oq~ op Þ b b b ^q^p ðxÞ; b D ^q^p ðxÞ ¼ dxAðxÞeUðih o q o p Þ D A ¼ dx DðxÞAðxÞ ¼ dxAðxÞe we obtain the next relation between the symbol A(x) and the symbol A^q^p ðxÞ in the ^ q^ p-quantization: eUðihoq op Þ AðxÞ ¼ A^q^p ðxÞ: All the results (24), P (26) and (27), obtained for one degree of freedom, can be generalized for s degrees of freedom by substituting oq op si¼1 oq i op i . If fk = vk/k!, then we obtain a class of linear quantizations where
374
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
b v ðxÞ ¼ eivhoq op D b ^q^p ðxÞ; D
b ^q^p ðxÞ: b v ðxÞ ¼ eivhoq op U U
ð28Þ
In particular if v = 1/2, then we obtain the well-known Wigner representation. Cases v = 0 and v = 1 lead to the ^ q^p- and p^^ q-quantizations respectively. Moyal brackets in the phase-space representations, determined by the operators (26) and (27), can be found by substituting (26) and (27) into the formulae (7) and (8). As a result we find ! ~~ Uðih~ oq~ Uðih~ oq o p Þ i ih o p~ ih o q o p op Þ oq fA; BgM ¼ e eUðihoq op Þ BðxÞ: AðxÞe e ð29Þ e h op ðABÞ ¼ Aðo q o p þ o q~ op þ o p~ oq þ ~ op ÞB, after some transformations the formula (29) in the linear oq~ Noticing that ~ oq~ quantization (28) can be led to the following simple form: i ihð1vÞ o p~oq þihv o q~op ~ ~ e fA; Bgv ¼ AðxÞ ð30Þ eihð1vÞ o q op þihv o p oq BðxÞ: h Of course, Eqs. (28) does not include all the linear phase-space representations of the type (13), (14). A more wide class of quantizations can be obtained as a correspondence between operators and symbols of the type b¼ A
1 X
Am;n K½ða^q þ b^pÞm ðc^q þ d^pÞn $ Aðq; pÞ ¼
m;n¼0
1 X
Am;n ðaq þ bpÞm ðcq þ dpÞn ;
ð31Þ
m;n¼0
where ad bc = 1. The same result we can obtain by fulfilling in the Eq. (24) a linear canonical transformation. Husimi-like representations can be obtained if we assign in the Eq. (31) s1 a ¼ c ¼ pffiffiffiffiffi ; 2h
is b ¼ pffiffiffiffiffi ; 2h
and d ¼ b :
4. Symbols of operators as functions of the symbols qt and pt b 0 ðnÞ be an operator in the Schro¨dinger representation. n is one or a set of parameters (in particular n = t). The Let A b 0 ðnÞ and its symbol A0(x, n) is given by the next formula: correspondence between the operator A ^ 0 ðnÞ ; b ðx0 ÞA A0 ðx0 ; nÞ ¼ tr U ^0 ¼ ð^ where x0, according to the Eq. (6), is the symbol of operator x q0 ; ^ p0 Þ in the Schro¨dinger representation. Evolution b t ðnÞ in the Heisenbeg representation is given by the formula of the operator A b t ðnÞ G b t ;t A b t ;t ; b t ðnÞ ¼ G A 2 2 1 1 1 2 b t ;t satisfies the equations where the operator G 2 1 b t ;t oG i b 2 1 b ¼ H t ðt 2 Þ G t2 ;t1 ; h 2 ot2
b t ;t oG i^ 2 1 b ¼ G t ;t H t ðt 1 Þ: h 2 1 1 ot1 b t ðnÞ, is related with the symbol A0(x0, n) through the propagator Symbol At2 ðx0 ; nÞ, corresponding to the operator A 2 G(x, tjx0, t0) in the phase-space representation: Z Z 0 0 b 0 b b b b b At ðx0 ; nÞ ¼ trð U ðx0 Þ A t ðnÞÞ ¼ dx0 trð U ðx0 Þ G t;t0 Dðx0 Þ G t0 ;t ÞAt0 ðx0 ; nÞ ¼ dx00 Gðx0 ; tjx00 ; t0 ÞAt0 ðx00 ; nÞ; ð32Þ where b t;t Dðx b t ;t Þ: b 0ÞG b ðxÞ G Gðx; tjx0 ; t0 Þ ¼ trð U 0 0 Because for an infinitesimal Dt b tþDt;t ¼ ^1 þ i H b t ðtÞDt þ oðDtÞ; G h
b t;tþDt ¼ ^1 i H b t ðtÞDt þ oðDtÞ; G h
then, for two next moments of time t + Dt and t we obtain b t ðtÞ; Dðx b ðxÞ i ½ H b 0 ÞÞ þ oðDtÞ: Gðx; t þ Dtjx0 ; tÞ ¼ dðx x0 Þ þ Dt trð U h R b 00 ÞH t ðx00 ; tÞ we obtain b t ðtÞ ¼ dx00 Dðx Substituting here H
ð33Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Gðx; t þ Dtjx0 ; tÞ ¼ dðx x0 Þ þ ¼ dðx x0 Þ þ
Z Z
dx00 dx00
Z Z
375
h i b ðxÞ i Dðx b 00 Þ H t ðx00 ; tÞdðx00 x0 Þ þ oðDtÞ b 00 Þ; Dðx dx00 Dt tr U h dx00 Dt Kðx; x00 ; x00 ÞH t ðx00 ; tÞdðx00 x0 Þ þ oðDtÞ
¼ ð1 Dt½H t ðtÞM Þdðx x0 Þ þ oðDtÞ ¼ e½H t ðtÞM Dt dðx x0 Þ þ oðDtÞ;
ð34Þ
where ½H M A ¼ fH ; AgM :
ð35Þ
Let us divide the interval of time [t0, t] on n small segments t0 = s0 < s1 < < sn = t. Then, taking into account the Eqs. (4) and (5) we find Z Y n1 n Y dxðjÞ G xðkÞ ; sk xðk1Þ ; sk1 ; Gðx; tjx0 ; t0 Þ ¼ j¼1
(0)
= x 0 and x
k¼1
(n)
= x. Substituting here the result (34) and passing to the limit max{Dtk} ! 0 we at last obtain Rt ½H s ðsÞM ds : Gðx; tjx0 ; t0 Þ ¼ Gt;t0 dðx x0 Þ; where Gt;t0 ¼ e t0
where x
Then, Eq. (32) takes the following form: At ðx0 ; nÞ ¼ Gt;0 A0 ðx0 ; nÞ:
ð36Þ
Symbol x0 at the initial moment of time is related with the symbol xt at the moment t by the same Eq. (36). From this equation we obtain that x0 ¼ G1 t;0 xt ¼ G0;t xt . Substituting this formula into the Eq. (36) we find the dependence of the symbol At at the moment of time t on the symbol xt in the x0-representation: At ðxt ; nÞ ¼ Gt;0 A0 ðG0;t xt ; nÞ:
ð37Þ
Eq. (37) can be generalized as follows: At2 ðxt2 ; nÞ ¼ Gt2 ;0 G0;t1 At1 ðGt2 ;0 G0;t1 Þ1 xt1 ; n ¼ Gt2 ;t1 At1 ðG1 t2 ;t1 xt1 ; nÞ:
5. Equations of motion in the phase-space representations Let us deduce Heisenberg equations in the phase-space representation. We assume that n = t. Mapping (2) transforms the increment of the operator At(t) into the increment of the corresponding symbol: b tþDt ðt þ DtÞ A b t ðtÞ7!AtþDt ðxt ; t þ DtÞ At ðxt ; tÞ ¼ AtþDt ðxt ; t þ DtÞ At ðxt ; t þ DtÞ þ At ðxt ; t þ DtÞ At ðxt ; tÞ: A The linear part of the increment of the symbol At(xt, t), in the right hand side of this equation, represents the absolute differential of this symbol. On the one hand we find AtþDt ðxt ; t þ DtÞ At ðxt ; t þ DtÞ ¼ ð1 ½H t ðtÞM Dt þ oðDtÞÞAt ðxt ; t þ DtÞ At ðxt ; t þ DtÞ ¼ ½H t ðtÞM DtAt ðxt ; t þ DtÞ þ oðDtÞ: On the other hand At ðxt ; t þ DtÞ At ðxt ; tÞ ¼
oAt ðxt ; tÞ Dt þ oðDtÞ: ot
Passing to the limit Dt ! 0 in the equation AtþDt ðxt ; t þ DtÞ At ðxt ; tÞ oAt ðxt ; tÞ oðDtÞ ¼ ½H t ðtÞM At ðxt ; t þ DtÞ þ þ Dt ot Dt and taking into account (35) we obtain the next equation: rt At ¼ fH t ; At gM þ
oAt ; ot
b t =dt has been transformed into the covariant derivative where the derivative of operator d A
ð38Þ
376
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
rt At ðxt ; tÞ lim
Dt!0
AtþDt ðxt ; t þ DtÞ At ðxt ; tÞ : Dt
Together with the covariant derivative $tAt(xt, t), we can consider the derivative dAt(xt, t)/dt which should be defined as follows: dAt ðxt ; tÞ At ðxtþDt ; t þ DtÞ At ðxt ; tÞ ¼ lim : Dt!0 dt Dt
ð39Þ
Because Gt+Dt,t = 1 Dt[Ht(t)] + o(Dt), then, taking into account the Eqs. (15) and (16) we can obtain oH t oH t xtþDt ¼ GtþDt;t xt ¼ xt þ DtfH t ðtÞ; xt gM þ oðDtÞ ¼ qt þ Dt þ oðDtÞ: ; pt Dt opt oqt Substituting this result into the Eq. (39) we obtain the formula for the total time derivative: dAt oH t oAt oH t oAt oAt oAt ¼ ¼ fH t ; At gcl þ : þ dt opt oqt oqt opt ot ot
ð40Þ
If we denote [C]At = [Ht]clAt [Ht]MAt, then the relation between the derivatives $tAt(xt,t) and dAt(xt,t)/dt can be written in the next form: rt At ¼
dAt þ ½CAt ; dt
ð41Þ
For the phase-space representations of the type (28) we can obtain k
nk 1 n
X X 1 ~ ~ ~ ~ ~ ~ ihð1 vÞo p oq þ i hvo q op i hð1 vÞo q op þ i hvo p oq At : ½CAt ¼ H t ðo p oq o q op Þ ðn þ 1Þ k¼0 n¼1
ð42Þ
In particular Eq. (38), in view of the conditions (15) and (16), leads to the equations of motion for the symbols qt and pt, which coincide with the equations of Hamilton in classical mechanics: t t rt qt ¼ dq ¼ fH t ; qt gM ¼ oH ; dt opt t t ¼ fH t ; pt gM ¼ oH : rt pt ¼ dp dt oqt
ð43Þ
b t ðtÞ in the Heisenberg representation can be calculated as follows: Quantum observable hAti of any operator A Z Z b t ðtÞÞ ¼ dx0 trð^ b 0 ÞÞAt ðx0 ; tÞ ¼ dx0 f 0 ðx0 ÞAt ðx0 ; tÞ: q0 A q0 Dðx ð44Þ hAt i ¼ trð^ b t ðtÞ, we should solve the Eq. In other words, formula (44) means that to calculate quantum average of the operator A (38) and obtain the symbol At(x0,t) as function of the initial phase-space coordinates x0, then calculate the integrals (44) over the initial coordinates x0 with the initial covariant distribution density f 0 ðx0 Þ. This situation is similar to the calculation of the classical statistical averages over the initial states in the phase-space. A complete description of any quantum dynamic system requires determination of all the symbols at every moment 0 of time. In other words, the calculation of the kernel G(x,tjx ,t0) of the propagator is inevitable. Nevertheless, there are problems in which knowledge of only two observables hqti and hpti is required. In this case, we should solve only the system of Eqs. (43) with some initial conditions x0 and substitute the solution into the Eq. (44). In the next sections we are applying this scheme to the investigation of nonlinear oscillations of a molecular dimer.
6. Coupled quasiparticles-oscillator system in the Wigner representation We now consider the spin-boson Hamiltonian (1). Let us obtain the system of equations of the type (43) for a system with the Hamiltonian (1) in the Wigner representation. In this representation both types of symbols A(x) and AðxÞ coincide. Substituting in (1) 1 ^aþ q1;2 i^p1;2 Þ; 1;2 ¼ pffiffiffi ð^ 2
1 ^a1;2 ¼ pffiffiffi ð^q1;2 þ i^p1;2 Þ 2
we come to the formula 1=2 b q2 þ ^ b 2 1 ð^q1 ^q2 þ ^p1 ^p2 Þ þ 1 p b ¼ 1 ^p2 þ r2 Q r Qð^ p22 ^ q21 ^ p21 Þ: H 2 2 2 2 2
ð45Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
377
b (45) in the Wigner representation has the form The symbol corresponding to the Hamiltonian H
Z Z Z n n n b n n1 n2 H ðQ; P ; q1 ; p1 ; q2 ; p2 Þ ¼ dn dn1 dn2 eiðP nþp1 n1 þp2 n2 Þ Q ; q1 1 ; q2 2 H Q þ ; q1 þ ; q2 þ 2 2 2 2 2 2 R3
1 1 1 p1=2 rQðq22 þ p22 q21 p21 Þ: ¼ ðP 2 þ r2 Q2 Þ ðq1 q2 þ p1 p2 Þ þ 2 2 2 2
ð46Þ
Then, the equations of motion for the symbols Q, P, q1,2, and p1,2 take the following simple form: oH rt Q ¼ Q_ ¼ ¼ P; oP rffiffiffi oH p 1 2 2 _ ¼ r Q r ðq þ p22 q21 p21 Þ; rt P ¼ P ¼ oQ 2 2 2 rffiffiffi oH 1 p rQp1 ; ¼ p2 rt q1 ¼ q_ 1 ¼ op1 2 2 rffiffiffi oH 1 p rQq1 ; rt p1 ¼ p_ 1 ¼ ¼ q þ oq1 2 2 2 rffiffiffi oH 1 p rQp2 ; ¼ p1 þ rt q2 ¼ q_ 2 ¼ op2 2 2 rffiffiffi oH 1 p rQq2 : rt p2 ¼ p_ 2 ¼ ¼ q oq2 2 1 2
ð47Þ
Let us introduce new symbols x ¼ q1 q2 þ p1 p2 ;
y ¼ q1 p2 þ q2 p1 ;
1 z ¼ ðq22 þ p22 q21 p21 Þ: 2
Equations of motion (47) in terms of new variables take the form: pffiffiffiffiffi x_ ¼ 2prQy; pffiffiffiffiffi y_ ¼ z þ 2prQx; z_ ¼ y; rffiffiffi € þ r2 Q ¼ r pz; Q 2
ð48Þ
ð49Þ ð50Þ ð51Þ ð52Þ
and the symbol (46) becomes 1 1 H ¼ ðQ_ 2 þ r2 Q2 Þ x þ 2 2
rffiffiffi p rQz: 2
ð53Þ
Moreover, it is easy to verify, using (42) for the Wigner representation (v = 1/2) that rt x ¼ x_ , rt y ¼ y_ , and rt z ¼ z_ . Also we can note that symbols x, y, and z satisfy to the next relations: fx; ygw ¼ 2z;
fy; zgw ¼ 2x;
fz; xgw ¼ 2y;
where f ; gw are the Moyal brackets in the Wigner representation. The explicit form of these brackets can be obtained from the formula (30) by assigning v = 1/2. b of number of excitons in the system. On the other hand, we can find the symbol that corresponds to the operator N b has the form The operator N 1 b ¼ ^aþ N a1 þ ^aþ a2 ¼ ð^q21 þ ^p21 þ ^q22 þ ^p22 Þ 1: 1^ 2^ 2 Then, the corresponding symbol is
Z Z n n b n1 n2 1 Nðq1 ; p1 ; q2 ; p2 Þ ¼ ¼ ðq21 þ p21 þ q22 þ p22 Þ 1: dn1 dn2 eiðp1 n1 þp2 n2 Þ q1 1 ; q2 2 N q1 þ ; q2 þ 2 2 2 2 2
ð54Þ
R2
From the definition of the variables x, y, and z (48) and the Eq. (54) we find x2 þ y 2 þ z2 ¼ ðN ðq1 ; p1 ; q2 ; p2 Þ þ 1Þ2 :
ð55Þ
378
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Here appears a difference between the formula (55) and the results of papers [5,7]. In the scheme realized in [5], all the operators has been changed to the respective C-numbers. It has led to the following relation between x, y, z and the Cb: number N which corresponds to the operator N x2 þ y 2 þ z2 ¼ N 2 þ 2N that differs from the result (55). Directly from the Eqs. (49)–(52) we can verify that d H ðQ; P ; q1 ; p1 ; q2 ; p2 Þ ¼ rt H ðQ; P ; q1 ; p1 ; q2 ; p2 Þ ¼ 0: dt
ð56Þ
Eq. (56) reveals that the symbol H(Q, P; q1, p1; q2, p2) is an integral of motion in quantum sense. Another quantum integral of motion is x2 + y2 + z2. From the system of Eqs. (49)–(52) follows that d 2 ðx þ y 2 þ z2 Þ ¼ rt ðx2 þ y 2 þ z2 Þ ¼ 0: dt
ð57Þ
^p^-quantization symbol H ^q^p ðQ; P ; q1 ; p1 ; q2 ; p2 Þ Here we should make a remark. Eq. (57) not always is satisfied. In the q has the same form that the corresponding symbol (46), and symbol x2 + y2 + z2 satisfies the equation d 2 ðx þ y 2 þ z2 Þ ¼ 0: dt However in the ^q^p-quantization symbol x2 + y2 + z2 is not integral of motion in quantum sense. Actually, from the Eqs. (41) and (42) we can obtain 2 !k !k 3
1 X 2 i X ði hÞk 4 X ~ 1 2 2 2 2 2 2 2 ~ q1 þ p21 þ q22 þ p22 rt ðx þ y þ z Þ ¼ ½Cðx þ y þ z Þ ¼ H o pj oqj o qj opj 5 h k¼2 k! 2 j j rffiffiffi
p 2 rQ q1 p21 q22 þ p22 6¼ 0: ¼ ih q1 q2 p1 p2 þ 2 To solve the system of Eqs. (49)–(52) we can use the conservation of the symbols (53) and (55) that is represented by the Eqs. (56) and (57). From the Eqs. (53) and (52) we immediately obtain pffiffiffiffiffi ð58Þ x ¼ ðQ_ 2 þ r2 Q2 bÞ þ 2przQ 2H ; sffiffiffi 1 2 € ðQ þ r2 QÞ: z¼ ð59Þ r p Substituting (58) into (50) we eliminate the symbols x and y from the Eqs. (51) and (52). As a result we come to the next equation: o pffiffiffiffiffi n pffiffiffiffiffi €z þ z ¼ 2pr Q_ 2 Q þ r2 Q3 þ 2przQ2 2HQ : ð60Þ Substituting (59) into (60) we finally obtain the following single nonlinear equation: :::: € þ r2 ð1 þ 2HpÞQ ¼ r2 p 2QQ € 2 Q_ 2 Q þ r2 Q3 : Q þð1 þ r2 ÞQ
ð61Þ
7. Asymptotic solutions Eq. (61) coincides with the corresponding equation in the quasiclassical approximation in the papers [5] and [7]. The stability properties of the solutions of this equation have been studied previously in [5], and all the fixed points have been classified. The asymptotic solution of the Eq. (61) has been obtained for the stable region H > 1/2p under the supposition of smallness of the parameter r. In this case the Eq. (61) has only one stable point Q = 0. As it has been shown in [5], it is possible to search the solution of the Eq. (61) in a quasiperiodic form as follows: X Qt ¼ Qðn1 ; n2 Þeiðn1 w1 þn2 w2 Þt ; where Qðn1 ; n2 Þ ¼ Q ðn1 ; n2 Þ; ð62Þ n1 ;n2
and the coefficients Q(n1, n2) and the frequencies w1,2 are to be searched as the next expansions in series in r:
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Qðn1 ; n2 Þ ¼
1 X
Qðf Þ ðn1 ; n2 Þrf ;
where w1;2 ¼
f ¼0
1 X
ðf Þ
w1;2 rf :
379
ð63Þ
f ¼0
The result of applying of the scheme (62),(63), expressed in terms of the C-numbers N and H, is presented in [7]. In virtue of the Eq. (55) the result of the paper [7] should be corrected. Finally the asymptotic solution of the Eq. (61) is the following: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ððN þ 1Þ2 ð2H Þ2 Þ Qt ¼ A cosðw2 t þ U2 Þ r2 A3 cosð3w2 t þ 3U2 Þ þ r 2 2
p p A2 p cosðw1 t þ U1 Þ þ rA2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cosððw1 þ 2w2 Þt þ U1 þ 2U2 Þ 4 1 þ 2Hp 2 8 1 þ 2Hp
2 p p A p rA2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ r cosððw1 2w2 Þt þ U1 2U2 Þ 4 1 þ 2Hp 2 8 1 þ 2Hp 2 4 p A cosðw1 t þ U1 Þ cosð4w2 t þ 4U2 Þ þ Oðr4 Þ: þr2 ð64Þ 64ð1 þ 2HpÞ where the frequencies w1 and w2 has been found in the form: w1 ¼ 1 r2 Hp þ Oðr4 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 ¼ r 1 þ 2Hp þ r3 1 þ 2Hp Hp 78 pA2 þ Oðr5 Þ; b and N b , that are given by the formulas (53) and (54). The momentum Pt and H and N are symbols of the operators H can be found as time derivative of the coordinate (64). To obtain the symbols qj(t) and pj(t) (j = 1, 2), we should consider the next complex vector: 1 q1 þ ip1 u ¼ pffiffiffi : 2 q2 þ ip2 Then, from the system of Eqs. (47) we obtain the following vectorial equation: rffiffiffi du 1 p rQ cu; i ¼ ru dt 2 2 t
ð65Þ
where r¼
0 1 ; 1 0
c¼
1 0 ; 0 1
and the symbol Qt is given in (64). We search the solution of the Eq. (65) in the form cos 2t i sin 2t : uðtÞ ¼ eirt=2 vðtÞ; where eirt=2 ¼ t t i sin 2 cos 2 Substituting this formula into (65), we come to the following equation: rffiffiffi dv p i ¼ rQt CðtÞv; dt 2 cos t i sin t : where CðtÞ ¼ eirt=2 ceirt=2 ¼ i sin t cos t Solution of the problem (66) with the respective initial condition v(0) = v0 has the form rffiffiffi Z t p dsi rQðsÞCðsÞ v0 ; vðtÞ ¼ T exp 2 0 where T is the time ordering. As a result we obtain up to the terms r2 uðtÞ ¼ eirt=2 u0 þ i
rffiffiffi rffiffiffi 2 Z Z t Z t1 p irt=2 t p e r eirt=2 dt1 Qt1 Cðt1 Þu0 þ i dt1 dt2 Qt1 Qt2 Cðt1 ÞCðt2 Þu0 þ Oðr3 Þ: 2 2 0 0 0
ð66Þ
380
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
8. Quantum observables To obtain the quantum observable of the symbol Qt we should express the solution (64) in terms of the initial coordinates and momenta x0 = (Q0,P0,x10,x20), where xj0 = (qj0,pj0). Then, according to the formula (44), Z ð67Þ hQt i ¼ dx0 fw ðx0 ÞQt ðx0 Þ; where the initial Wigner function Z Z Z 1 n n1 n2 n n1 n2 iðP nþp1 n1 þp2 n2 Þ ; q ; q fw ðxÞ ¼ ; q W ; q dndn dn e W Q Q þ þ þ 0 1 2 0 2 1 2 2 2 2 1 2 2 2 ð2pÞ6 R3
should be determined. Solution (64) is function of too much number of parameters: A, U1, U2, q10, p10, q20, and p20 (according to (46) and (54), H and N depend on the same parameters), i.e. formula (64) can be expressed as Qt ¼ Qt ðA; U1 ; U2 ; x10 ; x20 Þ:
ð68Þ
The parameters A, U1, and U2 should be eliminated from (68) and replaced by Q0 and P0. To make this, another two _ from which we obtain the relation equations like (68) are required. The first of them is given by the equation P ¼ Q, P t ¼ P t ðA; U1 ; U2 ; x10 ; x20 Þ:
ð69Þ
The second one can be obtained from the Eq. (52) by substituting (64) into the Eq. (52) and expanding the left-hand side of this equation in series in r. Then, searching U1 in the form U1 ¼ U10 þ rU11 þ r2 U12 þ Oðr3 Þ, we can obtain from the Eq. (52) the phase U1 as function of the parameters A, U2, q10, p10, q20, and p20. Substituting the resulting function U1(A, U2, x10, x20) into (68) and (69) we finally find the symbols Qt = Qt(A, U, x10, x20) and Pt = Pt(A, U, x10, x20), where we have denoted U2 U. For t = 0 we obtain the system of equations to express the parameters A and U through the initial conditions Q0, P0, q10, p10, q20, p20: Q0 ¼ Q0 ðA; U; x10 ; x20 Þ; P 0 ¼ P 0 ðA; U; x10 ; x20 Þ: Substituting this result into (67) we can calculate the quantum observable of the symbol Qt as follows: Z Z 1 Z Z 2p oA Q0 oA P 0 fw ðx0 ðA; U; x10 ; x20 ÞÞQ ðA; U; x10 ; x20 Þ: dA dU hQt i ¼ dx10 dx20 t oU Q0 oU P 0 0 0
ð70Þ
Solution (64) has been obtained under the condition H > 1/2p. This condition corresponds to the stable area S in the phase-space: S ¼ fx0 2 R6 jH ðx0 Þ > 1=2pg: The integration in (70) covers the entire phase-space, including areas, where the solution (64) becomes inapplicable, and also the unstable areas. This may result in a large calculation error in the quantum observable hQti. To avoid such an undesirable consequences, we should take the Wigner function entirely localized in the area S, i. e. the function fw(x0) should be finite and also sptfw(x0) S. We also can consider the initial Wigner function ‘‘almost’’ completely localized in the area S, and decreasing very fast outside of this area. In particular, we can take the initial Wigner function in the next form: fw ðx0 Þ ¼ lðx0 Þ expfk/ðx0 Þg; 0 2 S, where the function /(x0) has absolute minimum at a point x /ðx0 Þ ¼ /ðx0 Þ þ gðx0 x0 Þ þ hðx0 ; x0 Þ; where gðx0 x0 Þ is a positive quadratic form. Also we suppose that 9R > 0 and h > 0 such that jx0 x0 j < R;
lðx0 Þ ¼ c0 þ c1 ðx0 x0 Þ þ Oðjx0 x0 j2 Þ; hðx0 ; x0 Þ ¼ Oðjx0 x0 j3 Þ;
jx0 x0 j P R;
jhðx0 ; x0 Þj < gðx0 x0 Þ=2; /ðx0 Þ /ðx0 Þ P h:
Then, we can expect that the integration over the area R6 n S will not contribute any significant error in the calculation of quantum observables, and the integral (70) can be obtained asymptotically if k 1.
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
381
9. Mixed quantum–classical description In the mixed quantum–classical description, in which the symbols Qt and Pt are considered classical variables, the distribution function fw(Q0, P0; x10, x20) has the form: fw ðQ0 ; P 0 ; x10 ; x20 Þ ¼ dðQ0 Q0 ðA; U; x10 ; x20 ÞÞdðP 0 P 0 ðA; U; x10 ; x20 ÞÞ fw ðx10 ; x20 Þ;
ð71Þ
where function fw(x10, x20) is the Wigner distribution of the variables q10, p10, q20, p20. Substituting the function (71) into the formula (70) we obtain Z Z hQt i ¼ dx10 dx20 fw ðx10 ; x20 ÞQðt; Q0 ðA; U; x10 ; x20 Þ; P 0 ðA; U; x10 ; x20 Þ; x10 ; x20 Þ Z Z ¼ dx10 dx20 fw ðx10 ; x20 ÞQt ðA; U; x10 ; x20 Þ; ð72Þ i.e. in this case there is no necessity to express the solution (64) in terms of the initial conditions Q0 and P0. As a result we obtain from (72) and (64) * + * + A3 pr2 3iU 3iw2 t A2 p3=2 r2 2iU y 0 iz0 ið2w2 þw1 Þt A2 p3=2 r2 2iU y 0 þ iz0 ið2w2 w1 Þt pffiffiffi e pffiffiffi e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e e e hQt i ¼ 2Re 4 1 px0 1 px0 16 2 16 2
2 2 iApr iU x0 z0 A iw2 t iw2 t iApr iU x0 z0 iðw2 þw1 Þt iðw2 w1 Þt e þ e e ðy iz Þe e ðy þ iz Þe þ 0 0 2 4 y 2 þ z20 0 4 y 20 þ z20 0
pffiffiffi pffiffiffi 0 pffiffiffi
Ar p r p x 0 z0 ðy iz0 Þeiw1 t þ i 2 ðy 0 iz0 Þeiw1 t 16e2iU cos U 2 y 0 þ z20 0 4 8 * +#!! pffiffiffiffiffi y 0 iz0 iw1 t 4iU þ Oðr3 Þ; ð73Þ þ 2pAð1 þ e Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 1 px0 where the frequencies w1 and w2 are w1 ¼ 1 þ 12 px0 r2 þ Oðr3 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 r 2 cos Uz ffiffiffiffiffiffiffiffiffi 0 þ Oðr3 Þ: w2 ¼ r 1 px0 þ Appffiffip 2
ð74Þ
1px0
The observable (73), unlike the solution (64), is not quasiperiodic. To prove it, we should assume the opposite, i. e. that the function fw(x10, x20) is such that the quantum averages in the formula (73) are periodic or quasiperiodic functions of time. In particular, the quantum average heiw2 t i, which is the characteristic function of the random variable w2, is periodic or quasiperiodic function of time: Z þ1 þ1 X X iw t e 2 ¼ /w2 ðtÞ ¼ ak eix2k t ¼ dw2 eiw2 t ak dðw2 x2k Þ; ð75Þ k¼1
k¼1
Pþ1
where k¼1 ak ¼ 1, and Im ak = 0. From the Eq. (75) we conclude that the density function of the random variable w2 should be f ðw2 Þ ¼
þ1 X
ak dðw2 x2k Þ:
ð76Þ
k¼1
This distribution is a mixed state. In a pure state, the distribution function (76) should take the form f ðw2 Þ ¼ dðw2 hw2 iÞ; i.e. with the probability 1 the random variable w2 is equal to the mean value hw2i, and the variance is equal to zero: hw2 hw2ii2 = 0. Another type of averages in the formula (73) can be expressed as hAðx0 ; y 0 ; z0 Þeiðmw1 þnw2 Þt i; where A(x0, y0, z0) is a function of x0, y0, z0. If we assume that this average is a periodic function of time, we obtain in a pure state Aðx0 ; y 0 ; z0 Þeiðmw1 þnw2 Þt ¼ Ceihmw1 þnw2 it ;
382
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
i.e. the distribution of the random variable w2, as well as of the variable w1, is such that hw1 hw1ii2 = 0. Thus we come to the conclusion that the observable (73) has quasiperiodic structure if at least variables w1 and w2 are measurable simultaneously. However the uncertainty principle prohibits this, because as we can see from (74), fw1 ; w2 gw 6¼ 0: Let us calculate the observable (73) postulating the initial pure state as minimal wave packet. The corresponding Wigner function is ( ) 1 ðq hq1 iÞ2 ðq2 hq2 iÞ2 2 2 2 2 fw ðx10 ; x20 Þ ¼ 2 exp 1 ; ð77Þ 2r ðp hp iÞ 2r ðp hp iÞ 1 2 1 1 2 2 p 2r21 2r22 where the parameters of distribution r1, r2, hq1i, hq2i, hp1i, and hp2iare those, that the point (hq1i, hq2i, hp1i, hp2i) belongs to the area S and lays far enough from the border of the area S. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Inequality H(x0) > 1/2p in the limit r ! 0 turns into 1 px0 > 0. It makes the square root 1p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px0 a real number in the formulas (73) and (74). However integration over the full phase-space includes areas where 1 px0 is complex pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number. To avoid it, we assume that the parameter p is small enough to expand 1 px0 in series in px0 in the formulas (73) and (74): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px ð78Þ 1 px0 ¼ 1 þ 0 þ Oðp2 Þ: 2 pffiffiffi Then, if we take hq1i = hq2i = hp1i = hp2i = 0 and r1 ¼ r2 ¼ 1= 2, the error caused by the expansion (78), will not be too large. Let us introduce the new set of variables q, h, u1, u2, which are given by the equations pffiffiffi pffiffiffi p10 ¼ q sinðh=2Þ sinðu1 Þ; q10 ¼ q sinðh=2Þ cosðu1 Þ; pffiffiffi pffiffiffi q20 ¼ q cosðh=2Þ cosðu2 Þ; p20 ¼ q cosðh=2Þ sinðu2 Þ: In terms of these variables the function (77) takes the form fw ðx10 ; x20 Þ ¼ fw ðqÞ ¼
eq : p2
For the symbols x0, y0, z0 we obtain similarly 1 x0 ¼ q sin h cosðu1 u2 Þ; 2
1 y 0 ¼ q sin h sinðu1 u2 Þ; 2
1 z0 ¼ q cos h: 2
As x0, y0, and z0 depend on the variables u1 and u2 only in form of the difference u1 u2, for any symbol A(x0, y0, z0) = A(q, h, u1 u2) we have Z 2p Z 1 Z 2p Z p 1 eq sin hdh du1 du2 Aðq; h; u1 u2 Þ 2 qdq hAðx0 ; y 0 ; z0 Þi ¼ p 8 0 0 0 0 Z 1 Z p Z 2p 1 ¼ eq qdq sin hdh duAðq; h; uÞ: 4p 0 0 0 There are two types of quantum averages in the Eq. (73): heiwti and hKeiwti, where the frequencies w are of the form w = n1w1 + n2w2, n1;2 2 Z, and hKi = 0. Also we can verify directly that the frequencies w have the structure w = hwi + n, where is a small parameter. These averages can be calculated approximately. For the average heiwtiwe have the following identity: m X iwt ðitÞn n ¼ eizt eiðwzÞt ¼ eizt e hðw zÞ i þ Rm ðz; tÞ; n! n¼0
ð79Þ
where z is an arbitrary complex variable and Rm(z, t) is a residual term, 1 X ðitÞn n Rm ðz; tÞ ¼ eizt hðw zÞ i: n! n¼mþ1 Minimizing the residual term with respect to the variable z, we shell obtain a better approximation in the asymptotic formula (79), for a given number m. First of all we should note that the average (79) is bounded: iwt iwt e 6 e ¼ 1:
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
383
Secondly, for every time moment t > 0 m m X X ðitÞn tn izt n hðw zÞ i 6 sup etImz jhðw zÞn ij ¼ S < 1 e n¼0 n! n! t>0 n¼0 only if Im z > 0. In turn it means that the function Rm(z,t) is bounded only if Im z > 0: n X m m X ðitÞn ðitÞ iwt n n izt iwt izt e jRm ðz; tÞj ¼ e hðw zÞ i 6 e þ e hðw zÞ i 6 1 þ S < 1: n! n! n¼0 n¼0 As Rm(z, t) is an analytic function of the variable z = x + iy, then jRm(z, t)j has absolute minimum in one of the saddle points zk: oRm ðz; tÞ ðitÞmþ1 ¼ eizt hðw zÞm i ¼ 0; oz m! i. e. zk is a solution of the equation m X m hwmn izn ¼ 0: ð1Þn hðw zÞm i ¼ n n¼0
ð80Þ
We should take that solution zk (Im zk > 0) of the Eqs. (80), which makes the absolute value of the residual term Rm(z,t) minimal. Then, the average of the type (79) becomes m1 X iwt ðitÞn n ¼ eizk t e hðw zk Þ i þ Rm ðzk ; tÞ: n! n¼0
Let m = 2. Then, the unique solution z of the equation h (w z)2i = 0 with Im z > 0 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ hwi þ ig; g ¼ hw2 i hwi2 : Substituting this result into (79), we find that asymptotically D E iwt ¼ eihwit egt ð1 þ gtÞ þ O ðw hwi igÞ3 : e Basing on this formula we obtain iw t p2 rpt 29p2 rpt 29p2 rt pffiffiffi 1 þ 1 þ pffiffiffi 1 þ ; e 2 uw2 ðtÞ ’ exp i 1 16 64 64 2 2 2 2 3iw t e 2 ¼ uw2 ð3tÞ:
ð81Þ ð82Þ
Averages of the type hKeiwti, where hKi = 0, can be calculated similarly, i. e. by minimizing the residual term. We have the following identity: m X iwt ðitÞn n ~ m ðz; tÞ; ¼ eizt ð83Þ Ke hKðw zÞ i þ R n! n¼1 where the residual term is ~ m ðz; tÞ ¼ eizt R
1 X ðitÞn hKðw zÞn i: n! n¼mþ1
ð84Þ
The average (79) is bounded: iwt iwt Ke 6 Ke ¼ hjK ji < 1: ~ m ðz; tÞ is bounded too if Im z > 0. The minimization of the residual term (84) leads to the next Then the residual term R equation: m1 X m ð85Þ ð1Þn hKwmn izn ¼ 0: hKðw zÞm i ¼ n n¼0 As well as in the Eq. (80), we take that solution zk (Im zk > 0) of the Eqs. (85), which makes the absolute value of the ~ m ðz; tÞ minimal. Then, the average of the type (83) becomes residual term R
384
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
m1 X ðitÞn n ~ m ðzk ; tÞ: Keiwt ¼ eizk t hKðw zk Þ i þ R n n¼1
Let now m = 3. The solution z of the Eq. (85) with Im z > 0 has the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hKw2 i hKw3 i hKw2 i2 þ ig; where g ¼ : z¼ 2hKwi 3hKwi 4hKwi2 Then, the average (83) becomes * 4 +! iwt hKw2 i hKw2 i it gt hKwiitð1 þ gtÞ þ O K w ig Ke ¼ exp : 2hKwi 2hKwi This Formula leads to the following quantum observables: * + y 0 iz0 ið2w2 þw1 Þt Ap3=2 ð1 þ p2 Þr2 cos U pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e pffiffiffi ’t 1 px0 2
2 2 2 5p p r rþ it pð1 r2 Þt 1 þ pð1 r2 Þt ; exp 1 þ 2 4 2 * + 3=2 y þ iz0 Ap ð1 þ p2 Þr2 cos U p0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eið2w2 w1 Þt ’ t pffiffiffi 1 px0 2
2 2 2 5p pr r it pð1 þ r2 Þt 1 þ pð1 þ r2 Þt ; exp 1 þ 2 4 2
x 0 z0 1 iðw2 þw1 Þt ’ pð8 8r þ 3p2 Þrt ðy iz0 Þe 64 y 20 þ z20 0 3p2 r rpt rpt it ð1 rÞ 1 þ ð1 rÞ ; exp 1þr 8 2 2
x 0 z0 1 ðy þ iz0 Þeiðw2 w1 Þt ’ pð8 þ 8r þ 3p2 Þrt 64 y 20 þ z20 0 3p2 r rpt rpt it ð1 þ rÞ 1 þ ð1 þ rÞ ; exp 1 þ r 8 2 2
2 2 2 x 0 z0 pr t pr t pr t iw1 t exp it 1 þ ; ’ ðy iz Þe 0 0 8 2 2 y 20 þ z20 and in the same approximation we obtain * + y iz0 p0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eiw1 t 0; ðy 0 iz0 Þeiw1 t 0: 1 px0
ð86Þ
ð87Þ
ð88Þ
ð89Þ ð90Þ
ð91Þ
10. Conclusions Substituting the averages (81),(82), and (86),(91) into (73), we finally obtain the asymptotic formula for the observable hQti in the mixed quantum-classical description. The quantum observables of the symbols Pt, yt, and zt can be found relatively simply because of the linear relation between these symbols and the symbol Qt, and also owing to that _ and rt z ¼ z_ in the phase representation considered above. In the Figs. 1–6, the evolution of fact that rt x ¼ x_ , rt y ¼ y, the analytically calculated quantum averages hQti, hPti, hzti and hyti is presented for the parameters values A = 10, r = 0.05, p = 0.1, U = p/6. These figures are obtained in the mixed quantum-classical description, in which the variance of the parameters A and U has been considered zero. To obtain the ‘‘exact’’ observables, i.e. in the full quantized description, we should calculate the sixfold integral (70). If we need to calculate the quantum average of a symbol At(x) in a given representation, in which $tAt(x) 5 d At(x)/dt, then we have two options. In the first one we should calculate the propagator (33) to obtain the symbol at any moment of time through the formula (32), and then employ the formula (44) to calculate the respective observable.
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Fig. 1. Oscillations of the observable hQti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 2. Oscillations of the observable hPti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 3. Oscillations of the observable hzti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 4. Oscillations of the observable hzti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
385
386
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Fig. 5. Oscillations of the observable hyti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 6. Oscillations of the observable hyti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
In spite of that fact that the knowledge of the propagator allows to calculate any quantum observable, the exact calculation of the propagator for nonlinear quantum systems is difficult. Then, to avoid calculation of the full quantum propagator (33), we could pass from the set of phase-space variables x to a new phase-space variables y and a corresponding representation in which the relation $tAt(y) = dAt(y)/dt is satisfied. In other words, a canonical transformation x # y should be fulfilled. Then, according to (40) the equation, that describes the evolution of the symbol At(y), has classical form, and therefore can be solved as classical. However, to apply this method, a general form of the quantum canonical transformations should be established.
References [1] Takahashi K. Distribution functions in classical and quantum mechanics. Prog Theor Phys Suppl 1989;98:109–56. [2] Schanz H, Esser B. Mixed quantum–classical versus full quantum dynamics: Coupled quasiparticle-oscillator system. Phys Rev A 1997;55(5):3375–87. [3] Koch Ch, Esser B. Spin-boson Hamiltonian and optical absorption of molecular dimer. Phys Rev A 1999;61:0225XX. [4] Hennig D, Esser B. Transfer dynamics of a quasiparticle in a nonlinear dimer coupled to an intersite vibration: Chaos on Bloch sphere. Phys Rev A 1992;46(8):4569–76. [5] Kuzmenkov LS, Maximov SG, Guardado Zvala JL. On the asymptotic solutions of the coupled quasiparticle-oscillator system. Chaos, Solitons & Fractals 2003;15:597–610. [6] Berezin FA. Methods of Second Quantization. Berlin: Acad. Press; 1986. [7] Maximov SG, Kuzmenkov LS, Guardado Zvala JL. Nonlinear oscillations in the molecular dimer: asymptotic solutions. Int J Quantum Chem 2004;100:324–35.
Chaos, Solitons and Fractals 37 (2008) 369–386 www.elsevier.com/locate/chaos
On the full quantum description of nonlinear dynamic systems in the phase-space representation S. Maximov a
a,*
, L.S. Kuzmenkov
b
Instituto Tecnolo´gico de Morelia, PGIIE, Av. Tecnolo´gico No. 1500, Lomas de Santiaguito, C.P. 58120, Morelia, Mich., Me´xico b Moscow State University ‘‘M.V. Lomonosov’’, Vorobyevy gory, Moscow, 119899, Russia Accepted 1 September 2006
Communicated by Prof. M.S. El Naschie
Abstract An approach to the completely quantized description of nonlinear dynamic systems with chaos is considered. The method bases on the correspondence between operators and their symbols in a phase-space representation. Heisenberg equations in a wide class of phase-space representations are obtained. Nonlinear oscillations of the coupled quasiparticles-oscillator system are analyzed in the completely quantized description in terms of symbols of operators in the Wigner representation. The asymptotic solutions of the Heisenberg equations for the symbols Qt, Pt, etc. are obtained for a stable region of the phase-space, and the respective quantum observables are calculated. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The correspondence between classical nonlinear dynamics and its quantum counterpart is of interest for the class of problems denominated ‘‘quantum chaos’’. The analysis of completely quantized nonlinear dynamic systems is sometimes complicated. Therefore different kinds of approximations, such as quasiclassical approximation or Born–Oppenheimer approximation, are applied to treat these systems. Another approach to this problem consists in the analyzing the development of the density matrix in the Wigner or the Husimi representations [1]. Nevertheless, the problem of the completely quantized description of nonlinear dynamic systems remains actual. In the present paper, a possible approach to the fully quantized description of such systems is offered. The approach is based on the correspondence between operators and their symbols in a phase-space representation. This correspondence maps the dynamics of operators in the Heisenberg representation to the dynamics of symbols. Respective equations of the dynamics of symbols are obtained for a wide class of phase-space representations. A complete description of any quantum dynamic system requires determination of the propagator in the phase-space representation. For nonlinear dynamic systems with chaos this problem is very difficult due to non-integrabilities. Nevertheless, evolution of the symbols such as qt and pt, which correspond to the operators ^ qt and ^pt , does not require *
Corresponding author. Fax: +52 443 3171870. E-mail address: [email protected] (S. Maximov).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.040
370
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
calculation of the full quantum propagator. As a result, the quantum observables hqti and hpti can be calculated relatively simply. Nonlinear oscillations of the coupled quasiparticles-oscillator system are considered in the completely quantum description in terms of symbols of operators in one of the phase-space representations. The interest in the dynamic properties of this system is due to the applicability of the mathematical model to a wide range of physical phenomena. The model leads to the spin-boson Hamiltonian of the next form [2]: 1=2 b aþ ^ b 2 Þ 1 ð^aþ ^a2 þ ^aþ ^a1 Þ þ p b ¼ 1 ðP^ 2 þ r2 Q H a1 Þ; r Qð^ aþ ð1Þ 1 2 2 a2 ^ 1^ 2 2 2 b ¼ 1, ½^ak ; ^aþ ¼ dkj , ½^ak ; Q b ¼ ½^ak ; Pb ¼ 0. In the mixed quantum–classical description the dynamic where h 1, i½ Pb ; Q j properties of this system have been previously considered in the papers [2–4]. In [5] the quasiclassical approximation was applied to the quasiparticle-oscillator system to obtain asymptotic solutions. Heisenberg equations, with the Hamiltonian (1) in the quasiclassical approximation, was led to a single ordinary differential equation. The stability problem was investigated and the asymptotic solution around one of the stable points was found. b Pb , ^ In this paper the system of equations for the symbols of the operators Q, q1;2 , and ^ p1;2 is obtained. In the Wigner representation these equations are similar to the equations in the quasiclassical approximation, obtained in the paper [5]. The results of paper [5] are used to obtain the quantum solutions for this system. The respective quantum observables are calculated for a postulated initial density matrix in the Wigner representation.
2. Symbols of operators b B; b . . . in the Schro¨dinger representation and functions b . . . ; C; Let us consider a correspondence between operators A; A(x), B(x), . . . , C(x), . . . of a set of variables x=(x1, x2, . . . , xn): b $ AðxÞ; B b $ CðxÞ; . . . b $ BðxÞ; . . . ; C A b b ðxÞ as follows: This correspondence can be realized by means of the operators DðxÞ and U b b ðxÞ AÞ; AðxÞ ¼ trð U Z b ¼ dx DðxÞAðxÞ: b A
ð2Þ ð3Þ
b ðxÞ and DðxÞ b Eqs. (2) and (3) lead to the next two relations between the operators U b ðxÞ Dðx b 0 ÞÞ ¼ dðx x0 Þ; trð U Z 0 b 0 0 0 b dxha1 j DðxÞja 2 iha2 j U ðxÞja1 i ¼ dða1 a1 Þdða2 a2 Þ;
ð4Þ ð5Þ
where {jai} is a set of vectors orthogonal and complete. We call functions A(x), B(x), . . . , C(x),. . . symbols of operators b B; b . . .. b . . . ; C; A; Let the mapping (2) be such, that ð^q; ^pÞ7!ðq; pÞ, where at the initial moment of time q and p are eigenvalues of the operators ^q and ^p, i. e. ^qjqi ¼ qjqi, ^pjpi ¼ pjpi. Then, we can take (q, p) as the set of variables x, and the relation ^ ¼ ð^q; ^pÞ and the symbol x = (q, p) takes the next form: between the operator x Z b ðxÞ^ b ^ ¼ dx DðxÞx: x ¼ trð U xÞ; x ð6Þ Mapping (2) transforms commutators into Moyal brackets in the phase-space: Z Z i b b ½ A; B7!fAðxÞ; BðxÞgM ¼ dx1 dx2 Kðx; x1 ; x2 ÞAðx1 ÞBðx2 Þ; h where
b 1 Þ; Dðx b ðxÞ i ½ Dðx b 2 Þ : Kðx; x1 ; x2 Þ ¼ tr U h
b ðxÞ maps polynomial operators of ^ Let the operator U q and ^ p to the corresponding polynomials of q and p: 1 1 X X b¼ Am;n K½^qm ^pn 7!Aðq; pÞ ¼ Am;n qm pn : A m;n¼0
m;n¼0
ð7Þ
ð8Þ
ð9Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
371
Eq. (9) is a simple generalization of the well known ^q^p-, ^ p^ q- and Weyl-quantizations [6], in which polynomial operators b differ from each other only by means of the ordering K½^ pn of the operators ^ A qm ^ qy^ p: X X m n m n cP P½^q ^p ; cP ¼ 1; cP 2 R: ð10Þ K½^q ^p ¼ P
P
P is a permutation of the operators ^q and ^p in q^m ^pn : mþn X P½^qm ^pn ¼ ^qa1 ^p1a1 ^qa2 ^p1a2 ^qamþn ^p1amþn ; ak ¼ m;
ð11Þ
k¼1
where each ak takes the values 1 or 0. Excluding coefficients Amn from the formula (9), we find Z Z 1 X 1 omþn AðxÞ m n b ¼ dx b ^ p dðxÞ K½^ q ¼ dx DðxÞAðxÞ; A m!n! oqm opn m;n¼0 where b DðxÞ ¼
1 X ð1Þmþn ðmÞ d ðqÞdðnÞ ðpÞK½^qm ^pn : m!n! m;n¼0
ð12Þ
Defining explicitly the ordering (10) we completely determine the type of quantization. For example, any linear quantization leads to the correspondence of the next form [6]: b pa $ L2 a AðxÞ; ^qa A b qa $ L2 a AðxÞ; b $ L1 a AðxÞ; A^ b $ L1 a AðxÞ; A^ ^pa A ð13Þ p p q q a
where Lip are the first order differential operators s X a ab o ab o ab b b AðxÞ; aab q þ b p þ f þ g Lip AðxÞ ¼ i i i i oqb opb b¼1
ð14Þ
ab ab ab i ^b ¼ dab $ fpa ; qb gM ¼ dab . Let us pa ; q where the coefficients aab i ; bi ; fi ; g i are those to satisfy the correspondence h ½^ operate differently. We generalize the Eqs. (13) and (14) by imposing the next condition for the form of the operator b DðxÞ (12):
b i b o DðxÞ ½ DðxÞ; ^q ¼ ; h op b i b o DðxÞ ½ DðxÞ; ^p ¼ h oq
ð15Þ ð16Þ
b we obtain on the one hand b b ðxÞ. From the Eq. (15), for an arbitrary operator A, and search the operators DðxÞ and U i b b i ½ DðxÞ; b b ^q ¼ tr DðxÞ ½ A; ^q ¼ fA; qgM : tr A h h On the other hand we have ! b o DðxÞ o b b oAðxÞ b ¼ tr A DðxÞ ¼ : tr A op op op Similar manipulations we can fulfill for the Eq. (16). Thus the Eqs. (15) and (16) lead to the following relations for every symbol A(x): i b oA ½ A; ^q $ fA; qgM ¼ ; h op
i b oA ½ A; ^p $ fA; pgM ¼ : h oq
These relations are more general than the relations (13) and (14). b Operator DðxÞ in the Eq. (12) can be considered as the result of quantization of the classical microscopic distribution function d(x x(t)): 1 X ð1Þmþn ðmÞ d ðqÞdðnÞ ðpÞqm ðtÞpn ðtÞ dðx xðtÞÞ ¼ m!n! m;n¼0 pn of the operators ^ qm ^ q and ^ p. On in which the product of canonical variables qm(t)pn(t) is replaced by the ordering K½^ 0 ^ the other hand, Eq. (4) permits the next clear interpretation: function d(x x ) is the symbol of the operator DðxÞ. In
372
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
b other words, DðxÞ is the operator corresponding to the classical microscopic distribution function; for every phaseb space representation of quantum mechanics, symbol of the operator DðxÞ coincides with the classical microscopic distribution function. b is similar to the classical statistical average of a function A(x) with the distriQuantum average of the operator A bution function f ðx; tÞ: Z Z b ¼ dxtr q ^ ^ðtÞ A ^ðtÞDðxÞ hAi ¼ tr q AðxÞ ¼ dxf ðx; tÞAðxÞ; ð17Þ where ^ ^ðtÞDðxÞ f ðx; tÞ ¼ tr q :
ð18Þ
Eq. (17) shows the necessity of considering symbols of the type (18). Therefore, alongside with the symbols A(x), we shall consider dual symbols AðxÞ which are given by the next equations: Z ^ ðxÞ A ^ b ; A b ¼ dxDðxÞAðxÞ; AðxÞ ¼ tr U ^ ðxÞ and DðxÞ ^ ^ ðxÞ and DðxÞ b where the operators U are related with the operators U through the formulae ^ ðxÞ ¼ 2ph DðxÞ; b U
^ b ðxÞ: DðxÞ ¼ ð2phÞ1 U
b b 3. Structure of the operators DðxÞ and UðxÞ qmk ^ pnk . Such Commuting operators ^q and ^p, we expand the permutation (11) in series in terms of the type ði hÞk ^ terms appear after commuting k pares of operators ^q and ^ p. As a result we obtain K½^qm ^pn ¼
minfm;ng X
ðihÞk C km;n ^qmk ^pnk ;
ð19Þ
k¼0
P where C km;n 2 R. From the equation P cP ¼ 1 we obtain the initial condition for C km;n : C 0m;n ¼ 1. Substituting (19) into b (12) we come to the next formula for the operator DðxÞ: minfm;ng 1 X X ð1Þmþn ðmÞ qmk ^ pnk : d ðqÞdðnÞ ðpÞ ðihÞk C km;n ^ m!n! k¼0 m;n¼0 Pminfm;ng P we obtain Rearranging the summations 1 m;n¼0 and k¼0
b DðxÞ ¼
b DðxÞ ¼ ¼
1 X 1 X ð1Þmþn ðmÞ d ðqÞdðnÞ ðpÞðihÞk C km;n ^ qmk ^ pnk mn k¼0 m;n¼k 1 k X X ðihoq op Þk k¼0
m;n¼0
ð1Þmþn Ck dðmÞ ðqÞdðnÞ ðpÞ^ qm ^ pn : ðm þ kÞ!ðn þ kÞ! mþk;nþk
ð20Þ
b Coefficients C km;n completely determine the operator DðxÞ. To find these coefficients we substitute (20) into (15) and (16). The left-hand side of the Eq. (16) gives 1 1 X X ð1Þmþn C kmþkþ1;nþk ðm þ 1Þ ðmÞ i b ½ DðxÞ; ^p ¼ oq d ðqÞdðnÞ ðpÞ^ ðihoq op Þk qm ^ pn : h ðm þ k þ 1Þ!ðn þ kÞ! k¼0 m;n¼0
Substituting this result into the Eq. (16) we obtain the next equation: oq
1 1 X X ð1Þmþn C kmþkþ1;nþk ðm þ 1Þ ðmÞ pn ðihoq op Þk qm ^ d ðqÞdðnÞ ðpÞ^ ðm þ k þ 1Þ!ðn þ kÞ! k¼0 m;n¼0
¼ oq
1 k X X ðihoq op Þk k¼0
m;n¼0
ð1Þmþn Ck pn : dðmÞ ðqÞdðnÞ ðpÞ^ qm ^ ðm þ kÞ!ðn þ kÞ! mþk;nþk
On the other hand, substituting the formula (20) into the Eq. (15), we find
ð21Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
op
373
1 1 X X ð1Þmþn C kmþk;nþkþ1 ðn þ 1Þ ðmÞ d ðqÞdðnÞ ðpÞ^ pn ðihoq op Þk qm ^ ðm þ kÞ!ðn þ k þ 1Þ! k¼0 m;n¼0
¼ op
1 k X X ðihoq op Þk k¼0
m;n¼0
ð1Þmþn pn : dðmÞ ðqÞdðnÞ ðpÞ^ qm ^ Ck ðm þ kÞ!ðn þ kÞ! mþk;nþk
ð22Þ
Eqs. (21) and (22) result in the following system of recurrent equations for the coefficients C km;n with the respective initial condition 8 > Ck ¼ mþk C kmþk1;nþk ; > m < mþk;nþk C kmþk;nþk1 ; C kmþk;nþk ¼ nþk ð23Þ n > > : C 0 ¼ 1: m;n
From the first equation of the system (23) we obtain C kmþk;nþk ¼
ðm þ kÞ! k C k;nþk : k!m!
Substituting this result into the second equation of the system (23) we finally find C kk;nþk ¼
ðn þ kÞ! k ðn þ kÞ!k! C k;k ¼ fk ; n!k! n!
where C kk;k fk ðk!Þ2 :
The initial condition in the system (23) results in f0 ¼ C 00;0 ¼ 1. Finally the solution of the system of Eqs. (23) has the next form: C kmþk;nþk ¼
ðm þ kÞ!ðn þ kÞ! fk : m!n!
b Substituting this result into the Eq. (20) we find the operator DðxÞ: 1 X b b ^q^p ðxÞ; where Lx ¼ fk ðihoq op Þk : DðxÞ ¼ Lx D
ð24Þ
k¼0
b ^q^p ðxÞ corresponds to the ^q^p-quantization of quantum mechanics: D Z 1 X ð1Þmþn ðmÞ du dv iuðq^qÞ ivðp^pÞ b ^q^p ðxÞ ¼ e e ¼ jqihqjpihpj: d ðqÞdðnÞ ðpÞ^qm ^pn ¼ D 2 m!n! ð2pÞ m;n¼0
ð25Þ
^ If exists the inverse operator L1 x for the differential operator Lx , then the operator U ðxÞ has the form Z b ðxÞ ¼ dx0 Dðx; x0 Þ U b ^q^p ðx0 Þ; U b ^q^p ðxÞ can be obtained from (25) where D(x, x 0 ) satisfies the equations Lx Dðx0 ; xÞ ¼ Lx0 Dðx0 ; xÞ ¼ dðx x0 Þ. Operator U b ^q^p ðxÞ ¼ 2phjpihpjqihqj U to satisfy the Eqs. (4) and (5). P1 k UðzÞ , where U(z) is an analytic function of z that satisfies Let us suppose that converges the series k¼0 fk z ¼ e b b ðxÞ obtain the form U(0) = 0 (because f0 = 1). Then, the operators DðxÞ and U b ^q^p ðxÞ; b DðxÞ ¼ eUðihoq op Þ D Uði h o o Þb q p b ðxÞ ¼ e U ^q^p ðxÞ: U
ð26Þ ð27Þ
b Substituting the operator DðxÞ from the Eq. (26) into (3) and integrating by parts over the variables q and p, Z Z Z Uðih~ oq~ op Þ b b b ^q^p ðxÞ; b D ^q^p ðxÞ ¼ dxAðxÞeUðih o q o p Þ D A ¼ dx DðxÞAðxÞ ¼ dxAðxÞe we obtain the next relation between the symbol A(x) and the symbol A^q^p ðxÞ in the ^ q^ p-quantization: eUðihoq op Þ AðxÞ ¼ A^q^p ðxÞ: All the results (24), P (26) and (27), obtained for one degree of freedom, can be generalized for s degrees of freedom by substituting oq op si¼1 oq i op i . If fk = vk/k!, then we obtain a class of linear quantizations where
374
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
b v ðxÞ ¼ eivhoq op D b ^q^p ðxÞ; D
b ^q^p ðxÞ: b v ðxÞ ¼ eivhoq op U U
ð28Þ
In particular if v = 1/2, then we obtain the well-known Wigner representation. Cases v = 0 and v = 1 lead to the ^ q^p- and p^^ q-quantizations respectively. Moyal brackets in the phase-space representations, determined by the operators (26) and (27), can be found by substituting (26) and (27) into the formulae (7) and (8). As a result we find ! ~~ Uðih~ oq~ Uðih~ oq o p Þ i ih o p~ ih o q o p op Þ oq fA; BgM ¼ e eUðihoq op Þ BðxÞ: AðxÞe e ð29Þ e h op ðABÞ ¼ Aðo q o p þ o q~ op þ o p~ oq þ ~ op ÞB, after some transformations the formula (29) in the linear oq~ Noticing that ~ oq~ quantization (28) can be led to the following simple form: i ihð1vÞ o p~oq þihv o q~op ~ ~ e fA; Bgv ¼ AðxÞ ð30Þ eihð1vÞ o q op þihv o p oq BðxÞ: h Of course, Eqs. (28) does not include all the linear phase-space representations of the type (13), (14). A more wide class of quantizations can be obtained as a correspondence between operators and symbols of the type b¼ A
1 X
Am;n K½ða^q þ b^pÞm ðc^q þ d^pÞn $ Aðq; pÞ ¼
m;n¼0
1 X
Am;n ðaq þ bpÞm ðcq þ dpÞn ;
ð31Þ
m;n¼0
where ad bc = 1. The same result we can obtain by fulfilling in the Eq. (24) a linear canonical transformation. Husimi-like representations can be obtained if we assign in the Eq. (31) s1 a ¼ c ¼ pffiffiffiffiffi ; 2h
is b ¼ pffiffiffiffiffi ; 2h
and d ¼ b :
4. Symbols of operators as functions of the symbols qt and pt b 0 ðnÞ be an operator in the Schro¨dinger representation. n is one or a set of parameters (in particular n = t). The Let A b 0 ðnÞ and its symbol A0(x, n) is given by the next formula: correspondence between the operator A ^ 0 ðnÞ ; b ðx0 ÞA A0 ðx0 ; nÞ ¼ tr U ^0 ¼ ð^ where x0, according to the Eq. (6), is the symbol of operator x q0 ; ^ p0 Þ in the Schro¨dinger representation. Evolution b t ðnÞ in the Heisenbeg representation is given by the formula of the operator A b t ðnÞ G b t ;t A b t ;t ; b t ðnÞ ¼ G A 2 2 1 1 1 2 b t ;t satisfies the equations where the operator G 2 1 b t ;t oG i b 2 1 b ¼ H t ðt 2 Þ G t2 ;t1 ; h 2 ot2
b t ;t oG i^ 2 1 b ¼ G t ;t H t ðt 1 Þ: h 2 1 1 ot1 b t ðnÞ, is related with the symbol A0(x0, n) through the propagator Symbol At2 ðx0 ; nÞ, corresponding to the operator A 2 G(x, tjx0, t0) in the phase-space representation: Z Z 0 0 b 0 b b b b b At ðx0 ; nÞ ¼ trð U ðx0 Þ A t ðnÞÞ ¼ dx0 trð U ðx0 Þ G t;t0 Dðx0 Þ G t0 ;t ÞAt0 ðx0 ; nÞ ¼ dx00 Gðx0 ; tjx00 ; t0 ÞAt0 ðx00 ; nÞ; ð32Þ where b t;t Dðx b t ;t Þ: b 0ÞG b ðxÞ G Gðx; tjx0 ; t0 Þ ¼ trð U 0 0 Because for an infinitesimal Dt b tþDt;t ¼ ^1 þ i H b t ðtÞDt þ oðDtÞ; G h
b t;tþDt ¼ ^1 i H b t ðtÞDt þ oðDtÞ; G h
then, for two next moments of time t + Dt and t we obtain b t ðtÞ; Dðx b ðxÞ i ½ H b 0 ÞÞ þ oðDtÞ: Gðx; t þ Dtjx0 ; tÞ ¼ dðx x0 Þ þ Dt trð U h R b 00 ÞH t ðx00 ; tÞ we obtain b t ðtÞ ¼ dx00 Dðx Substituting here H
ð33Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Gðx; t þ Dtjx0 ; tÞ ¼ dðx x0 Þ þ ¼ dðx x0 Þ þ
Z Z
dx00 dx00
Z Z
375
h i b ðxÞ i Dðx b 00 Þ H t ðx00 ; tÞdðx00 x0 Þ þ oðDtÞ b 00 Þ; Dðx dx00 Dt tr U h dx00 Dt Kðx; x00 ; x00 ÞH t ðx00 ; tÞdðx00 x0 Þ þ oðDtÞ
¼ ð1 Dt½H t ðtÞM Þdðx x0 Þ þ oðDtÞ ¼ e½H t ðtÞM Dt dðx x0 Þ þ oðDtÞ;
ð34Þ
where ½H M A ¼ fH ; AgM :
ð35Þ
Let us divide the interval of time [t0, t] on n small segments t0 = s0 < s1 < < sn = t. Then, taking into account the Eqs. (4) and (5) we find Z Y n1 n Y dxðjÞ G xðkÞ ; sk xðk1Þ ; sk1 ; Gðx; tjx0 ; t0 Þ ¼ j¼1
(0)
= x 0 and x
k¼1
(n)
= x. Substituting here the result (34) and passing to the limit max{Dtk} ! 0 we at last obtain Rt ½H s ðsÞM ds : Gðx; tjx0 ; t0 Þ ¼ Gt;t0 dðx x0 Þ; where Gt;t0 ¼ e t0
where x
Then, Eq. (32) takes the following form: At ðx0 ; nÞ ¼ Gt;0 A0 ðx0 ; nÞ:
ð36Þ
Symbol x0 at the initial moment of time is related with the symbol xt at the moment t by the same Eq. (36). From this equation we obtain that x0 ¼ G1 t;0 xt ¼ G0;t xt . Substituting this formula into the Eq. (36) we find the dependence of the symbol At at the moment of time t on the symbol xt in the x0-representation: At ðxt ; nÞ ¼ Gt;0 A0 ðG0;t xt ; nÞ:
ð37Þ
Eq. (37) can be generalized as follows: At2 ðxt2 ; nÞ ¼ Gt2 ;0 G0;t1 At1 ðGt2 ;0 G0;t1 Þ1 xt1 ; n ¼ Gt2 ;t1 At1 ðG1 t2 ;t1 xt1 ; nÞ:
5. Equations of motion in the phase-space representations Let us deduce Heisenberg equations in the phase-space representation. We assume that n = t. Mapping (2) transforms the increment of the operator At(t) into the increment of the corresponding symbol: b tþDt ðt þ DtÞ A b t ðtÞ7!AtþDt ðxt ; t þ DtÞ At ðxt ; tÞ ¼ AtþDt ðxt ; t þ DtÞ At ðxt ; t þ DtÞ þ At ðxt ; t þ DtÞ At ðxt ; tÞ: A The linear part of the increment of the symbol At(xt, t), in the right hand side of this equation, represents the absolute differential of this symbol. On the one hand we find AtþDt ðxt ; t þ DtÞ At ðxt ; t þ DtÞ ¼ ð1 ½H t ðtÞM Dt þ oðDtÞÞAt ðxt ; t þ DtÞ At ðxt ; t þ DtÞ ¼ ½H t ðtÞM DtAt ðxt ; t þ DtÞ þ oðDtÞ: On the other hand At ðxt ; t þ DtÞ At ðxt ; tÞ ¼
oAt ðxt ; tÞ Dt þ oðDtÞ: ot
Passing to the limit Dt ! 0 in the equation AtþDt ðxt ; t þ DtÞ At ðxt ; tÞ oAt ðxt ; tÞ oðDtÞ ¼ ½H t ðtÞM At ðxt ; t þ DtÞ þ þ Dt ot Dt and taking into account (35) we obtain the next equation: rt At ¼ fH t ; At gM þ
oAt ; ot
b t =dt has been transformed into the covariant derivative where the derivative of operator d A
ð38Þ
376
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
rt At ðxt ; tÞ lim
Dt!0
AtþDt ðxt ; t þ DtÞ At ðxt ; tÞ : Dt
Together with the covariant derivative $tAt(xt, t), we can consider the derivative dAt(xt, t)/dt which should be defined as follows: dAt ðxt ; tÞ At ðxtþDt ; t þ DtÞ At ðxt ; tÞ ¼ lim : Dt!0 dt Dt
ð39Þ
Because Gt+Dt,t = 1 Dt[Ht(t)] + o(Dt), then, taking into account the Eqs. (15) and (16) we can obtain oH t oH t xtþDt ¼ GtþDt;t xt ¼ xt þ DtfH t ðtÞ; xt gM þ oðDtÞ ¼ qt þ Dt þ oðDtÞ: ; pt Dt opt oqt Substituting this result into the Eq. (39) we obtain the formula for the total time derivative: dAt oH t oAt oH t oAt oAt oAt ¼ ¼ fH t ; At gcl þ : þ dt opt oqt oqt opt ot ot
ð40Þ
If we denote [C]At = [Ht]clAt [Ht]MAt, then the relation between the derivatives $tAt(xt,t) and dAt(xt,t)/dt can be written in the next form: rt At ¼
dAt þ ½CAt ; dt
ð41Þ
For the phase-space representations of the type (28) we can obtain k
nk 1 n
X X 1 ~ ~ ~ ~ ~ ~ ihð1 vÞo p oq þ i hvo q op i hð1 vÞo q op þ i hvo p oq At : ½CAt ¼ H t ðo p oq o q op Þ ðn þ 1Þ k¼0 n¼1
ð42Þ
In particular Eq. (38), in view of the conditions (15) and (16), leads to the equations of motion for the symbols qt and pt, which coincide with the equations of Hamilton in classical mechanics: t t rt qt ¼ dq ¼ fH t ; qt gM ¼ oH ; dt opt t t ¼ fH t ; pt gM ¼ oH : rt pt ¼ dp dt oqt
ð43Þ
b t ðtÞ in the Heisenberg representation can be calculated as follows: Quantum observable hAti of any operator A Z Z b t ðtÞÞ ¼ dx0 trð^ b 0 ÞÞAt ðx0 ; tÞ ¼ dx0 f 0 ðx0 ÞAt ðx0 ; tÞ: q0 A q0 Dðx ð44Þ hAt i ¼ trð^ b t ðtÞ, we should solve the Eq. In other words, formula (44) means that to calculate quantum average of the operator A (38) and obtain the symbol At(x0,t) as function of the initial phase-space coordinates x0, then calculate the integrals (44) over the initial coordinates x0 with the initial covariant distribution density f 0 ðx0 Þ. This situation is similar to the calculation of the classical statistical averages over the initial states in the phase-space. A complete description of any quantum dynamic system requires determination of all the symbols at every moment 0 of time. In other words, the calculation of the kernel G(x,tjx ,t0) of the propagator is inevitable. Nevertheless, there are problems in which knowledge of only two observables hqti and hpti is required. In this case, we should solve only the system of Eqs. (43) with some initial conditions x0 and substitute the solution into the Eq. (44). In the next sections we are applying this scheme to the investigation of nonlinear oscillations of a molecular dimer.
6. Coupled quasiparticles-oscillator system in the Wigner representation We now consider the spin-boson Hamiltonian (1). Let us obtain the system of equations of the type (43) for a system with the Hamiltonian (1) in the Wigner representation. In this representation both types of symbols A(x) and AðxÞ coincide. Substituting in (1) 1 ^aþ q1;2 i^p1;2 Þ; 1;2 ¼ pffiffiffi ð^ 2
1 ^a1;2 ¼ pffiffiffi ð^q1;2 þ i^p1;2 Þ 2
we come to the formula 1=2 b q2 þ ^ b 2 1 ð^q1 ^q2 þ ^p1 ^p2 Þ þ 1 p b ¼ 1 ^p2 þ r2 Q r Qð^ p22 ^ q21 ^ p21 Þ: H 2 2 2 2 2
ð45Þ
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
377
b (45) in the Wigner representation has the form The symbol corresponding to the Hamiltonian H
Z Z Z n n n b n n1 n2 H ðQ; P ; q1 ; p1 ; q2 ; p2 Þ ¼ dn dn1 dn2 eiðP nþp1 n1 þp2 n2 Þ Q ; q1 1 ; q2 2 H Q þ ; q1 þ ; q2 þ 2 2 2 2 2 2 R3
1 1 1 p1=2 rQðq22 þ p22 q21 p21 Þ: ¼ ðP 2 þ r2 Q2 Þ ðq1 q2 þ p1 p2 Þ þ 2 2 2 2
ð46Þ
Then, the equations of motion for the symbols Q, P, q1,2, and p1,2 take the following simple form: oH rt Q ¼ Q_ ¼ ¼ P; oP rffiffiffi oH p 1 2 2 _ ¼ r Q r ðq þ p22 q21 p21 Þ; rt P ¼ P ¼ oQ 2 2 2 rffiffiffi oH 1 p rQp1 ; ¼ p2 rt q1 ¼ q_ 1 ¼ op1 2 2 rffiffiffi oH 1 p rQq1 ; rt p1 ¼ p_ 1 ¼ ¼ q þ oq1 2 2 2 rffiffiffi oH 1 p rQp2 ; ¼ p1 þ rt q2 ¼ q_ 2 ¼ op2 2 2 rffiffiffi oH 1 p rQq2 : rt p2 ¼ p_ 2 ¼ ¼ q oq2 2 1 2
ð47Þ
Let us introduce new symbols x ¼ q1 q2 þ p1 p2 ;
y ¼ q1 p2 þ q2 p1 ;
1 z ¼ ðq22 þ p22 q21 p21 Þ: 2
Equations of motion (47) in terms of new variables take the form: pffiffiffiffiffi x_ ¼ 2prQy; pffiffiffiffiffi y_ ¼ z þ 2prQx; z_ ¼ y; rffiffiffi € þ r2 Q ¼ r pz; Q 2
ð48Þ
ð49Þ ð50Þ ð51Þ ð52Þ
and the symbol (46) becomes 1 1 H ¼ ðQ_ 2 þ r2 Q2 Þ x þ 2 2
rffiffiffi p rQz: 2
ð53Þ
Moreover, it is easy to verify, using (42) for the Wigner representation (v = 1/2) that rt x ¼ x_ , rt y ¼ y_ , and rt z ¼ z_ . Also we can note that symbols x, y, and z satisfy to the next relations: fx; ygw ¼ 2z;
fy; zgw ¼ 2x;
fz; xgw ¼ 2y;
where f ; gw are the Moyal brackets in the Wigner representation. The explicit form of these brackets can be obtained from the formula (30) by assigning v = 1/2. b of number of excitons in the system. On the other hand, we can find the symbol that corresponds to the operator N b has the form The operator N 1 b ¼ ^aþ N a1 þ ^aþ a2 ¼ ð^q21 þ ^p21 þ ^q22 þ ^p22 Þ 1: 1^ 2^ 2 Then, the corresponding symbol is
Z Z n n b n1 n2 1 Nðq1 ; p1 ; q2 ; p2 Þ ¼ ¼ ðq21 þ p21 þ q22 þ p22 Þ 1: dn1 dn2 eiðp1 n1 þp2 n2 Þ q1 1 ; q2 2 N q1 þ ; q2 þ 2 2 2 2 2
ð54Þ
R2
From the definition of the variables x, y, and z (48) and the Eq. (54) we find x2 þ y 2 þ z2 ¼ ðN ðq1 ; p1 ; q2 ; p2 Þ þ 1Þ2 :
ð55Þ
378
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Here appears a difference between the formula (55) and the results of papers [5,7]. In the scheme realized in [5], all the operators has been changed to the respective C-numbers. It has led to the following relation between x, y, z and the Cb: number N which corresponds to the operator N x2 þ y 2 þ z2 ¼ N 2 þ 2N that differs from the result (55). Directly from the Eqs. (49)–(52) we can verify that d H ðQ; P ; q1 ; p1 ; q2 ; p2 Þ ¼ rt H ðQ; P ; q1 ; p1 ; q2 ; p2 Þ ¼ 0: dt
ð56Þ
Eq. (56) reveals that the symbol H(Q, P; q1, p1; q2, p2) is an integral of motion in quantum sense. Another quantum integral of motion is x2 + y2 + z2. From the system of Eqs. (49)–(52) follows that d 2 ðx þ y 2 þ z2 Þ ¼ rt ðx2 þ y 2 þ z2 Þ ¼ 0: dt
ð57Þ
^p^-quantization symbol H ^q^p ðQ; P ; q1 ; p1 ; q2 ; p2 Þ Here we should make a remark. Eq. (57) not always is satisfied. In the q has the same form that the corresponding symbol (46), and symbol x2 + y2 + z2 satisfies the equation d 2 ðx þ y 2 þ z2 Þ ¼ 0: dt However in the ^q^p-quantization symbol x2 + y2 + z2 is not integral of motion in quantum sense. Actually, from the Eqs. (41) and (42) we can obtain 2 !k !k 3
1 X 2 i X ði hÞk 4 X ~ 1 2 2 2 2 2 2 2 ~ q1 þ p21 þ q22 þ p22 rt ðx þ y þ z Þ ¼ ½Cðx þ y þ z Þ ¼ H o pj oqj o qj opj 5 h k¼2 k! 2 j j rffiffiffi
p 2 rQ q1 p21 q22 þ p22 6¼ 0: ¼ ih q1 q2 p1 p2 þ 2 To solve the system of Eqs. (49)–(52) we can use the conservation of the symbols (53) and (55) that is represented by the Eqs. (56) and (57). From the Eqs. (53) and (52) we immediately obtain pffiffiffiffiffi ð58Þ x ¼ ðQ_ 2 þ r2 Q2 bÞ þ 2przQ 2H ; sffiffiffi 1 2 € ðQ þ r2 QÞ: z¼ ð59Þ r p Substituting (58) into (50) we eliminate the symbols x and y from the Eqs. (51) and (52). As a result we come to the next equation: o pffiffiffiffiffi n pffiffiffiffiffi €z þ z ¼ 2pr Q_ 2 Q þ r2 Q3 þ 2przQ2 2HQ : ð60Þ Substituting (59) into (60) we finally obtain the following single nonlinear equation: :::: € þ r2 ð1 þ 2HpÞQ ¼ r2 p 2QQ € 2 Q_ 2 Q þ r2 Q3 : Q þð1 þ r2 ÞQ
ð61Þ
7. Asymptotic solutions Eq. (61) coincides with the corresponding equation in the quasiclassical approximation in the papers [5] and [7]. The stability properties of the solutions of this equation have been studied previously in [5], and all the fixed points have been classified. The asymptotic solution of the Eq. (61) has been obtained for the stable region H > 1/2p under the supposition of smallness of the parameter r. In this case the Eq. (61) has only one stable point Q = 0. As it has been shown in [5], it is possible to search the solution of the Eq. (61) in a quasiperiodic form as follows: X Qt ¼ Qðn1 ; n2 Þeiðn1 w1 þn2 w2 Þt ; where Qðn1 ; n2 Þ ¼ Q ðn1 ; n2 Þ; ð62Þ n1 ;n2
and the coefficients Q(n1, n2) and the frequencies w1,2 are to be searched as the next expansions in series in r:
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Qðn1 ; n2 Þ ¼
1 X
Qðf Þ ðn1 ; n2 Þrf ;
where w1;2 ¼
f ¼0
1 X
ðf Þ
w1;2 rf :
379
ð63Þ
f ¼0
The result of applying of the scheme (62),(63), expressed in terms of the C-numbers N and H, is presented in [7]. In virtue of the Eq. (55) the result of the paper [7] should be corrected. Finally the asymptotic solution of the Eq. (61) is the following: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ððN þ 1Þ2 ð2H Þ2 Þ Qt ¼ A cosðw2 t þ U2 Þ r2 A3 cosð3w2 t þ 3U2 Þ þ r 2 2
p p A2 p cosðw1 t þ U1 Þ þ rA2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cosððw1 þ 2w2 Þt þ U1 þ 2U2 Þ 4 1 þ 2Hp 2 8 1 þ 2Hp
2 p p A p rA2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ r cosððw1 2w2 Þt þ U1 2U2 Þ 4 1 þ 2Hp 2 8 1 þ 2Hp 2 4 p A cosðw1 t þ U1 Þ cosð4w2 t þ 4U2 Þ þ Oðr4 Þ: þr2 ð64Þ 64ð1 þ 2HpÞ where the frequencies w1 and w2 has been found in the form: w1 ¼ 1 r2 Hp þ Oðr4 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 ¼ r 1 þ 2Hp þ r3 1 þ 2Hp Hp 78 pA2 þ Oðr5 Þ; b and N b , that are given by the formulas (53) and (54). The momentum Pt and H and N are symbols of the operators H can be found as time derivative of the coordinate (64). To obtain the symbols qj(t) and pj(t) (j = 1, 2), we should consider the next complex vector: 1 q1 þ ip1 u ¼ pffiffiffi : 2 q2 þ ip2 Then, from the system of Eqs. (47) we obtain the following vectorial equation: rffiffiffi du 1 p rQ cu; i ¼ ru dt 2 2 t
ð65Þ
where r¼
0 1 ; 1 0
c¼
1 0 ; 0 1
and the symbol Qt is given in (64). We search the solution of the Eq. (65) in the form cos 2t i sin 2t : uðtÞ ¼ eirt=2 vðtÞ; where eirt=2 ¼ t t i sin 2 cos 2 Substituting this formula into (65), we come to the following equation: rffiffiffi dv p i ¼ rQt CðtÞv; dt 2 cos t i sin t : where CðtÞ ¼ eirt=2 ceirt=2 ¼ i sin t cos t Solution of the problem (66) with the respective initial condition v(0) = v0 has the form rffiffiffi Z t p dsi rQðsÞCðsÞ v0 ; vðtÞ ¼ T exp 2 0 where T is the time ordering. As a result we obtain up to the terms r2 uðtÞ ¼ eirt=2 u0 þ i
rffiffiffi rffiffiffi 2 Z Z t Z t1 p irt=2 t p e r eirt=2 dt1 Qt1 Cðt1 Þu0 þ i dt1 dt2 Qt1 Qt2 Cðt1 ÞCðt2 Þu0 þ Oðr3 Þ: 2 2 0 0 0
ð66Þ
380
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
8. Quantum observables To obtain the quantum observable of the symbol Qt we should express the solution (64) in terms of the initial coordinates and momenta x0 = (Q0,P0,x10,x20), where xj0 = (qj0,pj0). Then, according to the formula (44), Z ð67Þ hQt i ¼ dx0 fw ðx0 ÞQt ðx0 Þ; where the initial Wigner function Z Z Z 1 n n1 n2 n n1 n2 iðP nþp1 n1 þp2 n2 Þ ; q ; q fw ðxÞ ¼ ; q W ; q dndn dn e W Q Q þ þ þ 0 1 2 0 2 1 2 2 2 2 1 2 2 2 ð2pÞ6 R3
should be determined. Solution (64) is function of too much number of parameters: A, U1, U2, q10, p10, q20, and p20 (according to (46) and (54), H and N depend on the same parameters), i.e. formula (64) can be expressed as Qt ¼ Qt ðA; U1 ; U2 ; x10 ; x20 Þ:
ð68Þ
The parameters A, U1, and U2 should be eliminated from (68) and replaced by Q0 and P0. To make this, another two _ from which we obtain the relation equations like (68) are required. The first of them is given by the equation P ¼ Q, P t ¼ P t ðA; U1 ; U2 ; x10 ; x20 Þ:
ð69Þ
The second one can be obtained from the Eq. (52) by substituting (64) into the Eq. (52) and expanding the left-hand side of this equation in series in r. Then, searching U1 in the form U1 ¼ U10 þ rU11 þ r2 U12 þ Oðr3 Þ, we can obtain from the Eq. (52) the phase U1 as function of the parameters A, U2, q10, p10, q20, and p20. Substituting the resulting function U1(A, U2, x10, x20) into (68) and (69) we finally find the symbols Qt = Qt(A, U, x10, x20) and Pt = Pt(A, U, x10, x20), where we have denoted U2 U. For t = 0 we obtain the system of equations to express the parameters A and U through the initial conditions Q0, P0, q10, p10, q20, p20: Q0 ¼ Q0 ðA; U; x10 ; x20 Þ; P 0 ¼ P 0 ðA; U; x10 ; x20 Þ: Substituting this result into (67) we can calculate the quantum observable of the symbol Qt as follows: Z Z 1 Z Z 2p oA Q0 oA P 0 fw ðx0 ðA; U; x10 ; x20 ÞÞQ ðA; U; x10 ; x20 Þ: dA dU hQt i ¼ dx10 dx20 t oU Q0 oU P 0 0 0
ð70Þ
Solution (64) has been obtained under the condition H > 1/2p. This condition corresponds to the stable area S in the phase-space: S ¼ fx0 2 R6 jH ðx0 Þ > 1=2pg: The integration in (70) covers the entire phase-space, including areas, where the solution (64) becomes inapplicable, and also the unstable areas. This may result in a large calculation error in the quantum observable hQti. To avoid such an undesirable consequences, we should take the Wigner function entirely localized in the area S, i. e. the function fw(x0) should be finite and also sptfw(x0) S. We also can consider the initial Wigner function ‘‘almost’’ completely localized in the area S, and decreasing very fast outside of this area. In particular, we can take the initial Wigner function in the next form: fw ðx0 Þ ¼ lðx0 Þ expfk/ðx0 Þg; 0 2 S, where the function /(x0) has absolute minimum at a point x /ðx0 Þ ¼ /ðx0 Þ þ gðx0 x0 Þ þ hðx0 ; x0 Þ; where gðx0 x0 Þ is a positive quadratic form. Also we suppose that 9R > 0 and h > 0 such that jx0 x0 j < R;
lðx0 Þ ¼ c0 þ c1 ðx0 x0 Þ þ Oðjx0 x0 j2 Þ; hðx0 ; x0 Þ ¼ Oðjx0 x0 j3 Þ;
jx0 x0 j P R;
jhðx0 ; x0 Þj < gðx0 x0 Þ=2; /ðx0 Þ /ðx0 Þ P h:
Then, we can expect that the integration over the area R6 n S will not contribute any significant error in the calculation of quantum observables, and the integral (70) can be obtained asymptotically if k 1.
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
381
9. Mixed quantum–classical description In the mixed quantum–classical description, in which the symbols Qt and Pt are considered classical variables, the distribution function fw(Q0, P0; x10, x20) has the form: fw ðQ0 ; P 0 ; x10 ; x20 Þ ¼ dðQ0 Q0 ðA; U; x10 ; x20 ÞÞdðP 0 P 0 ðA; U; x10 ; x20 ÞÞ fw ðx10 ; x20 Þ;
ð71Þ
where function fw(x10, x20) is the Wigner distribution of the variables q10, p10, q20, p20. Substituting the function (71) into the formula (70) we obtain Z Z hQt i ¼ dx10 dx20 fw ðx10 ; x20 ÞQðt; Q0 ðA; U; x10 ; x20 Þ; P 0 ðA; U; x10 ; x20 Þ; x10 ; x20 Þ Z Z ¼ dx10 dx20 fw ðx10 ; x20 ÞQt ðA; U; x10 ; x20 Þ; ð72Þ i.e. in this case there is no necessity to express the solution (64) in terms of the initial conditions Q0 and P0. As a result we obtain from (72) and (64) * + * + A3 pr2 3iU 3iw2 t A2 p3=2 r2 2iU y 0 iz0 ið2w2 þw1 Þt A2 p3=2 r2 2iU y 0 þ iz0 ið2w2 w1 Þt pffiffiffi e pffiffiffi e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e e e hQt i ¼ 2Re 4 1 px0 1 px0 16 2 16 2
2 2 iApr iU x0 z0 A iw2 t iw2 t iApr iU x0 z0 iðw2 þw1 Þt iðw2 w1 Þt e þ e e ðy iz Þe e ðy þ iz Þe þ 0 0 2 4 y 2 þ z20 0 4 y 20 þ z20 0
pffiffiffi pffiffiffi 0 pffiffiffi
Ar p r p x 0 z0 ðy iz0 Þeiw1 t þ i 2 ðy 0 iz0 Þeiw1 t 16e2iU cos U 2 y 0 þ z20 0 4 8 * +#!! pffiffiffiffiffi y 0 iz0 iw1 t 4iU þ Oðr3 Þ; ð73Þ þ 2pAð1 þ e Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 1 px0 where the frequencies w1 and w2 are w1 ¼ 1 þ 12 px0 r2 þ Oðr3 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 r 2 cos Uz ffiffiffiffiffiffiffiffiffi 0 þ Oðr3 Þ: w2 ¼ r 1 px0 þ Appffiffip 2
ð74Þ
1px0
The observable (73), unlike the solution (64), is not quasiperiodic. To prove it, we should assume the opposite, i. e. that the function fw(x10, x20) is such that the quantum averages in the formula (73) are periodic or quasiperiodic functions of time. In particular, the quantum average heiw2 t i, which is the characteristic function of the random variable w2, is periodic or quasiperiodic function of time: Z þ1 þ1 X X iw t e 2 ¼ /w2 ðtÞ ¼ ak eix2k t ¼ dw2 eiw2 t ak dðw2 x2k Þ; ð75Þ k¼1
k¼1
Pþ1
where k¼1 ak ¼ 1, and Im ak = 0. From the Eq. (75) we conclude that the density function of the random variable w2 should be f ðw2 Þ ¼
þ1 X
ak dðw2 x2k Þ:
ð76Þ
k¼1
This distribution is a mixed state. In a pure state, the distribution function (76) should take the form f ðw2 Þ ¼ dðw2 hw2 iÞ; i.e. with the probability 1 the random variable w2 is equal to the mean value hw2i, and the variance is equal to zero: hw2 hw2ii2 = 0. Another type of averages in the formula (73) can be expressed as hAðx0 ; y 0 ; z0 Þeiðmw1 þnw2 Þt i; where A(x0, y0, z0) is a function of x0, y0, z0. If we assume that this average is a periodic function of time, we obtain in a pure state Aðx0 ; y 0 ; z0 Þeiðmw1 þnw2 Þt ¼ Ceihmw1 þnw2 it ;
382
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
i.e. the distribution of the random variable w2, as well as of the variable w1, is such that hw1 hw1ii2 = 0. Thus we come to the conclusion that the observable (73) has quasiperiodic structure if at least variables w1 and w2 are measurable simultaneously. However the uncertainty principle prohibits this, because as we can see from (74), fw1 ; w2 gw 6¼ 0: Let us calculate the observable (73) postulating the initial pure state as minimal wave packet. The corresponding Wigner function is ( ) 1 ðq hq1 iÞ2 ðq2 hq2 iÞ2 2 2 2 2 fw ðx10 ; x20 Þ ¼ 2 exp 1 ; ð77Þ 2r ðp hp iÞ 2r ðp hp iÞ 1 2 1 1 2 2 p 2r21 2r22 where the parameters of distribution r1, r2, hq1i, hq2i, hp1i, and hp2iare those, that the point (hq1i, hq2i, hp1i, hp2i) belongs to the area S and lays far enough from the border of the area S. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Inequality H(x0) > 1/2p in the limit r ! 0 turns into 1 px0 > 0. It makes the square root 1p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px0 a real number in the formulas (73) and (74). However integration over the full phase-space includes areas where 1 px0 is complex pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number. To avoid it, we assume that the parameter p is small enough to expand 1 px0 in series in px0 in the formulas (73) and (74): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px ð78Þ 1 px0 ¼ 1 þ 0 þ Oðp2 Þ: 2 pffiffiffi Then, if we take hq1i = hq2i = hp1i = hp2i = 0 and r1 ¼ r2 ¼ 1= 2, the error caused by the expansion (78), will not be too large. Let us introduce the new set of variables q, h, u1, u2, which are given by the equations pffiffiffi pffiffiffi p10 ¼ q sinðh=2Þ sinðu1 Þ; q10 ¼ q sinðh=2Þ cosðu1 Þ; pffiffiffi pffiffiffi q20 ¼ q cosðh=2Þ cosðu2 Þ; p20 ¼ q cosðh=2Þ sinðu2 Þ: In terms of these variables the function (77) takes the form fw ðx10 ; x20 Þ ¼ fw ðqÞ ¼
eq : p2
For the symbols x0, y0, z0 we obtain similarly 1 x0 ¼ q sin h cosðu1 u2 Þ; 2
1 y 0 ¼ q sin h sinðu1 u2 Þ; 2
1 z0 ¼ q cos h: 2
As x0, y0, and z0 depend on the variables u1 and u2 only in form of the difference u1 u2, for any symbol A(x0, y0, z0) = A(q, h, u1 u2) we have Z 2p Z 1 Z 2p Z p 1 eq sin hdh du1 du2 Aðq; h; u1 u2 Þ 2 qdq hAðx0 ; y 0 ; z0 Þi ¼ p 8 0 0 0 0 Z 1 Z p Z 2p 1 ¼ eq qdq sin hdh duAðq; h; uÞ: 4p 0 0 0 There are two types of quantum averages in the Eq. (73): heiwti and hKeiwti, where the frequencies w are of the form w = n1w1 + n2w2, n1;2 2 Z, and hKi = 0. Also we can verify directly that the frequencies w have the structure w = hwi + n, where is a small parameter. These averages can be calculated approximately. For the average heiwtiwe have the following identity: m X iwt ðitÞn n ¼ eizt eiðwzÞt ¼ eizt e hðw zÞ i þ Rm ðz; tÞ; n! n¼0
ð79Þ
where z is an arbitrary complex variable and Rm(z, t) is a residual term, 1 X ðitÞn n Rm ðz; tÞ ¼ eizt hðw zÞ i: n! n¼mþ1 Minimizing the residual term with respect to the variable z, we shell obtain a better approximation in the asymptotic formula (79), for a given number m. First of all we should note that the average (79) is bounded: iwt iwt e 6 e ¼ 1:
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
383
Secondly, for every time moment t > 0 m m X X ðitÞn tn izt n hðw zÞ i 6 sup etImz jhðw zÞn ij ¼ S < 1 e n¼0 n! n! t>0 n¼0 only if Im z > 0. In turn it means that the function Rm(z,t) is bounded only if Im z > 0: n X m m X ðitÞn ðitÞ iwt n n izt iwt izt e jRm ðz; tÞj ¼ e hðw zÞ i 6 e þ e hðw zÞ i 6 1 þ S < 1: n! n! n¼0 n¼0 As Rm(z, t) is an analytic function of the variable z = x + iy, then jRm(z, t)j has absolute minimum in one of the saddle points zk: oRm ðz; tÞ ðitÞmþ1 ¼ eizt hðw zÞm i ¼ 0; oz m! i. e. zk is a solution of the equation m X m hwmn izn ¼ 0: ð1Þn hðw zÞm i ¼ n n¼0
ð80Þ
We should take that solution zk (Im zk > 0) of the Eqs. (80), which makes the absolute value of the residual term Rm(z,t) minimal. Then, the average of the type (79) becomes m1 X iwt ðitÞn n ¼ eizk t e hðw zk Þ i þ Rm ðzk ; tÞ: n! n¼0
Let m = 2. Then, the unique solution z of the equation h (w z)2i = 0 with Im z > 0 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ hwi þ ig; g ¼ hw2 i hwi2 : Substituting this result into (79), we find that asymptotically D E iwt ¼ eihwit egt ð1 þ gtÞ þ O ðw hwi igÞ3 : e Basing on this formula we obtain iw t p2 rpt 29p2 rpt 29p2 rt pffiffiffi 1 þ 1 þ pffiffiffi 1 þ ; e 2 uw2 ðtÞ ’ exp i 1 16 64 64 2 2 2 2 3iw t e 2 ¼ uw2 ð3tÞ:
ð81Þ ð82Þ
Averages of the type hKeiwti, where hKi = 0, can be calculated similarly, i. e. by minimizing the residual term. We have the following identity: m X iwt ðitÞn n ~ m ðz; tÞ; ¼ eizt ð83Þ Ke hKðw zÞ i þ R n! n¼1 where the residual term is ~ m ðz; tÞ ¼ eizt R
1 X ðitÞn hKðw zÞn i: n! n¼mþ1
ð84Þ
The average (79) is bounded: iwt iwt Ke 6 Ke ¼ hjK ji < 1: ~ m ðz; tÞ is bounded too if Im z > 0. The minimization of the residual term (84) leads to the next Then the residual term R equation: m1 X m ð85Þ ð1Þn hKwmn izn ¼ 0: hKðw zÞm i ¼ n n¼0 As well as in the Eq. (80), we take that solution zk (Im zk > 0) of the Eqs. (85), which makes the absolute value of the ~ m ðz; tÞ minimal. Then, the average of the type (83) becomes residual term R
384
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
m1 X ðitÞn n ~ m ðzk ; tÞ: Keiwt ¼ eizk t hKðw zk Þ i þ R n n¼1
Let now m = 3. The solution z of the Eq. (85) with Im z > 0 has the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hKw2 i hKw3 i hKw2 i2 þ ig; where g ¼ : z¼ 2hKwi 3hKwi 4hKwi2 Then, the average (83) becomes * 4 +! iwt hKw2 i hKw2 i it gt hKwiitð1 þ gtÞ þ O K w ig Ke ¼ exp : 2hKwi 2hKwi This Formula leads to the following quantum observables: * + y 0 iz0 ið2w2 þw1 Þt Ap3=2 ð1 þ p2 Þr2 cos U pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e pffiffiffi ’t 1 px0 2
2 2 2 5p p r rþ it pð1 r2 Þt 1 þ pð1 r2 Þt ; exp 1 þ 2 4 2 * + 3=2 y þ iz0 Ap ð1 þ p2 Þr2 cos U p0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eið2w2 w1 Þt ’ t pffiffiffi 1 px0 2
2 2 2 5p pr r it pð1 þ r2 Þt 1 þ pð1 þ r2 Þt ; exp 1 þ 2 4 2
x 0 z0 1 iðw2 þw1 Þt ’ pð8 8r þ 3p2 Þrt ðy iz0 Þe 64 y 20 þ z20 0 3p2 r rpt rpt it ð1 rÞ 1 þ ð1 rÞ ; exp 1þr 8 2 2
x 0 z0 1 ðy þ iz0 Þeiðw2 w1 Þt ’ pð8 þ 8r þ 3p2 Þrt 64 y 20 þ z20 0 3p2 r rpt rpt it ð1 þ rÞ 1 þ ð1 þ rÞ ; exp 1 þ r 8 2 2
2 2 2 x 0 z0 pr t pr t pr t iw1 t exp it 1 þ ; ’ ðy iz Þe 0 0 8 2 2 y 20 þ z20 and in the same approximation we obtain * + y iz0 p0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eiw1 t 0; ðy 0 iz0 Þeiw1 t 0: 1 px0
ð86Þ
ð87Þ
ð88Þ
ð89Þ ð90Þ
ð91Þ
10. Conclusions Substituting the averages (81),(82), and (86),(91) into (73), we finally obtain the asymptotic formula for the observable hQti in the mixed quantum-classical description. The quantum observables of the symbols Pt, yt, and zt can be found relatively simply because of the linear relation between these symbols and the symbol Qt, and also owing to that _ and rt z ¼ z_ in the phase representation considered above. In the Figs. 1–6, the evolution of fact that rt x ¼ x_ , rt y ¼ y, the analytically calculated quantum averages hQti, hPti, hzti and hyti is presented for the parameters values A = 10, r = 0.05, p = 0.1, U = p/6. These figures are obtained in the mixed quantum-classical description, in which the variance of the parameters A and U has been considered zero. To obtain the ‘‘exact’’ observables, i.e. in the full quantized description, we should calculate the sixfold integral (70). If we need to calculate the quantum average of a symbol At(x) in a given representation, in which $tAt(x) 5 d At(x)/dt, then we have two options. In the first one we should calculate the propagator (33) to obtain the symbol at any moment of time through the formula (32), and then employ the formula (44) to calculate the respective observable.
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Fig. 1. Oscillations of the observable hQti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 2. Oscillations of the observable hPti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 3. Oscillations of the observable hzti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 4. Oscillations of the observable hzti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
385
386
S. Maximov, L.S. Kuzmenkov / Chaos, Solitons and Fractals 37 (2008) 369–386
Fig. 5. Oscillations of the observable hyti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
Fig. 6. Oscillations of the observable hyti: asymptotic solution with the parameters r = 0.07; p = 0.1; A = 10 and U = p/6.
In spite of that fact that the knowledge of the propagator allows to calculate any quantum observable, the exact calculation of the propagator for nonlinear quantum systems is difficult. Then, to avoid calculation of the full quantum propagator (33), we could pass from the set of phase-space variables x to a new phase-space variables y and a corresponding representation in which the relation $tAt(y) = dAt(y)/dt is satisfied. In other words, a canonical transformation x # y should be fulfilled. Then, according to (40) the equation, that describes the evolution of the symbol At(y), has classical form, and therefore can be solved as classical. However, to apply this method, a general form of the quantum canonical transformations should be established.
References [1] Takahashi K. Distribution functions in classical and quantum mechanics. Prog Theor Phys Suppl 1989;98:109–56. [2] Schanz H, Esser B. Mixed quantum–classical versus full quantum dynamics: Coupled quasiparticle-oscillator system. Phys Rev A 1997;55(5):3375–87. [3] Koch Ch, Esser B. Spin-boson Hamiltonian and optical absorption of molecular dimer. Phys Rev A 1999;61:0225XX. [4] Hennig D, Esser B. Transfer dynamics of a quasiparticle in a nonlinear dimer coupled to an intersite vibration: Chaos on Bloch sphere. Phys Rev A 1992;46(8):4569–76. [5] Kuzmenkov LS, Maximov SG, Guardado Zvala JL. On the asymptotic solutions of the coupled quasiparticle-oscillator system. Chaos, Solitons & Fractals 2003;15:597–610. [6] Berezin FA. Methods of Second Quantization. Berlin: Acad. Press; 1986. [7] Maximov SG, Kuzmenkov LS, Guardado Zvala JL. Nonlinear oscillations in the molecular dimer: asymptotic solutions. Int J Quantum Chem 2004;100:324–35.