A way to incorporation of thermodynamics into quantum theory

We suggest an approach allowing restrict a gap between macro- and micro descriptions of nature (i.e. between statistical thermodynamics and quantum mechanics) on the basis of a new heat bath model in the form of cold and heat vacua. Despite of standard quantum mechanics we start from microtheory in...

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Дата:	2012
Автори:	Golubjeva, O.N., Sukhanov, A.D.
Формат:	Стаття
Мова:	English
Опубліковано:	Peoples' Friendship University of Russia 2012
Назва видання:	Вопросы атомной науки и техники
Теми:	Section A. Quantum Field Theory
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/106985
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Цитувати:	A way to incorporation of thermodynamics into quantum theory / O.N. Golubjeva, A.D. Sukhanov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 70-73. — Бібліогр.: 4 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1069852016-10-11T03:02:17Z A way to incorporation of thermodynamics into quantum theory Golubjeva, O.N. Sukhanov, A.D. Section A. Quantum Field Theory We suggest an approach allowing restrict a gap between macro- and micro descriptions of nature (i.e. between statistical thermodynamics and quantum mechanics) on the basis of a new heat bath model in the form of cold and heat vacua. Despite of standard quantum mechanics we start from microtheory in the form of ħ,k-dynamics suggested by us earlier. Its concept, in general case, founds on use of complex wave function which amplitude and phase are depended on temperature. Мы предлагаем подход, позволяющий устранить разрыв между макро- и микроописаниями природы (т.е. между статистической термодинамикой и квантовой механикой) на основе новой модели термостата в форме холодного и теплого вакуумов. В отличие от стандартной квантовой механики мы исходим из микротеории в форме ħ,k-динамики, предложенной нами ранее. Ее концепция, в общем случае, основана на использовании комплексной волновой функции, амплитуда и фаза которой зависят от температуры. Ми пропонуємо підхід, що дозволяє усунути розрив між макро- й мікроописами природи (тобто між статистичною термодинамікою й квантовою механікою) на основі нової моделі термостата у формі холодного й теплого вакуумів. На відміну від стандартної квантової механіки ми виходимо з мікротеорії у формі ħ,k-динаміки, запропонованої нами раніше. Її концепція, у загальному випадку, заснована на використанні комплексної хвильової функції, амплітуда й фаза якої залежать від температури. 2012 Article A way to incorporation of thermodynamics into quantum theory / O.N. Golubjeva, A.D. Sukhanov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 70-73. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 03.65.Bz, 05.70, 05.30.d http://dspace.nbuv.gov.ua/handle/123456789/106985 en Вопросы атомной науки и техники Peoples' Friendship University of Russia
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Section A. Quantum Field Theory Section A. Quantum Field Theory
spellingShingle	Section A. Quantum Field Theory Section A. Quantum Field Theory Golubjeva, O.N. Sukhanov, A.D. A way to incorporation of thermodynamics into quantum theory Вопросы атомной науки и техники
description	We suggest an approach allowing restrict a gap between macro- and micro descriptions of nature (i.e. between statistical thermodynamics and quantum mechanics) on the basis of a new heat bath model in the form of cold and heat vacua. Despite of standard quantum mechanics we start from microtheory in the form of ħ,k-dynamics suggested by us earlier. Its concept, in general case, founds on use of complex wave function which amplitude and phase are depended on temperature.
format	Article
author	Golubjeva, O.N. Sukhanov, A.D.
author_facet	Golubjeva, O.N. Sukhanov, A.D.
author_sort	Golubjeva, O.N.
title	A way to incorporation of thermodynamics into quantum theory
title_short	A way to incorporation of thermodynamics into quantum theory
title_full	A way to incorporation of thermodynamics into quantum theory
title_fullStr	A way to incorporation of thermodynamics into quantum theory
title_full_unstemmed	A way to incorporation of thermodynamics into quantum theory
title_sort	way to incorporation of thermodynamics into quantum theory
publisher	Peoples' Friendship University of Russia
publishDate	2012
topic_facet	Section A. Quantum Field Theory
url	http://dspace.nbuv.gov.ua/handle/123456789/106985
citation_txt	A way to incorporation of thermodynamics into quantum theory / O.N. Golubjeva, A.D. Sukhanov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 70-73. — Бібліогр.: 4 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT golubjevaon awaytoincorporationofthermodynamicsintoquantumtheory AT sukhanovad awaytoincorporationofthermodynamicsintoquantumtheory AT golubjevaon waytoincorporationofthermodynamicsintoquantumtheory AT sukhanovad waytoincorporationofthermodynamicsintoquantumtheory
first_indexed	2025-07-07T19:17:15Z
last_indexed	2025-07-07T19:17:15Z
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fulltext	A WAY TO INCORPORATION OF THERMODYNAMICS INTO QUANTUM THEORY O.N. Golubjeva 1 and A.D. Sukhanov 1,2∗ 1Peoples’ Friendship University of Russia, Moscow, Russia 2Joint Institute for Nuclear Research, Dubna, Russia (Received December 5, 2011) We suggest an approach allowing restrict a gap between macro- and micro descriptions of nature (i.e. between statistical thermodynamics and quantum mechanics) on the basis of a new heat bath model in the form of cold and heat vacua. Despite of standard quantum mechanics we start from microtheory in the form of �, k-dynamics suggested by us earlier. Its concept, in general case, founds on use of complex wave function which amplitude and phase are depended on temperature. PACS: 03.65.Bz, 05.70, 05.30.d 1. STATEMENT OF THE PROBLEM At present, the interrelations between the macro- and microdescriptions of the nature attract special attention. Methods of equilibrium thermodynamics turn out to be very effective in the description of nano-objects. At the same time, methods of quan- tum mechanics are used more and more frequently to analyze macroscopic phenomena. Thus, the gap between the macro- and microtheories existing in the 19 and 20 centuries narrows continuously. In this connection, a number of researchers formu- lated the problem of possible coordination between these theories by incorporating thermodynamics into quantum theory [1]. This idea suggests adapting the apparatus of quantum theory to the description of thermal phenomena, thus generating an illusion about the complete reduction of the macrodescrip- tion to the microdescription. In our opinion, the idea of incorporating thermo- dynamics into quantum theory by combining statis- tical thermodynamics with quantum statistical me- chanics is not very promising. The reason for this is that the types of statistical ensembles used in them (the Gibbs ensemble and the Boltzmann assembly, respectively) do not coincide. We assume that it is much better to synthesize two other theories, namely, statistical thermodynam- ics without considering quantum effects and quantum mechanics without considering thermal effects. We are guided by the important fact that the same type of statistical collective (the Gibbs ensemble) is used in them. In the framework of the (�, k)-dynamics, proposed by us [2], we managed to construct a more or less holistic description of equilibrium quantum- thermal phenomena, including macroparameter fluc- tuations. Thus, we propose to modify both theories by moving in opposite directions. To do this, the c-number apparatus of statis- tical thermodynamics, which makes it possible to take equilibrium thermal macroparameter fluctua- tions into account, must be incorporated into quan- tum mechanics with its operator formalism. Intro- ducing the vacuum wave function at nonzero tem- peratures, we adapt quantum mechanics to the de- scription of thermal effects. From the other hand, the conceptual apparatus of statistical thermodynamics must be extended using c-number random quantities as previously. Matching the operator formalism with effective macroparame- ters [3], we adapt statistical thermodynamics to the description of quantum effects. 2. MODIFICATION OF QUANTUM MECHANICS FOR THE INCLUSION OF THERMAL EFFECTS In this paper, we refuse completely the density matrix and have a deal with a complex wave function depending on temperature. We renounce also an at- tempt to introduce the notion of temperature for iso- lated system and consider only a system in the heat bath because there are no isolated system in nature. We generalize the classical model of heat bath to the quantum one and interpret it as quantum-thermal vacuum. Generalizing the apparatus of quantum mechanics to take thermal effects into account, we assume that the zeroth law, i.e., the fundamental condition for the equilibrium of the object with the thermal-type stochastic environment, is initial in thermodynamics. Therefore, it would be natural to start with the de- termination of the quantum state that is adequate for ∗Corresponding author E-mail address: ogol@mail.ru 70 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 70-73. the equilibrium if the stochastic thermal action at the minimal value of effective (but not Kelvin) tempera- ture is additionally taken into account. To do this, we turn to the universal geometric properties of the Hilbert state space, in which the Cauchy-Bunyakovsky-Schwarz inequality (CBSI) for vectors \|δp〉 = δp̂\|ψ〉 and \|δq〉 = δq̂\|ψ〉 (1) holds. Here, δp̂ ≡ p̂− 〈ψ\|p̂\|ψ〉; δq̂ ≡ q̂ − 〈ψ\|q̂\|ψ〉 (2) are the operators of momentum and coordinate fluc- tuations. The CBSI for these vectors has the form of the most general Schrödinger uncertainties relation (SUR) [4]: Δp · Δq � ∣∣〈ψ\|δp̂ · δq̂\|ψ〉∣∣, (3) where (Δp)2 ≡ 〈ψ\|(δp̂)2\|ψ〉; (Δq)2 ≡ 〈ψ\|(δq̂)2\|ψ〉 (4) are the momentum and coordinate variances in the state \|ψ〉, and∣∣〈ψ\|δp̂ · δq̂\|ψ〉∣∣ = ∣∣〈δp\|δq〉∣∣ (5) has the meaning of the momentum and coordinate fluctuations correlator of the object in the same state expressed in terms of the fluctuation operators δp̂ and δq̂. In thermodynamics, the equilibrium state is sta- ble relative to small stochastic influence of environ- ment. The correlation between corresponding typical quantities is a mechanism for preserving this stabil- ity in this case. In other words, it is necessary that: (a) the correlator (5) was non equal to zero; (b) the SUR had the form of equality, i.e. became saturated. (Some heuristic considerations confirming this state- ments will be stated below.) Accordingly to the Schwarz-fonNeumann theo- rem the equality in the SUR (3) can be realized if the vectors \|δp〉 and \|δq〉 are proportional to each other, i.e. δp̂ ∣∣ψ〉 = (iγ · eiα)δq̂ ∣∣ψ〉. (6) Here, the real parameters γ > 0 and α � 0. To sim- plify calculations, we assume that the average mo- mentum and coordinate in this state are zero. It is interesting that the equation (p̂− iγeiαq̂)\|ψα〉 = 0, (7) can be obtained from formula (6); it resembles the result of the action of the annihilation operator â = p̂− iζq̂√ 2�ζ on the state \|ψα〉. In the coordinate representation, relation (7) takes the form of the differential equation for the unknown function ψα(q): � i d dq ψα − (iγeiα)q · ψα = 0. (8) Solving it and using the normalization condition, in the general case, we obtain the complex function ψα(q) = [ 2π(Δq0)2 1 cosα ]−1/4 (9) × exp { − q2 4(Δq0)2 eiα } , cosα �= 0 as the universal wave function ψα(q); here, (Δq0)2 = � 2γ , (10) and respectively (Δp0)2 = �γ 2 . The physical meaning of the function ψα(q) can be clarified if we take into account that, for α = 0, Eq. (8) for the function ψ0 is equivalent to the equa- tion for the state \|0〉, which is assumed to be called the cold vacuum. Accordingly, for arbitrary α �= 0, the state ψα describes an arbitrary vacuum. We note that the standard way of obtaining this state is related to the application of the Bogoliubov (u, v)- transformation. In the state of the arbitrary vacuum, the equali- ties (Δpα)2 = γ2(Δqα)2;∣∣〈δpα\|δqα〉 ∣∣ = ∣∣iγeiα ∣∣〈δqα\|δqα〉 = γ(Δqα)2. (11) hold for the average quantities. Substituting for- mulas (11) to SUR (3), we see that the correlated state \|ψα〉 is marked, because the SUR becomes sat- urated: Δpα · Δqα = ∣∣〈ψα\|p̂ · q̂\|ψα〉 ∣∣. (12) In the cold-vacuum state \|ψ0〉 (for α = 0), satu- rated SUR (12) transforms into the saturated Heisen- berg uncertainties relation U0 ω = Δp0 · Δq0 = ∣∣〈ψ0\|12[p̂, q̂]\|ψ0〉 ∣∣ = � 2 ≡ J0. (13) Here, J0 is the measure of the purely quantum en- vironmental action (for α = 0). Thus, for α = 0, the correlator ∣∣〈ψα\|p̂ · q̂\|ψα〉 ∣∣ has a minimum possi- ble value. As it was to be expected, the state \|ψ0〉, in really, has the sense of equilibrium state with the cold vacuum for it answers the minimal value of the vacuum energy U0 = �ω 2 . On this ground, from now on, we concede that saturated SUR (12) answers to equilibrium state with arbitrary vacuum \|ψα〉. It is assumed to regard relation (13) as a funda- mental equality reflecting the presence of unavoidable purely quantum effects in nature. This fact allows suggesting a hypothesis that the functions making it possible also to take into account thermal effects in addition to the quantum ones. They can be found 71 among the functions ψα(q) providing saturation of SUR (12) for α �= 0. To confirm it, we pass from ψ0 to the function ψα of the arbitrary vacuum. Then the correlator in the right-hand side of saturated SUR (12) can be repre- sented in the form ∣∣〈ψα\|p̂ · q̂\|ψα〉 ∣∣ = √ �2 4 tg2 α+ J2 0 = � 2 1 cosα ≡ Jα, (14) where the term in the radicand, which is additional if (14) is compared with (13), is related to the phase of the function ψα. The expression (14) depends on the quantity Jα, which generalizes the measure J0 of purely quantum effects. Under our assumption it is capable of tak- ing additional thermal effects into account too. To do this, it is necessary to relate the obtained expres- sions to the Kelvin temperature, which has no direct preimage in quantum mechanics. Of course, we did not infringe on the stability of the framework of quantum mechanics without ther- mal effects. We propose the solution only for a bor- derline area in which quantum and thermal effects cannot be neglected under equilibrium conditions. 3. INTERRELATION WITH KELVIN TEMPERATURE We introduce the concept of thermal equilibrium with the Kelvin temperature T , which is typical of this state. To do this, we start from the experimen- tally confirmed Planck expression for the average en- ergy of the quantum oscillator in a thermostat: UT ≡ EPl = �ω 2 coth ( κ ω T ) , κ = � 2kB , (15) where κ has the meaning of fundamental world con- stant reflecting the simultaneous stochastic action of the quantum and thermal types. Taking into account that, for the quantum oscil- lator in the equilibrium state, the average kinetic and potential energies coincide, we obtain the coordinate variance in this state: (ΔqT )2 = 2 ω2 · 1 2 EPl = � 2ω coth ( κ ω T ) , (16) where we set m = 1 without loss of generality. Simi- larly, we have (ΔpT )2 = �ω 2 coth ( κ ω T ) ; at m = 1 (17) for the momentum variance. In turn, calculating the coordinate variance in terms of the arbitrary-vacuum wave function ψα, we obtain (Δqα)2 = � 2γ 1 cosα . (18) We now compare two formulas (16) and (18) for the coordinate variance. They coincide if we set γ = ω; cosα = [ coth ( κ ω T )]−1 ; (19) sinα = [ ch ( κ ω T )]−1 . For such a choice of the parameter α �= 0, we choose only functions ψT such that correspond to the equi- librium with the environment at the temperature T : ψT (q) = [2π(ΔqT )2]−1/4 (20) × · exp { − q2 4(ΔqT )2 (1 − i 1 sh κ ω T ) } . They can be conditionally assumed to be the functions of the “thermal” vacuum. Accordingly, the correlator of coordinate and momentum fluctu- ations (14) becomes ∣∣〈ψT \|p̂ · q̂\|ψT 〉 ∣∣ ≡ JT = √ �2 4 sh2 κ ω T + �2 4 . (21) Here, the quantity (sh κ ω T )−1 in JT is directly related to the phase of the wave function ψT . We now pay our attention to the fact that the quantity JT can also be immediately obtained from the correlator through the variance of coordinate: ∣∣〈δpT \|δqT 〉 ∣∣ = ω(ΔqT )2 = JT = � 2 coth ( κ ω T ) . (22) Here, the dependence on the phase of the wave func- tion ψT is hidden. Together with the interrelation between momen- tum variance and coordinate variance (ΔpT )2 = ω2 · (ΔqT )2, (23) both sides of saturated ”momentum–coordinate” SUR becomes ΔpT · ΔqT = ω(ΔqT )2 = JT . (24) Thus, they are expressed only in terms of one quan- tity, namely, the coordinate variance (ΔqT )2; the ways of calculating it can be various. To clarify the physical meaning of the quantity JT , we rewrite saturated SUR (12) in the ”thermal- vacuum” state ψT (q) in the form UT ω = ΔpT · ΔqT = JT0 = � 2 coth ( κ ω T0 ) , (25) where we express the Kelvin thermostat temperature T0 explicitly. We stress that, in this relation, the left-hand side of the equality is expressed in terms of the object characteristics and the right-hand side, in terms of the environmental characteristics. At high temperatures, the saturated SUR Eq. (12) transforms into the equality T = T0 having the meaning of the standard zero law (without con- sidering temperature fluctuations), i.e., the thermal 72 equilibrium condition. This fact allows interpreting Eq. (12) as a generalized zero law in the case where the quantum and thermal effects are taken into ac- count simultaneously. This law is valid at any tem- perature. References 1. H. Umezava. Advanced Field Theory. Micro-, Macro-, and Thermal Physics. N.-Y.: AIP, 1993. 2. A.D. Sukhanov, O.N. Golubjeva. On quantum generalization of equilibrium statistical thermo- dynamics: (�, k) – dynamics // Teoret. Mat. Fiz. 2009, v. 160, N 2, p. 369 (in Russian). 3. A.D. Sukhanov, O.N. Golubjeva. Quantum gen- eralization of an equilibrium statistical thermo- dynamics: Effective macroparameters // Teoret. Mat. Fiz. 2008, v. 154, N 1, p. 185 (in Russian). 4. V.V. Dodonov, V.I. Man’ko. Generalizations of the uncertainty relations in quantum mechanics // Trudy Fiz. Inst. Lebedev. 1987, v. 183, p. 5 (in Russian). �� !�� " �� !�� , k�� #� ��$��$�� %�� !��$�� !�� & � �'��'� �� & �� '� �� '�� %�� '� �� '�� '�� ' ��( ��' �� !�� ' �� '�� )� �'� '�� '� ��( ��( ��'�� '��'( � !�� ' �, k�� '�� ( �� '*�� +( ��$��$'� � �� ' �� ( ��( !��$'( � ��'�� !�� ( �� '� �� ,-

A way to incorporation of thermodynamics into quantum theory

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