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Physical review. E

Thermodynamic framework for the ground state of a simple quantum system

André M. C. Souza, Fernando Dantas Nobre

Published Jan 9, 2017• 3 citations
Publisher

Abstract

The ground state of a two-level system (associated with probabilities $p$ and $1\ensuremath{-}p$, respectively) defined by a general Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}+\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ is studied. The simple case characterized by $\ensuremath{\lambda}=0$, whose Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}$ is represented by a diagonal matrix, is well established and solvable within Boltzmann-Gibbs statistical mechanics; in particular, it follows the third law of thermodynamics, presenting zero entropy (${S}_{\mathrm{BG}}=0$) at zero temperature ($T=0$). Herein it is shown that the introduction of a perturbation $\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ ($\ensuremath{\lambda}>0$) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy $S[p]$ (with $S[p]\ensuremath{\ne}{S}_{\mathrm{BG}}[p]$), if the Hermitian operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ is represented by a $2\ifmmode\times\else\texttimes\fi{}2$ matrix, defined by nonzero off-diagonal elements ${V}_{12}={V}_{21}=\ensuremath{-}z$, where $z$ is a real positive number. Hence, this new term in the Hamiltonian, presenting ${V}_{12}\ensuremath{\ne}0$, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature $\ensuremath{\theta}$ ($\ensuremath{\theta}\ensuremath{\propto}\ensuremath{\lambda}z$), which is shown to be a parameter conjugated to the entropy $S$. Based on this, one introduces an infinitesimal heatlike quantity, $\ensuremath{\delta}Q=\ensuremath{\theta}dS$, leading to a consistent thermodynamic framework, and by proposing an infinitesimal form for the first law, a Carnot cycle and thermodynamic potentials are obtained. All results found are very similar to those of usual thermodynamics, through the identification $T\ensuremath{\leftrightarrow}\ensuremath{\theta}$, and particularly the form for the efficiency of the proposed Carnot Cycle. Moreover, $S$ also follows a behavior typical of a third law, i.e., $S\ensuremath{\rightarrow}0$, when $\ensuremath{\theta}\ensuremath{\rightarrow}0$.

DOI
10.1103/physreve.95.012111

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The ground state of a two-level system (associated with probabilities ppp and 1\ensuremath−p1\ensuremath{-}p1\ensuremath−p, respectively) defined by a general Hamiltonian H\ifmmode^\elseˆ\fi=H\ifmmode^\elseˆ\fi0+\ensuremathλV\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}+\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}H\ifmmode^\elseˆ\fi​=H\ifmmode^\elseˆ\fi​0​+\ensuremathλV\ifmmode^\elseˆ\fi​ is studied. The simple case characterized by \ensuremathλ=0\ensuremath{\lambda}=0\ensuremathλ=0, whose Hamiltonian H\ifmmode^\elseˆ\fi0{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}H\ifmmode^\elseˆ\fi​0​ is represented by a diagonal matrix, is well established and solvable within Boltzmann-Gibbs statistical mechanics; in particular, it follows the third law of thermodynamics, presenting zero entropy (SBG=0{S}_{\mathrm{BG}}=0SBG​=0) at zero temperature (T=0T=0T=0). Herein it is shown that the introduction of a perturbation \ensuremathλV\ifmmode^\elseˆ\fi\ensuremath{\lambda}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}\ensuremathλV\ifmmode^\elseˆ\fi​ (\ensuremathλ>0\ensuremath{\lambda}>0\ensuremathλ>0) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy S[p]S[p]S[p] (with S[p]\ensuremath≠SBG[p]S[p]\ensuremath{\ne}{S}_{\mathrm{BG}}[p]S[p]\ensuremath=SBG​[p]), if the Hermitian operator V\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}V\ifmmode^\elseˆ\fi​ is represented by a 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 matrix, defined by nonzero off-diagonal elements V12=V21=\ensuremath−z{V}_{12}={V}_{21}=\ensuremath{-}zV12​=V21​=\ensuremath−z, where zzz is a real positive number. Hence, this new term in the Hamiltonian, presenting V12\ensuremath≠0{V}_{12}\ensuremath{\ne}0V12​\ensuremath=0, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature \ensuremathθ\ensuremath{\theta}\ensuremathθ (\ensuremathθ\ensuremath∝\ensuremathλz\ensuremath{\theta}\ensuremath{\propto}\ensuremath{\lambda}z\ensuremathθ\ensuremath∝\ensuremathλz), which is shown to be a parameter conjugated to the entropy SSS. Based on this, one introduces an infinitesimal heatlike quantity, \ensuremathδQ=\ensuremathθdS\ensuremath{\delta}Q=\ensuremath{\theta}dS\ensuremathδQ=\ensuremathθdS, leading to a consistent thermodynamic framework, and by proposing an infinitesimal form for the first law, a Carnot cycle and thermodynamic potentials are obtained. All results found are very similar to those of usual thermodynamics, through the identification T\ensuremath↔\ensuremathθT\ensuremath{\leftrightarrow}\ensuremath{\theta}T\ensuremath↔\ensuremathθ, and particularly the form for the efficiency of the proposed Carnot Cycle. Moreover, SSS also follows a behavior typical of a third law, i.e., S\ensuremath→0S\ensuremath{\rightarrow}0S\ensuremath→0, when \ensuremathθ\ensuremath→0\ensuremath{\theta}\ensuremath{\rightarrow}0\ensuremathθ\ensuremath→0.
We consider open quantum systems consisting of a finite system of independent fermions with arbitrary Hamiltonian coupled to one or more equilibrium fermion reservoirs (which need not be in equilibrium with each other). A strong form of the third law of thermodynamics, S(T) → 0 as T → 0, is proven for fully open quantum systems in thermal equilibrium with their environment, defined as systems where all states are broadened due to environmental coupling. For generic open quantum systems, it is shown that S(T) → g ln 2 as T → 0, where g is the number of localized states lying exactly at the chemical potential of the reservoir. For driven open quantum systems in a nonequilibrium steady state, it is shown that the local entropy Sx;T→0 as T(x) → 0, except for cases of measure zero arising due to localized states, where T(x) is the temperature measured by a local thermometer.
Historically statistical mechanics was developed for gasses for which there are very large numbers of molecules (i.e. on the order of 10 to 23rd power). As a result factorial schemes were developed to count the number of arrangements possible, the ln taken of this result and Stirling’s approximation for large N applied. The Shannon’s entropy form S/(-k) = Sum over i P(ei)ln(P(ei)) is consistent with this large N approach. Furthermore, large N statistical mechanics was linked to thermodynamics, in particular the first law dE=TdS - PdV. Surprisingly then, statistical mechanical results were applied to systems with small numbers of particles (e.g. a nucleus with about 100 nucleons etc). In Parts I and II we argued that the first law of thermodynamics describes two separate kinds of changes associated with energy changes. The second, work, is linked to the change of a mechanical parameter such as the volume of a box etc. The energy of a bound quantum particle in a box with an infinite potential is proportional to nn/LL. Thus we argued that a pure quantum bound state may also be associated with the first law of thermodynamics with changes in L with n, the energy level, held constant representing pure work. We further argued that for a bound state two classical like probabilities exists P(x)=W(x)W(x) and P(p)=a(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx) with P(x) being proportional to 1/L and P(p) to L. For a quantum adiabatic transformation (i.e. changing L with n staying constant) the term dS=0. If one wishes to create S from P(x) and P(p) and desires linearity i.e. S=Sp(n)+Sx, then Sp=Integral dp P(p)ln(P(p)) and Sx= Integral dx P(x) ln(P(x)) is a solution. If one next considers a quantum wavefunction for two levels W(x,t) = a(1)exp(-i E1 t) W1(x) + a(2) exp(-iE2 t) W2(x) with Eave = Sum over n en P(en) where Wn=a(n)a(n), one may again utilize the first law of thermodynamics. In Parts I and II we carried over the entropy form P(ei)ln(P(ei)). In this note we wish to argue that this form follows at the two energy level picture by considerations of a function which consists entirely of probabilities i.e. P(ei) and is linear in the sense that such a function for two identical two state systems should yield S1+S2 as the first law should apply to each system. Furthermore P(ei) should be a function of P(ei/T) where T is a parameter which behaves as 1/LL because ei is proportional to 1/LL. In this way any function of P(ei) remains unchanged under a quantum adiabatic transformation for which i’s stay constant, but L changes. As a result, the entropy at the quantum level seems to follow from such arguments and not from large N approximations of factorial counting approaches. In large N situations it is argued that P(ei)=C(T)exp(-ei/T) follows from maximizing S with respect to the form of P(ei) subject to constant average energy. For small N situations (e.g. the 2 level example) one may solve Sum over i ei d(a(i)a(i)) = T dS directly without maximization. The maximization approach follows directly from dE=TdS (work=0) because dE= Sum ei dP(ei) in such scenarios. This is of the form of a Lagrange constraint This has been pointed out in Part II. In this note, we argue that the maximization of entropy is not needed at all. For dE=Sum over i ei dP(ei) and the ln form of entropy P(ei) is the inverse function of ln. Thus we argue that linearity and not large N arguments are enough to obtain Shannon’s form of entropy and that this ln form suggests P(ei/T) = C(T)exp(-ei/T) if dE=Sum over ei dP(ei)= TdS (for dW=0). Thus maximization of entropy subject to constant average energy is also not needed (although it does follow mathematically from dE=TdS.)
We have developed a theoretical formalism to introduce temperature as a parameter into the framework of non-relativistic quantum mechanics using the laws of classical thermodynamics and the canonical ensemble scheme of statistical mechanics. A self-consistent Hamiltonian has then been constructed for a given quantum many-body system which includes the effect of temperature in the form of correction terms added to the corresponding zero-temperature Hamiltonian of the system. Investigating some quantum mechanical systems with exact zero-temperature solutions including the particle-in-a-box model, the free particle, and the harmonic oscillator within our finite-temperature approach up to the first order of self-consistency has led to temperature-dependent Hamiltonians describing these systems above absolute zero without encountering any physically unacceptable brand of behavior for their wave functions and energy spectra. Results firmly support the view that a quantum mechanical system at a finite temperature behaves as if it is in a zero-temperature excited state.
From the thermodynamic equilibrium properties of a two-level system with variable energy-level gap Delta\\DeltaDelta, and a careful distinction between the Gibbs relation dE=TdS+(E/Delta)dDeltadE = T dS + (E/\\Delta) d\\DeltadE=TdS+(E/Delta)dDelta and the energy balance equation $dE = \\delta Q^\\leftarrow - \\delta W^\\to$, we infer some important aspects of the second law of thermodynamics and, contrary to a recent suggestion based on the analysis of an Otto-like thermodynamic cycle between two values of Delta\\DeltaDelta of a spin-1/2 system, we show that a quantum thermodynamic Carnot cycle, with the celebrated optimal efficiency 1−(Tlow/Thigh)1 - (T_{low}/T_{high})1−(Tlow​/Thigh​), is possible in principle with no need of an infinite number of infinitesimal processes, provided we cycle smoothly over at least three (in general four) values of Delta\\DeltaDelta, and we change Delta\\DeltaDelta not only along the isoentropics, but also along the isotherms, e.g., by use of the recently suggested maser-laser tandem technique. We derive general bounds to the net-work to high-temperature-heat ratio for a Carnot cycle and for the 'inscribed' Otto-like cycle, and represent these cycles on useful thermodynamic diagrams.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
So far, we have proposed a method of expressing finite temperature in the form of ground state and pure state by changing the shape of the distribution function and using the step function. In this paper, we calculate the Hamiltonian and its eigenval- ues in the second quantized system in this format. Here, the Hamiltonian in the ground state is operated by the creation and annihilation operator in which the matrix elements are changed according to the change in the statistics. It also shows the basics of commutation relations of parastatistics and discusses generalized mathematics.