Thermodynamic framework for the ground state of a simple quantum system

The ground state of a two-level system (associated with probabilities $p$ and $1-p$, respectively) defined by a general Hamiltonian $\widehat{H}={\widehat{H}}_0+\lambda \widehat{V}$ is studied. The simple case characterized by $\lambda =0$, whose Hamiltonian ${\widehat{H}}_0$ 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 $\lambda \widehat{V}$ ($\lambda >0$) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy $S\left[p\right]$ (with $S\left[p\right]\ne S_{\mathrm{BG}}\left[p\right]$), if the Hermitian operator $\widehat{V}$ is represented by a $2\times 2$ matrix, defined by nonzero off-diagonal elements $V_{12}=V_{21}=-z$, where $z$ is a real positive number. Hence, this new term in the Hamiltonian, presenting $V_{12}\ne 0$, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature $\theta$ ($\theta \propto \lambda z$), which is shown to be a parameter conjugated to the entropy $S$. Based on this, one introduces an infinitesimal heatlike quantity, $\delta Q=\theta dS$, 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 $T\leftrightarrow \theta$, 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., $S\to 0$, when $\theta \to 0$.

Figure 1

(a) The internal energy of Eq. (3.7) (in units of $\mathrm{\Delta }$) is exhibited vs the dimensionless variable $\tau =(k\theta /\mathrm{\Delta })$; (b) the same representation is used for the dimensionless entropy of Eq. (3.8).

Figure 2

The rescaled dimensionless specific heat $c_{\mathrm{\Delta }}$ [defined in Eq. (4.17)] is represented vs the dimensionless variable $\tau =k\theta /\mathrm{\Delta }$.

Figure 3

The Carnot cycle $(A\to B\to C\to D\to A)$ is represented in the plane $\sigma$ (dimensionless) vs $\mathrm{\Delta }$ (dimensions of energy, e.g., Joules), with two isothermal and two adiabatic transformations intercalated. The transformations for $\sigma$ constant are adiabatic, and herein they were chosen to occur for $\sigma =0.25$ ($B\to C$) and $\sigma =0.15$ ($D\to A$). The isothermal transformations are obtained from the equation of state [cf. Eq. (4.4)] at conveniently chosen values of ${\theta }_1(A\to B)$ and ${\theta }_2(C\to D)$, with ${\theta }_1>{\theta }_2$. An amount of heat $Q_1$ is absorbed at ${\theta }_1$, and $Q_2$ is released at ${\theta }_2$; the work quantities $W_{AB}$ and $W_{BC}$ are negative, whereas $W_{CD}$ and $W_{DA}$ are positive, so that the area inside the cycle represents the total work $W$ done $\mathit{on}$ the system, being negative, as expected from Eq. (4.3). The cycle above holds for any system of units, e.g., by considering all quantities with dimensions of energy in Joules.