We study four distinct families of Gibbs canonical distributions defined on the standard complex, quaternionic, real and classical (nonquantum) two-level systems. The structure function or density of states for any two-level system is a simple power (1, 3, 0 or -1) of the length of its polarization vector, while the magnitude of the energy of the system, in all four cases, is the negative of the logarithm of the determinant of the corresponding two-dimensional density matrix. Functional relationships (proportional to ratios of gamma functions) are found between the average polarizations with respect to the Gibbs distributions and the effective polarization temperature parameters. In the standard complex case, this yields an interesting alternative, meeting certain probabilistic requirements recently set forth by Lavenda, to the more conventional (hyperbolic tangent) Brillouin function of paramagnetism (which, Lavenda argues, fails to meet such specifications).
We study four distinct families of Gibbs canonical distributions defined on the standard complex, quaternionic, real and classical (nonquantum) two-level systems. The structure function or density of states for any two-level system is a simple power (1, 3, 0 or -1) of the length of its polarization vector, while the magnitude of the energy of the system, in all four cases, is the negative of the logarithm of the determinant of the corresponding two-dimensional density matrix. Functional relationships (proportional to ratios of gamma functions) are found between the average polarizations with respect to the Gibbs distributions and the effective polarization temperature parameters. In the standard complex case, this yields an interesting alternative, meeting certain probabilistic requirements recently set forth by Lavenda, to the more conventional (hyperbolic tangent) Brillouin function of paramagnetism (which, Lavenda argues, fails to meet such specifications).
The three and five-dimensional convex sets of two-level complex and quaternionic quantum systems are studied in the Bayesian thermostatistical framework introduced by Lavenda. Associated with a given parameterization of each such set is a quantum Fisher (Helstrom) information matrix. The square root of its determinant (adopting an ansatz of Harold Jeffreys) provides a reparameterization-invariant prior measure over the set. Both such measures can be properly normalized and their univariate marginal probability distributions (which serve as structure functions) obtained. Gibbs (posterior) probability distributions can then be found, using Poisson's integral representation of the modified spherical Bessel functions. The square roots of the (classical) Fisher information of these Gibbs distributions yield (unnormalized) priors over the inverse temperature parameters.
Among the monotone metrics on the (n^{2} - 1)-dimensional convex set of n x n density matrices, as Petz and Sudar have recently elaborated, there are a minimal (Bures) and a maximal one. We examine the proposition that it is physically meaningful to treat the volume elements of these metrics as densities-of-states for thermodynamic purposes. In the n = 2 (spin-1/2) case, use of the maximal monotone metric, in fact, does lead to the adoption of the Langevin (and not the Brillouin) functions, thus, completely conforming with a recent probabilistic argument of Lavenda. Brody and Hughston also arrived at the Langevin function in an analysis based on the Fubini-Study metric. It is a matter of some interest, however, that in the first (subsequently modified) version of their paper, they had reported a different result, one fully consistent with the alternative use of the minimal monotone metric. In this part I of our investigation, we first study scenarios involving partially entangled spin-1/2 particles (n = 4, 6,...) and then a certain three-level extension of the two-level systems. In part II, we examine, in full generality, and with some limited analytical success, the cases n = 3 and 4.
In the Bayesian theory of statistical inference, as first suggested by Harold Jeffreys in highly influential work, one can employ the square root of the determinant of an n×n Fisher information matrix as a reparametrization-invariant prior (generally unnormalized) measure over an n-dimensional family (Riemannian manifold) of probability distributions. Jeffreys’ ansatz is adopted here to the quantum context, that is, with regard to density matrices rather than probability distributions, by computing the quantum Fisher information matrices (associated with Helstrom and Holevo) for the three-, five-, and eight-dimensional convex sets of two-level complex, two-level quaternionic, and three-level complex systems, respectively. In both the two-level cases, the priors have been normalized to probability distributions over the 2×2 density matrices, while, in the much more computationally demanding three-level situation, no such normalization has been accomplished. An argument is made for the general form, in terms of eigenvalues, that the (unnormalized) prior should assume over the (n2−1)-dimensional convex set of n×n density matrices.
Paramagnetic spin systems are typically analyzed using classical statistical mechanics, and the discussion is usually limited to the simplest, two-level, isolated spin-1/2 system. Here, I extend the analysis of these paramagnetic spin systems in two significant ways: first, I analyze a three-level, isolated spin-1 system, and second, I use quantum statistical mechanics for the analysis. In this way, I provide an introduction to quantum statistical mechanics, including the density operator and von Neumann entropy, but I do so using paramagnetic spin systems, which are quite familiar to students from their study of classical statistical mechanics. The high-field equilibrium density operator ρ̂eq of an isolated spin-1 system can be reincarnated as an angle-axis {Θ,ẑ} parameterized rotation operator ρ̂eq≃exp(βIz)=D̂(j)(Θ,ẑ)=exp(−iΘIz), where the modulus of the imaginary rotation angle Θ = iβ is the normalized inverse spin temperature |Θ| = β = ℏω0/(kBTS). Using the Chebyshev polynomial operators of a discrete real variable fλ(j)(Iz) as the density operator expansion basis, I show that all of the most significant quantum statistical mechanical properties of this paramagnetic spin-1 system, the quantum partition function Z=Tr[exp(βIz)], the average energy ⟨H⟩, the equilibrium von Neumann entropy SνNeq, the spin temperature TS, and the vector (dipole) and tensor (quadrupole) polarizations, can all be expressed in terms of the generalized characters χλ(j)(Θ)≡χλ(j)(iβ) of irreducible representations of the rotation group SO(3). Expanding the density operator ρ̂ in spin polarization moments (or statistical tensors) ρ10(1) and ρ20(1) defines the von Neumann entropy surface SνN(ρ10, ρ20), whose topography defined on the two-dimensional polarization moments landscape is used (i) to elicit the relation between the thermal equilibrium polarization moments ρ10(β) and ρ20(β), and (ii) to visualize the connection between entropy and spin temperature.
Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e., the wavefunction, of an isolated system is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger equation. On the other hand a mixed state, i.e., a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.
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 (SBG=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 $\ensuremath{\theta}\ensuremath{\rightarrow}0$.