System Of Quantum Particles Research Articles

Littlewood-Richardson coefficients via Yang-Baxter equation

The purpose of this paper is to present an interpretation for the decomposition of the tensor product of two or more irreducible representations of GL(N) in terms of a system of quantum particles. Our approach is based on a certain scattering matrix that satisfies a Yang-Baxter type equation. The corresponding piecewise-linear transformations of parameters give a solution to the tetrahedron equation. These transformation maps are naturally related to the dual canonical bases for modules over the quantum enveloping algebra Uq(sln)⁠. A byproduct of our construction is an explicit description for the cone of Kashiwara's parametrizations of dual canonical bases. This solves a problem posed by Berenstein and Zelevinsky. We present a graphical interpretation of the scattering matrices in terms of web functions, which are related to honeycombs of Knutson and Tao.

Renormalization group at finite temperature in quantum mechanics

We establish the exact renormalization group equation for the potential of a one quantum particle system at finite and zero temperature. As an example we use it to compute the ground state energy of the anharmonic oscillator. We comment on an improvement of the Feynman–Kleinert's variational method by the renormalization group.

Generalized blackbody distribution within the dilute gas approximation

We consider the application of Tsallis’ generalized statistical mechanics to systems of quantum particles (fermions or bosons). An approach due to Büyükkiliç et al. to find analytic formulae for quantum distribution functions and its application to generalized blackbody radiation is discussed. Generalized blackbody laws (Stefan–Boltzmann law, Rayleigh–Jeans law and Wien empirical distribution law) are proposed.

Quantum scattering theoretical description of responses of systems to thermodynamical gaps of reservoirs

A method of describing responses of systems to thermodynamical gaps of reservoirs is given by a quantum scattering theoretical approach based on the Liouville-von Neumann equation. We consider a quantum system of particles connected to thermodynamical reservoirs by leads. The effects of the reservoirs are introduced as an asymptotic condition at the end of the leads. We derive a simple expression for currents of conserved quantities. The Landauer formula and its generalizations are derived from this method.

On the generalized blackbody distribution

We consider an application of the generalized statistical mechanics due to Tsallis to systems of quantum particles (fermions or bosons). Tsallis' approximation to finding the analytic quantum distribution function and its application to blackbody radiation are discussed. A generalized Stefan-Boltzmann law is proposed for the q < 1 case and the q dependence of the Stefan constant is derived.

On the generalized distribution functions of quantum gases

Abstract The different formalisms of generalized fermion and boson statistics derived from Tsallis' statistical mechanics are discussed. It is concluded that, in spite of the nonextensitivity of the theoretical framework, simple, analytical expressions can be used, within a good approximation, for macroscopic systems of quantum particles.

Path integral simulations of atomic and molecular systems

The path integral picture of the statistical mechanics of quantum many-body systems is presented from the point of view of developing simulation algorithms. Monte Carlo and molecular dynamics techniques for systems of distinguishable quantum particles, bosons and fermions are reviewed. Path integral simulations of atomic liquids and solids, quantum clusters and solvated electrons are described and the usefulness of such techniques for understanding phenomena such as orientational transitions, surface adsorption and rates of quantum processes is discussed.

Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy).

Integrable Model of Interacting Fermions Confined by the Morse Potential

The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (su(n) spins) interacting with an external field with the Morse potential of an arbitrary strength τ2. These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form.

On Quantum Systems of Particles with Singular Magnetic Interaction

For systems of particles with singular magnetic interation for special choice of a selfadjoint extension of the Hamiltoniam equilibrium reduced density matrices are calculated in the thermodynamic limit for simplest pair magnetic potentials.

Relationship between the classical transport coefficients and hydrodynamic kernels for quantum-field models

A study is made of the classical limit of nonlocal hydrodynamics, describing quantum systems of many particles. The quantum-field model is not specified, it being assumed only that there are local laws governing the conservation of energy — momentum and the number of particles. Changing over to the classical limit corresponds in nonlocal hydrodynamics to the limit of slow processes and long waves. It is shown that a hydrodynamics with viscosity and self-diffusion but without heat conduction is obtained in this case because the velocity of the medium is determined in terms of the energy flux, which is natural for quantum-field systems. Heat conduction can be introduced if velocity is instead determined in terms of particle flux.

Exactly solvable model of interacting electrons confined by the Morse potential

The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU(n) spins) interacting with an external field with the Morse potential of an arbitrary strength τ2. These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞.

A stochastic Hamiltonian approach for quantum jumps, spontaneous localizations, and continuous trajectories

We give an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with "bubbles" which admit a continual counting observation. This model allows to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localisations of the momentums of the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singular interaction as a result of quantum filtering, i.e., a sequential time continuous conditioning of an input quantum process by the output measurement data. We show that in the case of indistinguishable particles or atoms the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schroedinger equation only in the mean field approximation, whereas the central limit yelds Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.

Universality in quantum chaos

Studies of the parametric dependence of energy levels and wavefunctions on changes in external perturbations reveal a new type of universality in quantum chaos. Statistical properties that characterise the universal behaviour display a striking connection to the dynamical properties of a strongly interacting one-dimensional quantum system of particles. We discuss the common mathematical structure which unites quantum chaos, matrix models, and an integrable quantum Hamiltonian.

Equilibrium in quantum systems of particles with magnetic interaction

We compute density matrices determining equilibrium states in the thermodynamic limit for a class of quantum systems. Interaction in these systems is described as a collective electromagnetic field whose scalar potential is equal to zero.

The SU( v) Calogero spin system

A one-dimensional quantum particle system in which particles with su( v) spins interact through inverse square interactions is introduced. We refer to it as the SU( v) Calogero spin system. Using the quantum inverse scattering method, we reveal algebraic structures of the system: hidden symmetry is the U( v) − SU( v) ⊗ U(1) current algebra. This is consistent with the fact that the ground-state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, known as the w 1 + ∞-algebra. With this W-algebra we have a unified viewpoint on the integrable quantum particle systems with long-range interactions such as the Calogero type ( 1 x 2 -interactions) and Sutherland type ( 1 sin 2 x -interactions). The Yangian symmetry is briefly discussed.

Fractional statistics in one dimension: view from an exactly solvable model

One-dimensional fractional statistics is studied using the Calogero-Sutherland model (CSM) which describes a system of non-relativistic quantum particles interacting with an inverse-square two-body potential on a ring. The inverse-square exchange can be regarded as a pure statistical interaction and this system can be mapped to an ideal gas obeying the fractional exclusion and exchange statistics. The details of the exact calculations of the dynamical correlation functions or this ideal system is presented in this paper. An effective low-energy one-dimensional “anyon” model is constructed; and its correlation functions are found to be in agreement with those in the CSM; and this agreement provides an evidence for the equivalence of the first- and the second-quantized construction of the 1D anyon model at least in the long-wavelength limit. Furthermore, the finite-size scaling applicable to the conformally invariant systems is used to obtain the complete set of correlation exponents for the CSM.

Chaos and quantum thermalization.

We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey {\it Berry's conjecture}. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell--Boltzmann, Bose--Einstein, or Fermi--Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon {\it eigenstate thermalization}. We show that a generic initial state will approach thermal equilibrium at least as fast as $O(\hbar/\Delta)t^{-1}$, where $\Delta$ is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a new foundation for quantum statistical mechanics.

FRACTIONAL STATISTICS IN ONE DIMENSION: MODELING BY MEANS OF 1/x2 INTERACTION AND STATISTICAL MECHANICS

The equivalence of a quantum system of particles interacting with a two-body inverse square potential to a system of noninteracting particles obeying 1D fractional statistics (1D anyons), stated by Polychronakos for particles on a line, is studied for the cases where the interacting system is placed (i) into a harmonic potential on a line, and (ii) on a ring, with imposing periodic boundary conditions. In the first case, reducibility of the interacting system to the Calogero system is used to explore the statistical distribution for free 1D anyons. On a ring, the thermodynamic limit is discussed in terms of the thermodynamic (asymptotic) Bethe ansatz. Yang and Yang’s integral equation is treated in this case as describing the statistical mechanics of free 1D anyons. It gives a functional equation for the statistical distribution of 1D anyons consistent with the harmonic potential approach. We show that on a ring of a finite circumference, a system of two free 1D anyons is equivalent to a system of two particles interacting with an inverse sine square potential (the Sutherland system). We also discuss the relation to the statistical mechanics of free anyons in 2+1 dimensions.

Quantum double-well chain: Ground-state phases and applications to hydrogen-bonded materials.

Extrapolating the results of hybrid quantum Monte Carlo simulations to the zero temperature and infinite-chain-length limits, we calculate the ground-state phase diagram of a system of quantum particles on a chain of harmonically coupled, symmetric, quartic double-well potentials. We show that the ground state of this quantum chain depends on two parameters, formed from the ratios of the three natural energy scales in the problem. As a function of these two parameters, the quantum ground state can exhibit either broken symmetry, in which the expectation values of the particle's coordinate are all nonzero (as would be the case for a classical chain), or restored symmetry, in which the expectation values of the particle's coordinate are all zero (as would be the case for a single quantum particle). In addition to the phase diagram as a function of these two parameters, we calculate the ground-state energy, an order parameter related to the average position of the particle, and the susceptibility associated with this order parameter. Further, we present an approximate analytic estimate of the phase diagram and discuss possible physical applications of our results, emphasizing the behavior of hydrogen halides under pressure.