Statistical dynamical feature of time development of one-dimensional harmonic lattice containing a few impurities is investigated. The fundamental solutions of equations of motion are obtained for a system of a finite number of particles as well as that of an infinite number of particles. The difference between those solutions is not appreciable at least within a range of time. Next, such an initial ensemble is introduced that a part of the lattice is canonically distributed with temperature T0 and the remaining part remains quite at rest. It is shown that the ensemble approaches a stationary state in a small sub-system for a more limited range of time. In the stationary state the mean energy of each particle is expressed as a function of characteristic parameters of the impurity particles.
Statistical dynamical feature of time development of one-dimensional harmonic lattice containing a few impurities is investigated. The fundamental solutions of equations of motion are obtained for a system of a finite number of particles as well as that of an infinite number of particles. The difference between those solutions is not appreciable at least within a range of time. Next, such an initial ensemble is introduced that a part of the lattice is canonically distributed with temperature T0 and the remaining part remains quite at rest. It is shown that the ensemble approaches a stationary state in a small sub-system for a more limited range of time. In the stationary state the mean energy of each particle is expressed as a function of characteristic parameters of the impurity particles.
linearSchrod- of coupled harmonic oscillators is investigated by means of the inger coordinates. Assuming such an initial canonical ensemble that half the system is at temperature T and other half at temperature zero, the correlation functions of particles in the system are calculated as functions of time. Thus, it is shown that the average kinetic and potential energies of each particle in the infinitely large system approach different stationary values in both sides of the impurity atom, and that the system approach a stationary state with a gap of energy distribution along the system at the impurity site. This result agrees with that obtained by Kashiwamura and Teramoto by means of trigonometric eigenfunctions of the dynamical system.The elements of position-velocity correlation matrix are also calculated. The energy flow in the large system is derived from these, and it is shown that even after a sufficiently long time the finite energy flow still exists at every point of the system though there is no temperature gradient at that point.It should be mentioned, however, that the method we used here is easier to calculate and simpler by far than that by means of the trigonometric eigenfunctions of the dynamical system.
Ensembles and expectation values Equations of motion provide predictions of time-evolving variables of a dynamical system, and thereby enable us to predict the changing behavior. These equations are characteristic of a particular type of system, say a classical harmonic oscillator or a two-state quantum system. To complete the description of behavior one must specify initial conditions that select, from the many possible solutions to the equations of motion, a particular realization. In the example of a classical point particle the equation is second order in time, and so we need two initial conditions, of position and velocity, to specify a particular solution. These two numbers, taken with the classical equation of motion, provide a complete description of the behavior of a specific harmonic oscillator. Similarly the two-state quantum system requires two initial values to complete its definition. The initial conditions are never defined with infinite precision, even for classical mechanics. More generally we deal with an ensemble, that is, a collection of similar dynamical systems (e.g. oscillators) that differ in their preparation. The individual cases share some common attributes, but not necessarily all. Thus we have only incomplete information about any single system. We might, for example, consider planetary motion in which we know only the mean energy and the orbital plane. The loci of positions of the planets then form ellipses that make up the ensemble. Although it is possible to prepare a single two-state quantum system in which the initial state is known (apart from an overall phase), often one deals with ensembles of such systems, each of which may differ in some uncontrollable property.
Abstract We present a comprehensive study of the static properties of a mobile impurity interacting with a bath with a few particles trapped in a one-dimensional harmonic trap. We consider baths with either identical bosons or distinguishable particles and we focus on the limiting case where the bath is non-interacting. We provide numerical results for the energy spectra and density profiles by means of the exact diagonalization of the Hamiltonian, and find that these systems show non-trivial solutions, even in the limit of infinite repulsion. A detailed physical interpretation is provided for the lowest energy states. In particular, we find a seemingly universal transition from the impurity being localized in the center of the trap to being expelled outside the majority cloud. We also develop an analytical ansatz and a mean-field solution to compare them with our numerical results in limiting configurations.
Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.
The nonequilibrium dynamics of an impurity immersed with a finite velocity into a one-dimensional system of weakly interacting bosons is studied within the framework of the time-dependent Gross-Pitaevskii equation. We uncover and characterize different regimes of relaxation dynamics. We find that the final impurity velocity remains constant in a large interval of sufficiently big and realistic initial velocities. The underlying physical mechanism is the emission of the dispersive density shock waves that carry away the excess of the initial impurity momentum, while locally the system remains in the same stationary state. In contrast, a heavy impurity with the same coupling constant relaxes differently and the regime of constant final velocity disappears. Furthermore, a fast heavy impurity exhibits damped velocity oscillations in time before reaching a stationary state. This process is accompanied by the oscillations of the local depletion of the boson density around the impurity, until their positions coincide and they continue the motion together. Decreasing the impurity-boson coupling or increasing the strength of repulsion between bosons, the oscillations get amplified. In the case of a heavy impurity with the mass bigger than the critical one, the ground-state energy as a function of momentum exhibits cusps and metastable branches. We show that they manifest themselves by a soliton emission, a considerable slowing down of the relaxation, and a change of the impurity direction of motion with respect to the initial one.