System Of Noninteracting Particles Research Articles

On the asymptotic behavior of line processes and systems of non-interacting particles

On the asymptotic behavior of line processes and systems of non-interacting particles

Bose particles in an external electromagnetic field as a one-particle problem

The statement is often made that the second quantized Hamiltonian for a system of noninteracting particles in an external weak electromagnetic field gives rise essentially to a one-particle problem, to which there corresponds a dressed vacuum different from the bare one. In this paper we implement that statement by explicitly constructing the dressed vacuum, together with the quasiparticle creation and annihilation operators associated with the one-particle problem. The process of diagonalizing the Hamiltonian is illustrated step by step in a simple model.

Cranking model calculation of the moment of inertia using the method of Dalgarno and Lewis

We present a calculation of the moment of inertia of a system of noninteracting particles moving in a common deformed, harmonic-oscillator potential well. Our calculation employs the cranking model formula, but instead of performing the usual sum over intermediate states, we use the method of Dalgarno and Lewis to replace that sum by an expectation value of appropriately defined operators. It is suggested that this calculation may serve as a useful example in a quantum mechanics or nuclear theory course.

Hohenberg-Kohn theorem for nonlocal external potentials

The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for $N$-representable density kernels, that any finite non-negative differentiable function is an $N$-representable particle density.

On the explicit-time-dependent invariance properties of quantum mechanical systems

We use the notion of split extension algebra to embody a given invariance algebra of a quantum mechanical system, a realization of which is known in terms of some variables, in a richer invariance algebra expressed in terms of the same variables. By applying this procedure to a free system of particles we show how to obtain the invariance under the Schrödinger algebra and we build a bigger invariance algebra which describes a system of noninteracting particles, for example the asymptotic states in a nonrelativistic scattering problem.

A curiosity concerning angular momentum

We show that for classical, relativistic particles (or systems of noninteracting particles) one can interpret the intrinsic (or spin) angular momentum as arising from a center of mass world line displaced from the real Minkowski space into complex Minkowski space. This interpretation is then connected with the Penrose theory of twistors. It is further shown that if the complex center of charge coincides with the complex center of mass, the resulting particle has the Dirac value of the gyromagnetic ratio, i.e., (e/mc). For massless particles, there is no unique complex line about which the angular momentum vanishes-instead, there is a complex 2-surface, which can be considered to be a Penrose twistor.

Multiparticle States with Spin-Independent Poincaré Transformation Properties

We consider systems of noninteracting particles. We define, in terms of standard on-mass-shell states and direct-product helicity states, three complete classes of multiparticle states with spin-independent Poincar\'e transformation properties. No auxiliary spin group is used, and the new "scalar spin" state labels, which are eigenvalues of generalized covariant helicity operators, have a direct physical interpretation in terms of single-particle quantum numbers. In the two-particle case we compare our states with Jacob and Wick's c.m. angular momentum states.

Temporal Evolution and the Approach to Equilibrium of a System of Noninteracting Quantum Particles

A study is made of the temporal evolution and its implications in regards to the approach to equilibrium of a system of noninteracting particles that has undergone a sudden expansion. It is assumed that the system is initially in thermal equilibrium. It is shown that the particle density approaches an equilibrium density essentially as $\frac{1}{t}$ for times shortly after expansion up to one-half the characteristic period. This period is the same as for a single-particle system. An expression is derived for the particle density that is valid for Bose, Fermi, and Boltzmann particles. The density for Bose and Fermi particles is compared with that found for Boltzmann particles.

Ergodicity and the Lorentz transformation of time-based probabilities

One-particle systems and systems of non-interacting particles, free or confined to a container, are considered in this paper. The systems are assumed to have discrete states available to them, but the particles constituting them are considered as point particles. It is shown that a necessary condition for the ergodicity of a confined system to be a Lorentz- invariant concept is the introduction of a new kind of ensemble whose systems can be in motion but are on average at rest in one and the same inertial frame. The discrete probabilities of statistical mechanics are shown to lack Lorentz invariance (contrary to accepted ideas), and a transformation formula is derived for them.

Maximum entropy arguments in relativistic statistical mechanics

The special relativistic statistical mechanics of systems of non-interacting particles is investigated by means of a maximum entropy argument. Particular attention is paid here to ensuring that the constraints required for this process satisfy the principle of covariance. The results of the preceding paper are used to determine the distribution of probabilities in any inertial frame of reference. Assuming an invariant entropy and using an entropy maximization technique, the probability distribution which is found differs from the usual one. This is discussed.

Nonequilibrium Thermodynamics of the Relaxing System Subject to Multinomial Distribution

Abstract It has been shown that the relaxation process in a system of noninteracting particles in contact with a heat bath, on the assumption that the system is in local equilibrium, can be described by the Markov process homogeneous in time, and that the solution of the Markov process is a multinomial distribution. The nonequilibrium thermodynamics for such a relaxing system was formulated by means of the stochastic entropy production which can be derived by the probabilistic version of classical entropy production. As a special case, the stochastic nonequilibrium thermodynamics for the relaxing system of identical harmonic oscillators was discussed.

Note on Statistical Mechanics of Random Systems

We discuss some equilibrium properties of random systems, i.e., systems whose Hamiltonian depends on some random variables y with a distribution P(y) which is independent of the dynamic state of the system. For a system of noninteracting particles which interact with randomly distributed scattering centers, the important quantity is the average density of states of a single particle per unit volume, n(E). Feynman's path-integral formulation of quantum statistics is used to derive some properties of the average partition function for one particle 〈Z1〉, which is the Laplace transform of n(E). In particular, it is shown that 〈Z1〉 is an analytic function of the density of scatterers ρ for a wide class of particle-scattering center potentials V(r), including those with hard cores and those with negative parts. The analyticity in ρ of the equilibrium properties of these systems is very remarkable and is in contrast to the conjectured nonanalytic behavior of their transport (i.e., diffusion) coefficients. We find also upper and lower bounds on 〈Z1〉 for a particle acted upon by a random potential V(r) obeying Gaussian statistics with 〈V(r)V(r′)〉∼exp [−α2(r−r′)2]. In the limit of ``white noise,'' 〈V(r)V(r′)〉∼δ(r−r′), 〈Z1〉 is shown to diverge in two and three dimensions but remains finite in one dimension. This agrees with approximate results on the density of states. In appendices we prove the existence, in the thermodynamic limit, of the free-energy density of a system with random scatterers and also of the frequency distribution and, thus, the free-energy density for a random harmonic crystal.

Infinite Systems of Classical Particles

In this paper, infinite translation-invariant and periodic systems of classical particles are treated directly—not as limits of finite systems. A formalism of classical creation and annihilation operators that create and destroy classical identical particles at points of phase space is employed. By use of this formalism, classical dynamical variables are expressed without reference to the canonical coordinates and momenta of individual particles, or to the number of particles. Different physical situations are described by different representations of the algebra of creation and annihilation operators. The concept of thermal equilibrium is generalized so as to be meaningful in infinite systems. Stationary states and thermal-equilibrium states for an infinite system of noninteracting particles (possibly in a periodic external potential) are exhibited explicitly.

New Approach to Statistical Mechanics

An elementary mathematical method is introduced for evaluating the average energy-state occupation numbers of a system of noninteracting particles in thermal equilibrium. Conceptualization of the problem is identical with that of Darwin and Fowler; but the execution avoids, without sacrifice of rigor or generality, the use of their profound analytic-function-theoretic machinery. The case of Maxwell-Boltzmann statistics is initially handled separately, but a more general unified treatment is also provided for Fermi-Dirac, Bose-Einstein, and Maxwell-Boltzmann systems as special cases. The three familiar distribution functions, with a known general formula for their mean-square fluctuations, are derived. The methods are also applied to mixtures of systems, with and without the possibility of combination and dissociation.

Quantum Correlation Function for the Noninteracting Particle System

A method for determing the quantum correlation functions of the noninteracting particle system in thermal equilibrium is developed. It is designed to reduce the labor involved in treating the large number of permutation operators of the symmetric group that occurs. An alternate form of the n-particle correlation function is obtained in order to simplify computation. The London-Placzek formula is derived as a check. The error in Kirkwood's superposition approximation for this system is investigated, and the exact relationship between the two- and three-particle correlation functions is found. Finally, a method determining the pair correlation function in the Hartree-Fock approximation of a pair-interacting particle system at a finite temperature is presented.

Perturbation Theory for an Infinite Medium of Fermions. III. Derivation of the Landau Theory of Fermi Liquids

The methods employed in the previous papers of this series for the ground-state energy of a normal Fermion system are extended to finite temperatures. The basic assumption is that the density matrix of the actual system in thermodynamic equilibrium can be derived by adiabatic transformation from the density matrix of system of noninteracting particles at the same temperature, volume, and chemical potential. However, the spectrum of the noninteracting system must be dependent on the intensive parameters. The self-consistency of the scheme is exhibited by demonstrating that the effective potential characterizing this system can be related to the self-energy operator of the actual system. Two equivalent versions of the theory are derived. In one of these the basic equations are precisely those of the Landau theory of Fermi liquids with the difference, however, that all quantities are given explicitly from first principles.

Relativistic Particle Dynamics. II

The relativistic dynamics for a system of non-interacting particles in Hamiltonian form is separated by a contact transformation into motion of their center of mass and internal motion. Interaction at a distance between them is then introduced into the expression for the rest-mass in terms of the internal variables. This gives a dynamics for which invariance over space displacements and rotation is trivial and which is rigorously invariant over Lorentz transformations. Earlier approximate treatments may be reduced by contact transformations to special cases of the general treatment.