Jack Wong

Linear transport theory in a random medium

The time‐independent linear transport problem in a purely absorbing (no scattering) random medium is considered. A formally exact equation for the ensemble averaged distribution function 〈Ψ〉 is derived. Under the assumption of a two‐fluid statistical mixture, with the transition from one fluid to the other assumed to be determined by a Markov process, an exact solution to this equation for 〈Ψ〉 is obtained. In the source‐free case, this solution is shown to agree with the result obtained by ensemble averaging simple exponential attenuation. Several approximations to the exact equation for 〈Ψ〉 are considered, and numerical results given to assess the accuracy of these approximations.

Renewal theory for transport processes in binary statistical mixtures

Renewal theory is used to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non‐Markovian) fluid chord length distributions. One conclusion (for unimodal distributions) that is drawn is that the mean and variance of the chord length distributions through each fluid is sufficient knowledge of the statistics to give a reasonably accurate description of the ensemble averaged intensity. Expressions for effective cross sections and an effective source to be used in the usual deterministic transport equation are also obtained. The use of these effective quantities allows statistical information to be introduced very simply into a standard transport equation. An analysis is given which shows how the transport description, including scattering, in a Markovian mixture can be modified to yield an approximate description of transport in a non‐Markovian mixture. Numerical results are given to assess the accuracy of this model...

Canonically Conjugate Pairs, Uncertainty Relations, and Phase Operators

Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCRs). Examples are given for which the uncertainty principle does not follow from the CCRs. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.

Results on Certain Non‐Hermitian Hamiltonians

We present a few results on the spectral properties of a class of physically reasonable non‐Hermitian Hamiltonians. These theorems relate the spectral properties of a non‐self‐adjoint operator (of the aforementioned class) in terms of that of a self‐adjoint operator. These theorems can be specialized to yield conditions under which the perturbed eigenvalues (of the above class of operators) vary continuously from the eigenvalues of the unperturbed operators. If the Schrodinger equation has to be solved numerically, a knowledge of the spectral properties of the non‐Hermitian Hamiltonian would insure when the eigensolutions exist.

Absence of Long‐Range Order in Thin Films

Thin films are described as idealized systems having finite extent in one direction but infinite extent in the other two. For systems of particles interacting through smooth potentials (e.g., no hard cores), it is shown that an equilibrium state for a homogeneous thin film is necessarily invariant under any continuous internal symmetry group generated by a conserved density. For short‐range interactions it is also shown that equilibrium states are necessarily translation invariant. The absence of long‐range order follows from its relation to broken symmetry. The only properties of the state required for the proof are local normality, spatial translation invariance, and the Kubo‐Martin‐Schwinger boundary condition. The argument employs the Bogoliubov inequality and the techniques of the algebraic approach to statistical mechanics. For inhomogeneous systems, the same argument shows that all order parameters defined by anomalous averages must vanish. Identical results can be obtained for systems with infinit...

GALILEAN RELATIVITY, LOCALITY, AND QUANTUM HYDRODYNAMICS.

In this work we study the consequences of locality and Galilean covariance for the operators that occur in Landaus quantum hydrodynamics. We specifically consider the following requirements: (1) Galilean covariance of the velocity field, (2) locality of the velocity field, and (3) Landaus assumption that the momentum density is a symmetrized product of the velocity and density operators. It is demonstrated that the density‐velocity commutation relation of the Landau theory is essentially a direct consequence of (1) and (2). The addition of (3) is sufficient to determine the velocity‐velocity commutation relation, also in agreement with Laudau. We further show that the density‐velocity commutation relation, independent of (3) or any specific form for the velocity field, is inconsistent with the nonnegative character of the local density.

On a Condition for Completeness

It is shown that a completeness relation for the eigensolutions of a non‐Hermitian operator H can be derived even if the resolvent operator R(H) of H is allowed to have poles of higher order than just simple poles, as required by Fonda, Ghirardi, Weber, and Rimini. A class of operators satisfying the requirements of this note is cited.

Scaling laws for saturation in free electron lasers

Abstract Saturation in free electron laser amplifiers is described by the critical intensity at which the saturated gain length equals the electron synchrotron wavelength. This yields approximate scaling laws agreeing with simulation results within ±10%.