No abstract

In 1926, E. Schrödinger published a paper solving his new time dependent wave equation for a displaced ground state in a harmonic oscillator (now called a coherent state). He showed that the parameters describing the mean position and mean momentum of the wave packet obey the equations of motion of the classical oscillator while retaining its width. This was a qualitatively new kind of correspondence principle, differing from those leading up to quantum mechanics. Schrödinger surely knew that this correspondence would extend to an N -dimensional harmonic oscillator. This Schrödinger Correspondence Principle is an extremely intuitive and powerful way to approach many aspects of harmonic solids including anharmonic corrections.

A potential for propagation of a wave in two-dimension is constructed from a random superposition of plane waves around all propagation angles. Surprisingly, despite the lack of periodic structure, sharp Bragg diffraction of the wave is observed, analogous to powder diffraction pattern. The scattering is partially resonant, so Fermi's golden rule does not apply. This phenomenon would be experimentally observable by sending an atomic beam into a chaotic cavity populated by a single mode laser.

We describe two approximation methods designed to capture the leading behavior of the lowenergy tail of the momentum-dependent spectral density A(k, E) and the tunneling density of states D(E) for an injected particle, such as an electron or an exciton, interacting with a bath of phonons at a non-zero initial temperature T , including quantum corrections due to the non-zero frequencies of the relevant phonons. In our imaginary-time-dependent Hartree (ITDH) approximation, we consider a situation where the particle is injected into a specified coherent state of the phonon system, and we show how one can use the ITDH approximation to obtain the correlation function C(τ ) for that initial state. The thermal average C(τ ) is obtained, in principle, by integrating the result over all possible initial phonon coherent states, weighted by a thermal distribution. However, in the low-energy tail, one can obtain a good first approximation by considering only initial states near the one that maximizes the integrand. Our second approximation, the fixed-wave-function (FWF) approximation, assumes that the wave function of the injected particle evolves instantaneously to a wave function which then is independent of time, while the phonon system continues to evolve due to interaction with the particle. We discuss how to invert the Laplace transform and how to obtain A(k, E) as well as D(E) from the imaginary-time analysis. The FWF approximation is used to calculate D(E) for a one-dimensional continuum model of a particle interacting with acoustic phonons, and effects due to the quantum motion of phonons are observed. In the classical phonon limit, where the nuclear mass is taken to infinity while the elastic constants and other parameters are held fixed, the dominant behaviors of both the ITDH and FWF approximations in the low-energy tail reduce to that found in the past for a particle in a random potential with a Gaussian statistical distribution.