Scaled Fourier Transforms and Heisenberg’s Uncertainty Principle

At the recent QSCP XIX, the author claimed a procedure of using a scaled Fourier transform (the scaling being determined by the detailed interaction and particle mass for a harmonic oscillator) to achieve simultaneous resolution of position and momentum greater than the standard Heisenberg value of 1/2. The procedure, while correct for digital signal processing, is, in fact, invalid for quantum mechanics. The purpose of this paper is simply to give the correct analysis of the uncertainty product, thereby clarifying the error made.     

On stability of self-assembled nanoscale patterns

We conduct linear and nonlinear stability analyses on a paradigmatic model of nanostructure self-assembly. We focus on the spatio-temporal dynamics of the concentration field of deposition on a substrate. The physical parameter of interest is the mean concentration C₀ of the monolayer. Linear stability analysis of the system shows that a homogeneous monolayer is unstable when C₀ lies within a band symmetric about . On increasing C₀ from zero, the homogeneous solution destabilizes to a hexagonal array, which then transitions to stripes. Transitions to and from the hexagonal state are subcritical. Square patterns are unstable for all values of C₀ transitioning either to hexagons or stripes. Further, we present stability maps for striped arrays by considering possible instabilities. The analytical results are confirmed by numerical integrations of the Suo-Lu model. Our formalism provides a theoretical framework to understand guided self-assembly of nanostructures.     

Eigenstate expansions in a soluble model of identical particle scattering

Solutions to the Schrödinger equation and the inhomogeneous equation for the case of two identical particles interacting with a center of force are studied. Eigenstate expansions for solving each equation are explicitly introduced and their properties discussed. The case when the interparticle interaction v₁₂ is zero is then examined; this is a completely soluble problem. The eigenstate expansion solutions for the Schrödinger and inhomogeneous equations are used to explore the means by which the correct solution is obtained. Finally, approximate solutions, obtained by truncating the eigenfunction expansions, are introduced. It is seen that both methods lead to the correct amplitude when τ₁₂ = 0, even though the approximate solution to the inhomogeneous equation does not lead, in the end, to an antisymmetric solution.     

A new method in the theory of many-body scattering

New coupled equations for the transition (T) and reactance (K) operators for N-channel, many-body, rearrangement scattering are derived. The key to the new method is the channel coupling array W, which links the various rearrangement channels together. By specializing W to the class of channel permuting arrays, it is shown that the (N−1)st iterate of the kernel of the coupled equations is connected.     

Rotational-translattonal energy exchange in molecule-surface collisions

Reported are calculations of rotational energy transfer in collisions of Cl₂ molecules with a rigid, smooth surface, using both sudden approximations and quasi-classical trajectories. High probabilities are found for very large Δj transitions compared to corresponding gas-phase collisions: thus molecular collisions with walls can be extremely efficient in producing rotational transitions.