Superadiabatic transitions in quantum molecular dynamics

We study the dynamics of a molecule’s nuclear wave function near an avoided crossing of two electronic energy levels for one nuclear degree of freedom. We derive the general form of the Schrödinger equation in the nth superadiabatic representation for all n є N. Using these results, we obtain closed formulas for the time development of the component of the wave function in an initially unoccupied energy subspace when a wave packet travels through the transition region. In the optimal superadiabatic representation, which we define, this component builds up monotonically. Finally, we give an explicit formula for the transition wave function away from the avoided crossing, which is in excellent agreement with high-precision numerical calculations

Determination of Non-Adiabatic Scattering Wave Functions in a Born-Oppenheimer Model

We study non--adiabatic transitions in scattering theory for the time dependent molecular Schroedinger equation in the Born--Oppenheimer limit. We assume the electron Hamiltonian has finitely many levels and consider the propagation of coherent states with high enough total energy. When two of the electronic levels are isolated from the rest of the electron Hamiltonian's spectrum and display an avoided crossing, we compute the component of the nuclear wave function associated with the non--adiabatic transition that is generated by propagation through the avoided crossing. This component is shown to be exponentially small in the square of the Born--Oppenheimer parameter, due to the Landau-Zener mechanism. It propagates asymptotically as a free Gaussian in the nuclear variables, and its momentum is shifted. The total transition probability for this transition and the momentum shift are both larger than what one would expect from a naive approximation and energy conservation

Indirect Evidence for L\'evy Walks in Squeeze Film Damping

Molecular flow gas damping of mechanical motion in confined geometries, and its associated noise, is important in a variety of fields, including precision measurement, gravitational wave detection, and MEMS devices. We used two torsion balance instruments to measure the strength and distance-dependence of `squeeze film' damping. Measured quality factors derived from free decay of oscillation are consistent with gas particle superdiffusion in L\'evy walks and inconsistent with those expected from traditional Gaussian random walk particle motion. The distance-dependence of squeeze film damping observed in our experiments is in agreement with a parameter-free Monte Carlo simulation. The squeeze film damping of the motion of a plate suspended a distance d away from a parallel surface scales with a fractional power between 1/d and 1/d^2.Comment: 5 pages 5 figures accepted for PRD; typo in equation 3 and figure 1 fixe

A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates

We present the construction of an exponentially accurate time-dependent Born-Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to $\epsilon^{-4}$, where $\epsilon$ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time-dependent Schr\"odinger equation that agree with exact normalized solutions up to errors whose norms are bounded by \ds C \exp(-\gamma/\epsilon^2), for some C and $\gamma>0$

Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering and More General States

We prove six theorems concerning exponentially accurate semiclassical quantum mechanics. Two of these theorems are known results, but have new proofs. Under appropriate hypotheses, they conclude that the exact and approximate dynamics of an initially localized wave packet agree up to exponentially small errors in $\hbar$ for finite times and for Ehrenfest times. Two other theorems state that for such times the wave packets are localized near a classical orbit up to exponentially small errors. The fifth theorem deals with infinite times and states an exponentially accurate scattering result. The sixth theorem provides extensions of the other five by allowing more general initial conditions

Semiclassical Dynamics with Exponentially Small Error Estimates

We construct approximate solutions to the time--dependent Schr\"odinger equation $i \hbar (\partial \psi)/(\partial t) = - (\hbar^2)/2 \Delta \psi + V \psi$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and growth hypotheses and $|t|\le T$, these solutions agree with exact solutions up to errors whose norms are bounded by $C \exp{-\gamma/\hbar}$, for some $C$ and $\gamma>0$. Under more restrictive hypotheses, we prove that for sufficiently small $T', |t|\le T' |\log(\hbar)|$ implies the norms of the errors are bounded by $C' \exp{-\gamma'/\hbar^{\sigma}}$, for some $C', \gamma'>0$, and $\sigma>0$