3,489 research outputs found
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 , where 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
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
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 for small values of . If satisfies appropriate analyticity and
growth hypotheses and , these solutions agree with exact solutions up
to errors whose norms are bounded by , for some and
. Under more restrictive hypotheses, we prove that for sufficiently
small implies the norms of the errors are bounded
by , for some , and
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