146 research outputs found
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
A note on the switching adiabatic theorem
We derive a nearly optimal upper bound on the running time in the adiabatic
theorem for a switching family of Hamiltonians. We assume the switching
Hamiltonian is in the Gevrey class as a function of time, and we
show that the error in adiabatic approximation remains small for running times
of order . Here denotes the minimal spectral
gap between the eigenvalue(s) of interest and the rest of the spectrum of the
instantaneous Hamiltonian.Comment: 20 pages, no figures, to appear in JM
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Molecular Quantum Dynamics
We provide a brief introduction to some basic ideas of Molecular Quantum Dynamics. We discuss the scope, strengths and main applications of this field of science. Finally, we also mention open problems of current interest in this exciting subject
Exponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension
We study the time behavior of wave functions involved in tunneling through a
smooth potential barrier in one dimension in the semiclassical limit. We
determine the leading order component of the wave function that tunnels. It is
exponentially small in . For a wide variety of incoming wave packets,
the leading order tunneling component is Gaussian for sufficiently small
. We prove this for both the large time asymptotics and for moderately
large values of the time variable
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