110 research outputs found
Feynman's Path Integrals and Bohm's Particle Paths
Both Bohmian mechanics, a version of quantum mechanics with trajectories, and
Feynman's path integral formalism have something to do with particle paths in
space and time. The question thus arises how the two ideas relate to each
other. In short, the answer is, path integrals provide a re-formulation of
Schroedinger's equation, which is half of the defining equations of Bohmian
mechanics. I try to give a clear and concise description of the various aspects
of the situation.Comment: 4 pages LaTeX, no figures; v2 shortened a bi
Time of Arrival from Bohmian Flow
We develop a new conception for the quantum mechanical arrival time
distribution from the perspective of Bohmian mechanics. A detection probability
for detectors sensitive to quite arbitrary spacetime domains is formulated.
Basic positivity and monotonicity properties are established. We show that our
detection probability improves and generalises earlier proposals by Leavens and
McKinnon. The difference between the two notions is illustrated through
application to a free wave packet.Comment: 18 pages, 8 figures, to appear in Journ. Phys. A; representation of
ref. 5 improved (thanks to Rick Leavens
Hypersurface Bohm-Dirac models
We define a class of Lorentz invariant Bohmian quantum models for N entangled
but noninteracting Dirac particles. Lorentz invariance is achieved for these
models through the incorporation of an additional dynamical space-time
structure provided by a foliation of space-time. These models can be regarded
as the extension of Bohm's model for N Dirac particles, corresponding to the
foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to
more general foliations. As with Bohm's model, there exists for these models an
equivariant measure on the leaves of the foliation. This makes possible a
simple statistical analysis of position correlations analogous to the
equilibrium analysis for (the nonrelativistic) Bohmian mechanics.Comment: 17 pages, 3 figures, RevTex. Completely revised versio
Times of arrival: Bohm beats Kijowski
We prove that the Bohmian arrival time of the 1D Schroedinger evolution
violates the quadratic form structure on which Kijowski's axiomatic treatment
of arrival times is based. Within Kijowski's framework, for a free right moving
wave packet, the various notions of arrival time (at a fixed point x on the
real line) all yield the same average arrival time. We derive an inequality
relating the average Bohmian arrival time to the one of Kijowksi. We prove that
the average Bohmian arrival time is less than Kijowski's one if and only if the
wave packet leads to position probability backflow through x. Otherwise the two
average arrival times coincide.Comment: 9 page
Locality and Causality in Hidden Variables Models of Quantum Theory
Motivated by Popescu's example of hidden nonlocality, we elaborate on the
conjecture that quantum states that are intuitively nonlocal, i.e., entangled,
do not admit a local causal hidden variables model. We exhibit quantum states
which either (i) are nontrivial counterexamples to this conjecture or (ii)
possess a new kind of more deeply hidden irreducible nonlocality. Moreover, we
propose a nonlocality complexity classification scheme suggested by the latter
possibility. Furthermore, we show that Werner's (and similar) hidden variables
models can be extended to an important class of generalized observables.
Finally a result of Fine on the equivalence of stochastic and deterministic
hidden variables is generalized to causal models.Comment: revised version, 21 pages, submitted to Physical Review
A microscopic derivation of the quantum mechanical formal scattering cross section
We prove that the empirical distribution of crossings of a "detector''
surface by scattered particles converges in appropriate limits to the
scattering cross section computed by stationary scattering theory. Our result,
which is based on Bohmian mechanics and the flux-across-surfaces theorem, is
the first derivation of the cross section starting from first microscopic
principles.Comment: 28 pages, v2: Typos corrected, layout improved, v3: Typos corrected.
Accepted for publication in Comm. Math. Phy
On the exit statistics theorem of many particle quantum scattering
We review the foundations of the scattering formalism for one particle
potential scattering and discuss the generalization to the simplest case of
many non interacting particles. We point out that the "straight path motion" of
the particles, which is achieved in the scattering regime, is at the heart of
the crossing statistics of surfaces, which should be thought of as detector
surfaces. We sketch a proof of the relevant version of the many particle flux
across surfaces theorem and discuss what needs to be proven for the foundations
of scattering theory in this context.Comment: 15 pages, 4 figures; to appear in the proceedings of the conference
"Multiscale methods in Quantum Mechanics", Accademia dei Lincei, Rome,
December 16-20, 200
Quantile Motion and Tunneling
The concepts of quantile position, trajectory, and velocity are defined. For
a tunneling quantum mechanical wave packet, it is proved that its quantile
position always stays behind that of a free wave packet with the same initial
parameters. In quantum mechanics the quantile trajectories are mathematically
identical to Bohm's trajectories. A generalization to three dimensions is
given.Comment: 13 pages, LaTeX, elsart, 3 ps figures, submitted to Phys. Lett.
Bohmian transmission and reflection dwell times without trajectory sampling
Within the framework of Bohmian mechanics dwell times find a straightforward
formulation. The computation of associated probabilities and distributions
however needs the explicit knowledge of a relevant sample of trajectories and
therefore implies formidable numerical effort. Here a trajectory free
formulation for the average transmission and reflection dwell times within
static spatial intervals [a,b] is given for one-dimensional scattering
problems. This formulation reduces the computation time to less than 5% of the
computation time by means of trajectory sampling.Comment: 14 pages, 7 figures; v2: published version, significantly revised and
shortened (former sections 2 and 3 omitted, appendix A added, simplified
mathematics
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