14 research outputs found
Implications of Lorentz covariance for the guidance equation in two-slit quantum interference
It is known that Lorentz covariance fixes uniquely the current and the
associated guidance law in the trajectory interpretation of quantum mechanics
for spin particles. In the non-relativistic domain this implies a guidance law
for the electron which differs by an additional spin-dependent term from that
originally proposed by de Broglie and Bohm. In this paper we explore some of
the implications of the modified guidance law. We bring out a property of
mutual dependence in the particle coordinates that arises in product states,
and show that the quantum potential has scalar and vector components which
implies the particle is subject to a Lorentz-like force. The conditions for the
classical limit and the limit of negligible spin are given, and the empirical
sufficiency of the model is demonstrated. We then present a series of
calculations of the trajectories based on two-dimensional Gaussian wave packets
which illustrate how the additional spin-dependent term plays a significant
role in structuring both the individual trajectories and the ensemble. The
single packet corresponds to quantum inertial motion. The distinct features
encountered when the wavefunction is a product or a superposition are explored,
and the trajectories that model the two-slit experiment are given. The latter
paths exhibit several new characteristics compared with the original de
Broglie-Bohm ones, such as crossing of the axis of symmetry.Comment: 27 pages including 6 pages of figure
Spin dependent observable effect for free particles using the arrival time distribution
The mean arrival time of free particles is computed using the quantum
probability current. This is uniquely determined in the non-relativistic limit
of Dirac equation, although the Schroedinger probability current has an
inherent non-uniqueness. Since the Dirac probability current involves a
spin-dependent term, an arrival time distribution based on the probability
current shows an observable spin-dependent effect, even for free particles.
This arises essentially from relativistic quantum dynamics, but persists even
in the non-relativistic regime.Comment: 5 Latex pages, 2.eps figures; discussions sharpened and references
added; accepted for publication in Physical Review
Almost uniform sampling via quantum walks
Many classical randomized algorithms (e.g., approximation algorithms for
#P-complete problems) utilize the following random walk algorithm for {\em
almost uniform sampling} from a state space of cardinality : run a
symmetric ergodic Markov chain on for long enough to obtain a random
state from within total variation distance of the uniform
distribution over . The running time of this algorithm, the so-called {\em
mixing time} of , is , where
is the spectral gap of .
We present a natural quantum version of this algorithm based on repeated
measurements of the {\em quantum walk} . We show that it
samples almost uniformly from with logarithmic dependence on
just as the classical walk does; previously, no such
quantum walk algorithm was known. We then outline a framework for analyzing its
running time and formulate two plausible conjectures which together would imply
that it runs in time when is
the standard transition matrix of a constant-degree graph. We prove each
conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac
Bell-Type Quantum Field Theories
In [Phys. Rep. 137, 49 (1986)] John S. Bell proposed how to associate
particle trajectories with a lattice quantum field theory, yielding what can be
regarded as a |Psi|^2-distributed Markov process on the appropriate
configuration space. A similar process can be defined in the continuum, for
more or less any regularized quantum field theory; such processes we call
Bell-type quantum field theories. We describe methods for explicitly
constructing these processes. These concern, in addition to the definition of
the Markov processes, the efficient calculation of jump rates, how to obtain
the process from the processes corresponding to the free and interaction
Hamiltonian alone, and how to obtain the free process from the free Hamiltonian
or, alternatively, from the one-particle process by a construction analogous to
"second quantization." As an example, we consider the process for a second
quantized Dirac field in an external electromagnetic field.Comment: 53 pages LaTeX, no figure