76 research outputs found
Elementary Quantum Mechanics in a Space-time Lattice
Studies of quantum fields and gravity suggest the existence of a minimal
length, such as Planck length \cite{Floratos,Kempf}. It is natural to ask how
the existence of a minimal length may modify the results in elementary quantum
mechanics (QM) problems familiar to us \cite{Gasiorowicz}. In this paper we
address a simple problem from elementary non-relativistic quantum mechanics,
called "particle in a box", where the usual continuum (1+1)-space-time is
supplanted by a space-time lattice. Our lattice consists of a grid of
rectangles, where , the lattice
parameter, is a fundamental length (say Planck length) and, we take to
be equal to . The corresponding Schrodinger equation becomes a
difference equation, the solution of which yields the -eigenfunctions and
-eigenvalues of the energy operator as a function of . The
-eigenfunctions form an orthonormal set and both -eigenfunctions and
-eigenvalues reduce to continuum solutions as
The corrections to eigenvalues because of the assumed lattice is shown to be
We then compute the uncertainties in position and momentum,
for the box problem and study the consequent modification
of Heisenberg uncertainty relation due to the assumption of space-time lattice,
in contrast to modifications suggested by other investigations such as
\cite{Floratos}
Consistent patterns of common species across tropical tree communities
Trees structure the Earth’s most biodiverse ecosystem, tropical forests. The vast number of tree species presents a formidable challenge to understanding these forests, including their response to environmental change, as very little is known about most tropical tree species. A focus on the common species may circumvent this challenge. Here we investigate abundance patterns of common tree species using inventory data on 1,003,805 trees with trunk diameters of at least 10 cm across 1,568 locations1,2,3,4,5,6 in closed-canopy, structurally intact old-growth tropical forests in Africa, Amazonia and Southeast Asia. We estimate that 2.2%, 2.2% and 2.3% of species comprise 50% of the tropical trees in these regions, respectively. Extrapolating across all closed-canopy tropical forests, we estimate that just 1,053 species comprise half of Earth’s 800 billion tropical trees with trunk diameters of at least 10 cm. Despite differing biogeographic, climatic and anthropogenic histories7, we find notably consistent patterns of common species and species abundance distributions across the continents. This suggests that fundamental mechanisms of tree community assembly may apply to all tropical forests. Resampling analyses show that the most common species are likely to belong to a manageable list of known species, enabling targeted efforts to understand their ecology. Although they do not detract from the importance of rare species, our results open new opportunities to understand the world’s most diverse forests, including modelling their response to environmental change, by focusing on the common species that constitute the majority of their trees
Introduction to numerical analysis
x, 511 p.; 23 cm
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