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}
Introduction to numerical analysis
x, 511 p.; 23 cm
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