  A simple method for transmitting quantum states within a quantum computer is
via a quantum spin chain---that is, a path on $n$ vertices. Unweighted paths
are of limited use, and so a natural generalization is to consider weighted
paths; this has been further generalized to allow for loops (\emph{potentials}
in the physics literature). We study the particularly important situation of
perfect state transfer with respect to the corresponding adjacency matrix or
Laplacian through the use of orthogonal polynomials. Low-dimensional examples
are given in detail. Our main result is that PST with respect to the Laplacian
matrix cannot occur for weighted paths on $n\geq 3$ vertices nor can it occur
for certain symmetric weighted trees. The methods used lead us to a conjecture
directly linking the rationality of the weights of weighted paths on $n>3$
vertices, with or without loops, with the capacity for PST between the end
vertices with respect to the adjacency matrix.
