  We describe the "Feynman diagram" approach to nonrelativistic quantum
mechanics on R^n, with magnetic and potential terms. In particular, for each
classical path \gamma connecting points q_0 and q_1 in time t, we define a
formal power series V_\gamma(t,q_0,q_1) in \hbar, given combinatorially by a
sum of diagrams that each represent finite-dimensional convergent integrals. We
prove that exp(V_\gamma) satisfies Schr\"odinger's equation, and explain in
what sense the t\to 0 limit approaches the \delta distribution. As such, our
construction gives explicitly the full \hbar\to 0 asymptotics of the
fundamental solution to Schr\"odinger's equation in terms of solutions to the
corresponding classical system. These results justify the heuristic expansion
of Feynman's path integral in diagrams.
