Foundations of Physics, Vol. 31, No. 4, 2001 Beyond Quantum Mechanics: Insights from the Work of Martin Gutzwiller Daniel Kleppner 1 and John B. Delos 2 Received December 13, 2000 A complete quantum solution provides all possible knowledge of a system, whereas semiclassical theory provides at best approximate solutions in a limited region. Nevertheless, semiclassical methods based on the work of Martin Gutzwiller can provide stunning physical insights in regimes where quantum solutions are opaque. Furthermore, they can provide a unique bridge between the quantum and classical worlds. We illustrate these ideas with an account of a theoretical and experimental attack on the paradigm problem of the hydrogen atom in strong magnetic and electric fields. 1. BACKGROUND Martin Gutzwiller's seminal paper on periodic orbit theory was published in 1971. (1) At about that time atomic physicists started to develop experi- mental methods for studying Rydberg atoms. Because Rydberg atoms by definition involve states with large quantum numbers, there might well have been an immediate convergence with semiclassical theory. However, the main thrust of the early work was based on traditional quantum theory, and it was not until a decade later that experiments on Rydberg atoms in strong fields started to converge with theoretical descriptions stemming from Gutzwiller's ideas. The authorsone theoretical (JBD), the other experimental (DK)who participated in some of these developments, describe how their work converged, and how this helped to lead to radi- cally new points of view. 593 0015-9018010400-059319.500  2001 Plenum Publishing Corporation 1 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. 2 Department of Physics, College of William and Mary, Williamsburg, Virginia 23187.