Variational principles and Heisenberg matrix mechanics

If in Heisenberg's equations of motion for a problem in quantum mechanics (or quantum field theory) one studies matrix elements in the energy representation and by use of completeness conditions expresses the equations solely in terms of matrix elements of the canonical variables, and if one does likewise with the associated kinematical constraints (commutation relations), one arrives at a formulation - largely unexplored hitherto - which can be exploited for both practical and theoretical development. In this contribution, the above theme is developed within the framework of one-dimensional problems. It is shown how this formulation, both dynamics and kinematics, can be derived from a new variational principle, indeed from an entire class of such principles. A powerful method of diagonalizing the Hamiltonian by means of computations utilizing these equations is described. The variational method is shown to be particularly useful for the study of the regime of large quantum numbers. The usual WKB approximation is seen to be contained as well as a basic for the study of systematic corrections to it. Further applications in progress are mentioned.