Matrix mechanics as a practical tool in quantum theory: the anharmonic oscillator

For an anharmonic oscillator, it follows from the nodal structure of the exact wave functions as functions of the quantum number n that the matrix elements [n parallel x parallel n'] and [n parallel p parallel n'] should be rapidly decreasing functions of parallel n-n' parallel. Matrix elements of polynomials in x and p should therefore be well approximated by a finite number of terms in their sum rule decomposition. From the matrix elements of the equations of motion for x and for p and of the commutator [x,p], one thereby obtains closed sets of nonlinear algebraic equations to characterize subspaces of the Hilbert space of exact eigenfunctions. The approximations are also derived from a novel variational principle, and numerous variant approximation schemes are suggested. Essentially exact numerical results are obtained and compared with previous work. The broad applicability of the techniques is emphasized. (U.S.)