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

Abstract

For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number $n$ that the matrix elements $〈n\left|x\right|\mathrm{}{n}^{\prime }〉$ and $〈n\left|p\right|\mathrm{}{n}^{\prime }〉$ should be rapidly decreasing functions of $|n-n^{\prime }|$. 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 $\mathrm{}|x,p|\mathrm{}$, 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.