Matrix mechanics as a practical tool in quantum theory: The anharmonic oscillator
- 15 October 1975
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review D
- Vol. 12 (8), 2311-2324
- https://doi.org/10.1103/physrevd.12.2311
Abstract
For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number that the matrix elements and should be rapidly decreasing functions of . Matrix elements of polynomials in and 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 and for and of the commutator , 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.
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