DOI: 10.1119/5.0332743 ISSN: 0002-9505
A shortcut to the eigenvalues of the harmonic oscillator and the hydrogen atom
J. M. Zhang, R. K. LinTextbook solutions for the harmonic oscillator and the hydrogen atom using the Sommerfeld polynomial method often appear tedious and formidable to students. However, the differential operators acting on the reduced wave functions possess a convenient algebraic property: they preserve the subspace of polynomials of degree at most k. This observation makes the derivation of energy eigenvalues transparent and direct, without the need for the full power series recurrence relations.