For Schrödinger, the atom was an oscillating system — like the string of a musical instrument — for which there exist a number of modes of oscillation (fundamentals and overtones) which are interpreted as the atom’s energy states.

Armin Hermann. Werner Heisenberg 1901-1976.

This set of slides is from one of my courses in Computational Chemistry. It covers the one-dimensional solution of the Schrödinger equation using Numerov’s method. The material for this article was mainly taken from the excellent Levin textbook in the reference.
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Reference

Ira N. Levine. Quantum Chemistry. 6th edition. 2009. Pearson International edition