A harmonic oscillator is an event generator that only produces signals an integer multiples of a fundamental frequency.
This is in contrast to an event generator that produces non integer multiples of a fundamental frequency or produces random events.

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The Quantum Harmonic Oscillator: Dirac’s Approach
How this fundamental model in physics becomes quantised and how we can apply Dirac’s ‘ladder method’ to find its energy, eigenstates, and eigenvalues.

Web site https://www.cantorsparadise.com/the-quantum-harmonic-oscillator-diracs-approach-f847569b9df0

Table of Contents
Introducing the Harmonic Oscillator
Dirac’s Ladder Method
a. Annihilation and Creation Operators
b. Number Operator
c. Energy Eigenvalues
d. Back to the Annihilation and Creation Operators…
e. Commuting with the Hamiltonian
f. Commuting with the Number Operator
g. Creation and Annihilation Operator Formal Definitions
Wave Functions of the Quantum Harmonic Oscillator
a. Ground State Wave Function
b. Excited State Wave Functions