Solving the quantum harmonic oscillator with ladder operators

The quantum harmonic oscillator is one of the most important and ubiquitous model systems in quantum mechanics, which features equally spaced energy levels. In this video, we shall solve this quantum harmonic oscillator using a very elegant approach based on the so-called ladder operators. Starting from the classical harmonic oscillator, we discuss the mapping to obtain the Hamiltonian for the quantum harmonic oscillator. We then introduce the ladder operators and rewrite the Hamiltonian in terms of them. We discuss the utility of the ladder operators and their action on the eigenstates, and derive their eigenfunctions in terms of the ground state wavefunction. We derived the explicit expressions for the wavefunctions in this system, in terms of the Hermite polynomials.

Technical Content: Tony Low
Video Production: Katie Low, Tony Low

Erratum.
In 9:42, 1/(sqrt(n)-1) should be 1/sqrt(n-1). Thanks to @JohnVKaravitis.