 Quantum harmonic oscillator

This equation is a subcase of Equation "Schrödinger equation for a one dimensional particle" with $V(x)=x^2$.

We get the time-independent Schrödinger equation by substituting this $V$ into Equation "time-independent Schrödinger equation for a one dimensional particle":

Now, there are two ways to go about this.

The first is the stupid "here's a guess" + "hey this family of solutions forms a complete basis"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.

The second is the much celebrated ladder operator method.

The quantum harmonic oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle subject to a restoring force that is proportional to its displacement from an equilibrium position. This model is essential for understanding various physical systems, such as vibrations in molecules, phonons in solid-state physics, and quantum field theory. Here's a detailed overview of the quantum harmonic oscillator: ### 1.