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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>:
$$
\left[- \frac{\hbar}{2m} \pdv{^2}{x} + x^2 \right]\psi = E \psi(x)
$$

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 <solving partial differential equations with the Fourier series>{full}, except that now the complete basis are the <Hermite functions>.

The second is the much celebrated <ladder operator> method.


Quantum harmonic oscillator