Finding a complete basis such that each vector solves a given <differential equation> is the basic method of solving <partial differential equation> through <separation of variables>.

The first example of this you must see is <solving partial differential equations with the Fourier series>.

Notable examples:
* <Fourier series> for the <heat equation> as shown at <fourier basis is complete for l2> and <solving partial differential equations with the Fourier series>
* <Hermite functions> for the <quantum harmonic oscillator>
* <Legendre polynomials> for <Laplace's equation> in <spherical coordinates>
* <Bessel function> for the <2D wave equation on a circular domain> in <polar coordinates>


Complete basis

{wiki}

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.


Ciro Santilli @cirosantilli 34

 Incoming links: Hermite functions