This equation is a subcase of Equation 31. "Schrödinger equation for a one dimensional particle" with $V(x)=x_{2}$.

We get the time-independent Schrodinger equation by substituting this $V$ into Equation 32. "time-independent Schrodinger equation for a one dimensional particle":

$[−2mℏ ∂x∂_{2} +x_{2}]ψ=Eψ(x)$

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 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.

- Solutions of the Schrodinger equation | 34, 2k, 35
- Schrödinger equation | 555, 5k, 60
- Non-relativistic quantum mechanics | 26, 5k, 66
- Quantum mechanics | 169, 20k, 270
- Particle physics | 137, 32k, 512
- Physics | 276, 46k, 825
- Natural science | 0, 61k, 2k
- Science | 0, 66k, 2k
- Ciro Santilli's Homepage | 262, 218k, 4k

- Complete basis | 150
- Hermite polynomials | 43, 55, 1
- Mathematical formulation of quantum mechanics | 490, 1k, 6
- Lecture 4 | 86
- The wave equation can be seen as infinitely many infinitesimal coupled oscillators | 67