# Numerov method for Schrodinger equation

While learning about numerical methods for solving the Schrödinger equation I came across Numerov's method.

I want to get the solution for the harmonic oscillator by alreading giving the eigenvalues. The algorithm requires that I know two initial values of $$\psi$$ while I only know that $$\psi$$ must vanish at thr boundary. How are those initial values found? Also it would be great if someone could refer me to some resources for this method.

Denoting your grid points by $$x_1, x_2, ... x_N$$ it is correct that you need the values of $$\Psi(x_1)$$ and $$\Psi(x_2)$$ to execute Numerov's algorithm. As you write in your post, $$\Psi(x_1)$$ is given by the boundary condition. For $$\Psi(x_2)$$ you can take any non zero value you like, as the correct value can always be computed by normalizing $$\Psi$$ once you have obtained it at all grid points.
• That is not entirely correct. The sign of $\Psi(x_2)$ determines the parity of the wavefunction and cannot be chosen arbitrarily. – Henri Menke Jul 29 '19 at 3:13
• I think it is indeed correct. Multiplying $\Psi(x)$ by a constant does not change its parity and you can therefore just multiply it by $-1$ if you like. – Mathias Klahn Jul 29 '19 at 6:38