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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.

nicoguaro
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ben tenyson
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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.

Mathias Klahn
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  • 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 03:13
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    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 06:38