Recently, I came across a question which asks to solve the Schrödinger equation for a harmonic oscillator on $ [a, b] $ :
$-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = E \psi$
I can use the 'shooting method' to find $ E $ for which the BVP $ \psi(a) = 0 $ and $\psi(b) = 0 $ has a solution. This can be done by using the Numerov algorithm to find $ \psi $ at $ x = b $ starting from $ \psi(a) = 0 $ and $ \psi(a+\delta x) = c $ where $ c $ is an arbitrary constant, and then varying the parameter $E$ by the secant method until the BVP is solved. But, in addition, what if I also want to ensure that the final $E$ value will be the ground state energy level of the oscillator ?
One method would be to pick up an initial guess of $ E $ very close to the ground energy, so that the shooting method iterations converge to the ground energy level, but that would require a priori guessing of the same.
Another problem is, even if the ground state is found, how do I know that the solution is really the ground state solution ? Note that the ground state is defined to be the minimum allowed energy state of a particle in a given potential field.
Thanks in advance !