I am trying to derive the eigenfunctions from $$-\frac{1}{2}\partial_x^2\phi_k(x)+\frac{1}{2}x^2\phi_k(x)=\lambda_k\phi_k(x).$$
I got stuck here. I don't know which method I need to use.
Actually, I have $$\partial_x^2\phi_k(x)=(x^2-2\lambda_k)\phi_k(x).$$ Any hints are welcome!
This is the quantum harmonic oscillator equation. If you start with the anzatz that $$ \phi_k(x) = H_k(x) e^{-ax^2} $$ where $H_k$(x) is required to be a polynomial, you will find that $\{\lambda_m\}$ are all in a linear progression, and the $H_k$ are the Hermite polynomials.