# Can anyone explain the harmonic oscillator (in context to quantum mechanics) 2.3 (Griffiths) using Taylor series?

At the end he concludes $V(x) = V''(x_0)(x-x_0)^2$. How does he get to know that the rest are $0$? How does he conclude $V''(x_0) = k$. Please try to explain in easy ways and tough vocabulary. I don't even known what a Hilbert space is.

He chooses $V(x_0)=0$, so $x_0$ as our reference point. And note that $V^\prime (x_0)=0$ since the potential is at a minimum at $x_0$. This gives you to second order (ignoring higher order terms):
$V(x) \approx \frac{1}{2} k (x-x_0)^2$.
You can just call $k=V''(x_0)$ since it is a constant. After using the differential equation $F=-kx=m\ddot{x}$ we see that $\omega = \sqrt{\frac{k}{m}}$.