I was just getting into quantum mechanics. But I'm having a bit of trouble following Griffiths for the analytic method. It goes like so:
The Schrodinger equation is:
$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = E \psi $$
Griffiths expresses $\xi$ as:
$$\xi = \sqrt{\frac{m \omega}{\hbar}}x$$
and $K$ as:
$$K=\frac{2E}{\hbar \omega}$$
Ultimately, leading to the equation:
$$\frac{d^2\psi}{d\xi^2}=(\xi^2-K) \psi$$
I've tried to rearrange on my own, but:
- I do not understand why $\xi$ equals the square root, except for $x$. $\xi$ is eventually squared.