I was revising the harmonic oscillator for my intro to quantum course and realised I'd sort of accepted a change of variable result without actually being able to get to it. It says:

The stationary state Schrodinger equation of energy $E$ is


The first thing to do is to redefine variables so as to remove the various physical constants:

$$\epsilon=\frac{2E}{\hbar\omega}, \xi=\sqrt{\frac{m\omega}{\hbar}}x$$

so that (5.2) becomes

$$-\frac{d^{2}\chi}{d\xi^{2}}+\xi^{2}\chi=\epsilon\chi$$ where $$\psi(x)=\chi(\xi)=\chi(\sqrt{\frac{m\omega}{\hbar}}x).$$

So, I've tried working with the algebra but can't seem to get to this. I'm probably missing something really obvious, but it's getting quite frustrating! Can anyone help?

  • $\begingroup$ Let $x=a\xi$ and $E=b\epsilon$, do that substitution & reduction and figure out what a and b need to be. It will be the variables as given. $\endgroup$ – Kyle Kanos Apr 16 '14 at 12:02

You should show your work, but my guess is that you have to notice the change of variables:


You need to do this a second time (using the derivate of a product.

See if you can continue from there.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.