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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

$$-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}\psi=E\psi\tag{5.2}$$

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?

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

$$\frac{d\chi}{dx}=\frac{d\chi}{d\xi}\frac{d\xi}{dx}$$

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

See if you can continue from there.

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