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I am following a quantum mechanics text book which uses a simple looking substitution in a derivative.

The substitution is $\xi=\gamma x\tag1$

It then says that $\frac{d\psi}{dx}=\frac{d\psi}{d\xi}\frac{d\xi}{dx}=\gamma\frac{d\psi}{d\xi}\tag2$

So far so good. Now comes the part I don't follow. It says: $\frac{d^2\psi}{dx^2}=\gamma^2\frac{d^2\psi}{d\xi^2}\tag3$

I don't know how they get this. I tried $\frac{d^2\psi}{dx^2}=\frac{d^2\psi}{d\xi^2}\frac{d^2\xi}{dx^2}\tag4$ but I get $\frac{d^2\xi}{dx^2}=0$ by using the substitution in equation (1). I'd appreciate it if someone could explain this to me.


Another thing I tried is (suggested by comment below): $\frac{d}{dx}\left(\frac{d\psi}{dx}\right)=\frac{d}{dx}\left(\frac{d\psi}{d\xi}\frac{d\xi}{dx}\right)=\frac{d^2\psi}{dxd\xi}\frac{d\xi}{dx}+\frac{d\psi}{d\xi}\frac{d^2\xi}{dx^2}\tag5$

Now since $\frac{d^2\xi}{dx^2}=0$ equation (5) reduces to:$\frac{d}{dx}\left(\frac{d\psi}{dx}\right)=\frac{d}{dx}\left(\frac{d\psi}{d\xi}\frac{d\xi}{dx}\right)=\gamma\frac{d^2\psi}{dxd\xi}$

Once again, I'm stuck..

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    (4) is wrong. You need to use (2) again, but with $d\psi/dx$ in $\psi$'s place.2012-11-28

1 Answers 1

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Ok, I got it.

$\gamma\frac{d^2\psi}{dxd\xi}=\gamma\frac{d^2\psi}{d\xi dx}=\gamma \frac{d}{d\xi}\frac{d\psi}{dx}$

$\gamma \frac{d}{d\xi}\frac{d\psi}{dx}=\gamma \frac{d}{d\xi}\left(\gamma\frac{d\psi}{d\xi}\right)=\gamma^2\frac{d^2\psi}{d\xi^2}$