Schrodinger equation reduction using substitution

1. Apr 16, 2009

phagist_

SOLVED: Schrodinger equation reduction using substitution

1. The problem statement, all variables and given/known data
Given

$$\frac{d^2 \psi}{dx^2} - Ax\psi + B\psi = 0$$

make a substitution using
$$w= A^{1/3} (x - \frac{B}{A})$$

to get
$$\frac{d^2 \psi}{dw^2} - w\psi = 0$$

2. Relevant equations

3. The attempt at a solution
I use

$$\frac{d\psi}{dx} = \frac{d\psi}{dw} \frac{dw}{dx}$$

then
$$\frac{d^2\psi}{dx^2} = \frac{d}{dx}[ \frac{d\psi}{dw} \frac{dw}{dx}]$$

then

$$\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx} + \frac{d^2w}{dx^2} \frac{d\psi}{dw}$$

but the $$\frac{d^2w}{dx^2}$$ equals zero, since x is linear in w.

which implies

$$\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx}$$

and $$\frac{dw}{dx} = A^{1/3}$$

but I'm not sure how to evaluate the

$$\frac{d\psi}{dwdx}$$ term (I'm not sure If it should even be there.. did I use the chain rule correctly?)

Then I'll sub in $$\frac{d\psi}{dwdx} A^{1/3}$$ for $$\frac{d^2 \psi}{dx^2}$$ and hopefully it all works out.

Any help would be greatly appreciated.

Edit: SOLVED

Last edited: Apr 16, 2009