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phagist_
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SOLVED: Schrodinger equation reduction using substitution
Given
[tex] \frac{d^2 \psi}{dx^2} - Ax\psi + B\psi = 0[/tex]
make a substitution using
[tex]w= A^{1/3} (x - \frac{B}{A})[/tex]
to get
[tex] \frac{d^2 \psi}{dw^2} - w\psi = 0[/tex]
I use
[tex] \frac{d\psi}{dx} = \frac{d\psi}{dw} \frac{dw}{dx}[/tex]
then
[tex]\frac{d^2\psi}{dx^2} = \frac{d}{dx}[ \frac{d\psi}{dw} \frac{dw}{dx}][/tex]
then
[tex]\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx} + \frac{d^2w}{dx^2} \frac{d\psi}{dw}[/tex]
but the [tex]\frac{d^2w}{dx^2}[/tex] equals zero, since x is linear in w.
which implies
[tex]\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx}[/tex]
and [tex]\frac{dw}{dx} = A^{1/3}[/tex]
but I'm not sure how to evaluate the
[tex]\frac{d\psi}{dwdx}[/tex] term (I'm not sure If it should even be there.. did I use the chain rule correctly?)
Then I'll sub in [tex]\frac{d\psi}{dwdx} A^{1/3}[/tex] for [tex] \frac{d^2 \psi}{dx^2}[/tex] and hopefully it all works out.
Any help would be greatly appreciated.
Edit: SOLVED
Homework Statement
Given
[tex] \frac{d^2 \psi}{dx^2} - Ax\psi + B\psi = 0[/tex]
make a substitution using
[tex]w= A^{1/3} (x - \frac{B}{A})[/tex]
to get
[tex] \frac{d^2 \psi}{dw^2} - w\psi = 0[/tex]
Homework Equations
The Attempt at a Solution
I use
[tex] \frac{d\psi}{dx} = \frac{d\psi}{dw} \frac{dw}{dx}[/tex]
then
[tex]\frac{d^2\psi}{dx^2} = \frac{d}{dx}[ \frac{d\psi}{dw} \frac{dw}{dx}][/tex]
then
[tex]\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx} + \frac{d^2w}{dx^2} \frac{d\psi}{dw}[/tex]
but the [tex]\frac{d^2w}{dx^2}[/tex] equals zero, since x is linear in w.
which implies
[tex]\frac{d^2\psi}{dx^2} = \frac{d\psi}{dwdx} \frac{dw}{dx}[/tex]
and [tex]\frac{dw}{dx} = A^{1/3}[/tex]
but I'm not sure how to evaluate the
[tex]\frac{d\psi}{dwdx}[/tex] term (I'm not sure If it should even be there.. did I use the chain rule correctly?)
Then I'll sub in [tex]\frac{d\psi}{dwdx} A^{1/3}[/tex] for [tex] \frac{d^2 \psi}{dx^2}[/tex] and hopefully it all works out.
Any help would be greatly appreciated.
Edit: SOLVED
Last edited: