- #1
Exulus
- 50
- 0
Hi guys, hoping someone can help with this manipulation. I need to transform this:
[tex] \frac{-\hbar^2}{2m}\frac{d^2}{dx^2}u(x) + \frac{1}{2}m\omega^2 x^2 u(x) = Eu(x)[/tex]
Into its dimensionless form:
[tex]\frac{d^2}{dy^2}u(y) + (2\epsilon - y^2)u(y) = 0[/tex]
I have the following info:
[tex]E = \epsilon\hbar\omega[/tex]
[tex]x = y\sqrt{\frac{\hbar}{m\omega}}[/tex]
Heres what I've done so far:
[tex] \frac{-\hbar^2}{2m}\frac{d^2}{dx^2}u(x) + \frac{1}{2}m\omega^2 y^2 \frac{\hbar}{m\omega} u(x) = \epsilon\hbar\omega u(x)[/tex]
[tex] \frac{-\hbar}{m}\frac{d^2}{dx^2}u(x) + \omega^2 y^2 u(x) = 2\epsilon\omega u(x)[/tex]
[tex] \frac{\hbar}{m}\frac{d^2}{dx^2}u(x) + 2\epsilon\omega u(x) - \omega^2 y^2 u(x) = 0[/tex]
[tex] \frac{\hbar}{m}\frac{d^2}{dx^2}u(x) + (2\epsilon - y^2)\omega u(x) = 0[/tex]
But i can't see where to go next..i know i must be close to the end though..surely!
[tex] \frac{-\hbar^2}{2m}\frac{d^2}{dx^2}u(x) + \frac{1}{2}m\omega^2 x^2 u(x) = Eu(x)[/tex]
Into its dimensionless form:
[tex]\frac{d^2}{dy^2}u(y) + (2\epsilon - y^2)u(y) = 0[/tex]
I have the following info:
[tex]E = \epsilon\hbar\omega[/tex]
[tex]x = y\sqrt{\frac{\hbar}{m\omega}}[/tex]
Heres what I've done so far:
[tex] \frac{-\hbar^2}{2m}\frac{d^2}{dx^2}u(x) + \frac{1}{2}m\omega^2 y^2 \frac{\hbar}{m\omega} u(x) = \epsilon\hbar\omega u(x)[/tex]
[tex] \frac{-\hbar}{m}\frac{d^2}{dx^2}u(x) + \omega^2 y^2 u(x) = 2\epsilon\omega u(x)[/tex]
[tex] \frac{\hbar}{m}\frac{d^2}{dx^2}u(x) + 2\epsilon\omega u(x) - \omega^2 y^2 u(x) = 0[/tex]
[tex] \frac{\hbar}{m}\frac{d^2}{dx^2}u(x) + (2\epsilon - y^2)\omega u(x) = 0[/tex]
But i can't see where to go next..i know i must be close to the end though..surely!