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-[tex]\frac{\hbar^{2}}{2m}[/tex][tex]\frac{d^{2}u}{dx^{2}}[/tex] + [tex]\frac{1}{2}[/tex]m[tex]\omega_{0}^{2}[/tex]x^{2}u = Eu

Now my lecture notes say that it is convenient to define scaled variables

y = [tex]\sqrt{\frac{m\omega_{0}}{\hbar} x}[/tex]

and [tex]\alpha[/tex] = [tex]\frac{2E}{\hbar\omega_{0}}[/tex]

Hence

[tex]\frac{d}{dx}[/tex] = [tex]\sqrt{\frac{\hbar}{m\omega_{0}} x}[/tex] [tex]\frac{d}{dy}[/tex]

so [tex]\frac{2}{\hbar\omega_{0}}[/tex] times the TISE can be written as

[tex]\frac{d^{2}u}{dy^{2}}[/tex] + ([tex]\alpha[/tex] - y[tex]^{2}[/tex])u= 0

now, it is not at all obvious to me where these scaled variables came from? I know this leads on to deriving hermite polynomials, but I'm just wondering if there is some triviality behind the scaled variables. It really helps me to know the order in which things were derived/formulated. It might have been the case that these scaled variables were used after much painstaking trial and error, or some mathematician just noticed that it would be easier to write the equation this way.

Does anyone have any background information on this? Is it usual to scale the variables in this way?

Thanks