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In Griffith's derivation of the quantum SHO, he uses some funny math:

first he considers asymptotic behavior to get ψ=Ae-(ε^2/2)

then he 'peels off the exponential part' to say that ψ=h(ε)e-(ε^2/2)

then he hopes that h(ε) will have simpler form than ψ(ε)

I can kind of understand the first part, but I have no clue what he means by the second part, i dont understand the motivation for this step...given this ODE, i would not know to proceed this way.

and idk what reason we have to 'hope' that h(ε) will be simple, just from the above data.

I am not familiar at all with this method of solving differential equation and i cannot find any resource on it...does anyone know of a better explanation? everything i have found merely copies word for word griffith's derivation.

I also have liboff's and mahan's book and they are even worse at this explanation

first he considers asymptotic behavior to get ψ=Ae-(ε^2/2)

then he 'peels off the exponential part' to say that ψ=h(ε)e-(ε^2/2)

then he hopes that h(ε) will have simpler form than ψ(ε)

I can kind of understand the first part, but I have no clue what he means by the second part, i dont understand the motivation for this step...given this ODE, i would not know to proceed this way.

and idk what reason we have to 'hope' that h(ε) will be simple, just from the above data.

I am not familiar at all with this method of solving differential equation and i cannot find any resource on it...does anyone know of a better explanation? everything i have found merely copies word for word griffith's derivation.

I also have liboff's and mahan's book and they are even worse at this explanation

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