Transformation of energy space to momentum space

[queries]

I have learned that to transform from one space to another, we can use
g(e) = g(p)/f’, where de/dp = f’

Can we use this relation to transform wavefunctions of energy space to momentum space?
If not, why?
If so, that's very strange as E= p^2/2m and dE/dp= p/m and put into
|psi>=exp(iEt/hbar) ==>|psi>= exp(ipt/mhbar)??

 

[exponent137]

phase space has three dimensions and momentum space has three dimensions. So, transformation is possible. Energy space is one dimensional, so ... it is not possible.

 

[queries]

I see.. Thanks.. So can I do the same for position space and momentum space as they both have three dimensions?