Transformation of energy space to momentum space

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SUMMARY

The discussion centers on the transformation of wavefunctions from energy space to momentum space using the relation g(e) = g(p)/f’, where de/dp = f’. Participants explore the implications of this transformation, particularly questioning whether it can be applied to wavefunctions given the dimensional differences between energy and momentum spaces. The conclusion drawn is that while momentum and position spaces are three-dimensional, the one-dimensional nature of energy space limits the possibility of such transformations.

PREREQUISITES
  • Understanding of wavefunctions in quantum mechanics
  • Familiarity with the concepts of energy space and momentum space
  • Knowledge of the relation E = p²/2m
  • Basic grasp of phase space dimensions in physics
NEXT STEPS
  • Research the mathematical framework of wavefunction transformations in quantum mechanics
  • Study the implications of dimensionality in quantum state spaces
  • Learn about the role of phase space in quantum mechanics
  • Explore the relationship between position space and momentum space transformations
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Students and professionals in physics, particularly those focused on quantum mechanics, wavefunction analysis, and the mathematical foundations of state transformations.

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

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