I'm attempting to do two things. One making a well posed question, and two answering that question.

Here is the question: I'm trying to show that the operator $- i \frac{\partial }{ \partial x } $ (a cleaner version of the momentum operator from QM) is the unique linear operator $L$ such that

$$ L[e^{i px } ] = p e^{i px} $$

for all complex functions of 1 real variables $x$ , of the form: $e^{i px}$ indexed by real valued parameter $p$.

Why the question may not be well posed:

It's possible that there are a plethora of linear operators (i.e. functions sending complex valued functions to complex valued functions) that satisfy that condition. In that case, I probably need to add additional restrictions ex: defining a notion of a "continuous" or "smooth" operator and then asking for the "unique, continuous, smooth, linear operator" and I'm not sure what those restrictions should be so that the question becomes one of good substance.


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