I understand from some studies in mathematics, that the generator of translations is given by the operator ddx.
Similarly, I know from quantum mechanics that the momentum operator is −iℏddx.
Therefore, we can see that the momentum operator is the generator of translations, multiplied by −iℏ.
I however, am interested in whether an argument can be made along the lines of "since ddx is the generator of translations, then the momentum operator must be proportional to ddx". If you could outline such an argument, I believe this will help me understand the physical connection between the generator of translations and the momentum operator in quantum mechanics.
Answer
In the position representation, the matrix elements (wavefunction) of a momentum eigenstate are ⟨x|p⟩=ψp(x)=eipx
For our momentum eigenstate, if I spatially shift it by an infinitesimal amount ϵ, it becomes ψ(x+ϵ)=eip(x+ϵ)=eipϵeipx=(1+iϵp+...)eipx
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