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Hey i was wondering how to express the time evolution operator U(t,to) to a momentum eigen state |p> for a particle moving in the xdirection under a zero potential, V= 0. The reason i need this is that iam told the only way to get the matrix element of the time evolution operator using position eigen states <x|U(t,to)|x2> is to first apply the Time evolution operator on a momentum basis and use a linear transformation to position eigenstates, Iam not quite sure how to apply the time evolution operator on the momentum eigenstate though.

Thanks any help would be appreciated

My attempt:

I thought of trying to expand the exponential time evolution operator using

e^{x}=\\sum_n x^n/n!

but then your just left with the exact same problem which is i can't seem to be able to use the operator on the momentum eigenstate, not quite sure how to begin...

Thanks any help would be appreciated

My attempt:

I thought of trying to expand the exponential time evolution operator using

e^{x}=\\sum_n x^n/n!

but then your just left with the exact same problem which is i can't seem to be able to use the operator on the momentum eigenstate, not quite sure how to begin...

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