Hydrogen Transitions/TD Pert Theory

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Homework Statement


I'm looking at two different time dependent perturbation solutions to transitions in hydrogen, and I'm wondering where the discrepancy comes from.


Homework Equations


See the attached documents- problem 2 in solutions5.pdf and exercise 2.10 in 2s decay.pdf. You'll notice different equations for the matrix elements (specifically the denominators). I'm trying to figure out why these two are different, if they actually are. I'm probably just missing something obvious that makes them equivalent.


The Attempt at a Solution


 

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