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For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My question is about what is inside the red box. SG says that "because the wave functions vanish at infinity, the first term (green box) does no contribute, and the integral gives..."

If the waves functions vanish at infinity (and also their derivatives), why doesn't the second term (blue box) vanish aswell?

Thank you for your time.

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# I Square integrable wave functions vanishing at infinity

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