I got this problem from a book called Introduction to quantum mechanics, griffin 2nd edition. and I did not get why the solution says
first term integrates to zero, integration by parts twice?!
Please see the solution below! Thanks everyone for helping!
Answer
He uses
∫(∂∂xf(x)) g(x) dx=∫ f(x)(−∂∂xg(x)) dx,
to get
∫(∂2∂x2Ψ∗)∂∂xΨ dx=∫(∂∂xΨ∗)(−∂2∂x2Ψ) dx=∫Ψ∗∂3∂x3Ψ dx.
The fact that you can shift the sqare of the derivative ∂2∂x2 around like this is already suggested by Δ=∑3n=1∂2∂x2n being an observable in several contexts. If it's an observable, then it's hermitean and ⟨ΔΦ|Φ⟩=⟨Φ|ΔΦ⟩.
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