Integrate Laplacian operator by parts

Eric_J · Jan 18, 2016

Two equations from a textbook page: top, ih dφ(p,t)/dt equals 1/(2πħ)^(3/2) times the integral of exp(-i p·r/ħ) times the bracket of -ħ²/(2m) ∇² + V(r), times ψ(r,t) d³r; bottom, boxed equation (3.21) with p²/2m φ(p,t) plus a Fourier-transformed potential term involving V(r) and φ(p′,t).

This is the key step to transform from position space Schrödinger equation to its counterpart in momentum space.
How is the first equation transformed into 3.21?
To be more specific, how to integral Laplacian term by parts?

dextercioby · Jan 18, 2016

Do you know tensor tensor calculus or at least the Gauss-Ostrograski theorem in 3D?

Eric_J · Jan 18, 2016

dextercioby said:

Do you know tensor tensor calculus or at least the Gauss-Ostrograski theorem in 3D?

Handwritten derivation on paper: a prefactor of -ħ²/2m over (2πħ)^(3/2) times an integral of e^(-i p·r/ħ) times the Laplacian of ψ(r,t), integrated by parts to a surface term vanishing sufficiently fast minus a volume integral, ending at p²/2m times φ(p,t).

Thanks for the hint!

Integrate Laplacian operator by parts

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