Quantum Mechanics and Momentum
Part 3′ (Mathematical Addendum): Fourier Transforms and Derivatives.

Again, we have our Fourier transform . Now, let us consider the transform of the derivative of a function, . We can use integration by parts on this: with , ; we have , and , and so
.

Now, if we require  (as any meaningful signal, or quantum wavefunction, must be), the first term on the right side of the above vanishes, and thus:
.

This will prove important in later posts.