Let (x) and (k) be Fourier transforms pairs so The averages of the operator O with respect

Let ƒ(x) and ƒ̂(k) be Fourier transforms pairs so

The averages of the operator O with respect to the distributions f(x) and ƒ̂(k) are

(a) Show that ĥ(k) = ik ƒ̂(k) if h(x) = df/dx.
(b) Prove Parseval’s theorem,

(c) Show that (k)k = −i(d/dx)x and (k²)k = −(d²/dx²)x.

(d) Let ΔO = √(O²)_x – (O)²_x and reproduce a proof of the (Hermitian) operator identity

from any (quantum mechanics) text in enough detail to demonstrate that you understand it.
(e) Let Δk = (k²)_k – (k)²_k and use all of the above to prove that 

This shows that the width of a wave packet in real space increases as the wave vector content of the packet decreases.

Transcribed Image Text:

f(x) = ·Ï dk 2л f(k)eikx and ƒ (k) = [ dxf [ dxf(x)e-ikx