Question

Show that the uncertainty in the measurement of a physical quantity "f" is equal to the following expression:

$Δf=√(⟨f^2 ⟩-⟨f⟩^2 )$

Answer 1

Assume

Displacement operator

We know that displacement operator

$∆f=\frac{\hbar}{mw}(n+\frac{1}{2})$

<f>=0

$<f^2>=\frac {\hbar}{mw}(n+\frac{1}{2})$

$∆f=\sqrt{<f^2>-<f>^2}$

Put value in eqution

$∆f=\sqrt{[\frac{\hbar}{mw}(n+\frac{1}{2})]^2-0}$

$∆f=\frac{\hbar}{mw}(n+\frac{1}{2})$

Left hand side = right hand side

Hence proved