Uncertainty principle from analogical classical wave frequency
Let f1 and f2 be two frequencies which differ by Δf when measured over Δt.
Observing time for a beat is thus, 1/Δf second.
A single beat may certainly be observed for Δt/Δf,
this implies that ΔtΔf ≥ 1.
Presume the distance traveled by the wave in time Δt be Δx = v Δt
this leads to Δx ≥ v/Δf.
From f = v/λ
Thus, Δx Δλ ≥ λ2
When λ is measured over distance Δx, the wavelength is uncertain by Δλ.
Also from relation of momentum and wavelength,
λ = h/p
or Δλ = Δp
or Δx Δp ≥ λ2
or Δx h Δp ≥
Thus, Δx Δp ≥ h
This leads to yet another interesting and significant deduction that since energy is associated with frequency, the frequency uncertainty leads to energy uncertainty.
ΔE = h Δf
Δf Δt ≥ 1
ie ΔE Δt ≥ h
Therefore, a particle having an energy E for a time interval Δt will have its energy uncertain by ΔE. This also proves the analogy of the classical wave mechanics with the uncertainty principle.