Uncertainty principle from analogical classical wave frequency

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
ie           
or                 

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

ie           

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.

Further,

ΔE = h Δf

But,

                Δf Δt ≥ 1
                Δf ≥

So,

                ΔE ≥

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.

 

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