Tuesday, January 09, 2007

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


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
                Δf ≥


                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.


EkendraLamsal.com, mail@EkendraLamsal.com

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