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The Uncertainty Principle for Energy and Time

The Uncertainty Principle says that the product of the unceratinty Δx of the location of a particle and the uncertainty of the momentum Δpx can be no smaller than ½h, where h is Planck's constant divided by 2π; i.e.,

Δx·Δpx ≥ ½h

It is also true that the product of the uncertainty in the energy ΔE of a particle and the uncertainty concering time Δt must be no smaller than ½h. Thus

ΔE·Δt ≥ ½h

A Quick Derivation of the Energy-Time Uncertainty Principle

The energy E of a particle with mass m and velocity v is ½mv². Its momentum p is mv. Therefore the energy expressed in terms of momentum is

E = p²/(2m)

The change in energy δE resulting from a change δp in momentum is given approximately by

δE = pδp/m = (p/m)δp

Thus the uncertainty ΔE in energy is given by

ΔE = pΔp/m = (p/m)Δp

The uncertainty in time Δt is given by

Δt = Δx/v

Thus

ΔE·Δt = (p/m)Δp·(Δx/v)
= (p/(mv))Δp·Δx
and hence
ΔE·Δt ≥ ½h
since p=mv and Δp·Δx≥½h


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