Question #101226
derive uncertainty principle using gaussian wave packet
1
Expert's answer
2020-01-22T04:23:37-0500

It is known that gaussian wave packet happen to be minimum uncertainly wave packets. In this case, if the uncertainty of coordinates Δx and the uncertainty of wave number Δk are taken as the standard deviations then this minimum value is 12.\frac{1}{2}.

In this way, we can write

ΔxΔk12(1)ΔxΔk≥\frac{1}{2} (1)

Δx is the uncertainty of coordinates, Δk is the uncertainty of wave number


The wave number is equal to

k=2πph(2)k=\frac{2πp}{h} (2)

where h is the Planck constant, p is the momentum


Using (2) we have

Δk=2πΔph(3)Δk=\frac{2πΔp}{h} (3)

Where Δp is the uncertainty of momentum


We put (3) in (1)

ΔxΔpħ2ΔxΔp≥\frac{ħ}{2}

where ħ=h2πħ=\frac{h}{2π}



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