Answer to Question #101226 in Quantum Mechanics for SAHIL

Question

derive uncertainty principle using gaussian wave packet

Answer 1

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 $\frac{1}{2}.$

In this way, we can write

$ΔxΔk≥\frac{1}{2} (1)$

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

The wave number is equal to

$k=\frac{2πp}{h} (2)$

where h is the Planck constant, p is the momentum

Using (2) we have

$Δk=\frac{2πΔp}{h} (3)$

Where Δp is the uncertainty of momentum

We put (3) in (1)

$ΔxΔp≥\frac{ħ}{2}$

where $ħ=\frac{h}{2π}$