Question

SHOW THAT minimum value of product occurs for a 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

$ΔxΔk=\frac{1}{2} (1)$

Using (1) we have

$Δx=\frac{1}{2Δk} (2)$

Where k is the wave number

The wave number is equal to

$k=\frac{2πp}{h} (3)$

Using (3) we have

$Δp=\frac{Δkh}{2π} (4)$

Using (2) and (4) we find product

$ΔxΔp=\frac{ħ}{2} (5)$

Where $ħ==\frac{h}{2π}$

Product has the minimum possible value

Gaussian wave function is a minimum uncertainty wave packet.