Minimum uncertainty wave packet
A wave packet for which minimum uncertainty in both the position and the momentum is attained.
In one dimension it satisfies
, where
is the root-mean-square deviation from the mean position and
is the root-mean-square from the mean momentum.
The solution to the mathematical problem of finding the wave function
that satisfies is left as an exercise.
The result is
where
is the root-mean-square deviation from the mean
position 
and
is the mean momentum of the particle.
The initial probability distribution
is given by 
a Gaussian function. is a positive real number. The Gaussian function
is shown on the right. The curve has the same shape as the normal distribution with a standard deviation
.
In two or three dimensions the minimum uncertainty wave packet takes the form of a product of one-dimensional minimum uncertainty wave packets.
For a particle moving in a plane the probability distribution looks like the one shown in the picture on the right.