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 Dx Dpx = h / 2, 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 Dx Dpx = h / 2 is left as an exercise. The result is
where s is the root-mean-square deviation from the mean position x0 and hk0 is the mean momentum of the particle. The initial probability distribution P( x , 0 ) is given by
a Gaussian function. P( x , 0 ) 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 s. 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.
