Quantum Mechanics

Motion in a potential: Free particle

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.