Suppose we have an experiment which produces a probability wave packet
describing a particle of mass m in free space. At time t=0, the
packer has a width 
centered about <X> and travels at a
velocity <V>. By the uncertainty principle, we know that there is
an uncertainty in the momentum of the packet exceeding 
. Let us suppose that we begin with a packet
with the absolute minimum uncertainty. (We will be precise here with
the uncertainty principle because we are now using precise
mathematical definitions for 
and 
and because we
later will be able to compare our result here with a calculation using
the full formal wave theory.) Associated with the uncertainty in
momentum, there is also an uncertainty in the velocity of the
particle, 
.
Because we are in free space and there is no force to change the velocity of the particle, if we wait a time t>0 the velocity of the particle will not change
and its new position will be given by the usual formula
The uncertainty (standard deviation) in 
now comes from
uncertainties both in X and V. Assuming that X and V are not
correlated and are independent, we then have
So that,
Note that 
does not change with time as the packet
spreads because there is no force acting to change the velocity of the
particle in each measurement. This does not violate the uncertainty
principle because 
,
which grows with time, continues to exceed 
. The uncertainty
principle is an inequality, not an equality!