Heisenberg's Uncertainty Principle

In 1928, Heisenberg (1901-1976) stated that it was impossible to measure both the momentum and position of a particle (electron) simultaneously with accuracy less than Plank's constant, ie. there must be uncertainty. If we could stop the particle long enough to measure its position, the information on its momentum is lost and conversely if we allow the particle to continue moving so that we measure its momentum, it is everywhere.

The same principle of uncertainty applies in other settings such as the trend of $\mbox{CO}_2$ measurements over time at Mauna Loa (Hawaii) which is plotted in Figure 1. The systematic part appears to be an upwards trend with oscillations due to annual effects, with some irregularities. These irregularities may be due to instrument error, unmeasured atmospheric influences, or pure randomness.

Figure: $\mbox{CO}_2$ readings on Mauna Loa, Hawaii.

observations	trend
annual	random

The interest is of course the broad trend so we smooth out the random fluctuations. If we wanted to focus on the fine scale detail, we would have to subtract the broad trend - ie we can have trend or detail but not both simultaneously. So if we want an exact description of the instantaneous location, we have to forego the momentum of observations that preceded it and if we want the systematic trend, we shall not have fine detail about the position. A prediction that incorporates both trend and fine detail will be a narrow range of possible $\mbox{CO}_2$ concentrations, ie. with probability.