2.C: Trajectories and Probability
To see how probability can describe classical motion, take the simple example of a particle bouncing back and forth between two walls. The top animation on this page is a particle executing this motion according to the familiar laws of classical mechanics.
If our class of 100 students were to measure the position of the particle at various times they would obtain a distribution of measured positions. The bottom figure shows the motion of the particle plus the probability distribution obtained by our hypothetical class at various times. The motion shows how the probability distribution changes with time.
The center of the probability distribution traces out the trajectory of the particle predicted by classical mechanics. It is the crucial connection between probability and the laws of mechanics.
In classical mechanics the width of the probability distribution is all due to experimental error: the different ways people hold and read rulers, whether they were conscientious or careless in their measurements, etc. The better the measuring equipment and the more careful the experiments, the narrower the distribution of measurements will be. According to classical mechanics we can always expect that the probability distribution can be made more and more narrow with better experiments. This is not so in quantum mechanics. In quantum mechanics there is a fundamental limit to the precision of a measurement. No matter how much care and equipment is invested in the experiment, under the laws of quantum mechanics the width of the probability distribution can never be made less than a fundamental limit. This is the uncertainty principle of quantum mechanics.