According to Heisenberg, it is impossible to know a particle's momentum and position at the same time. One explanation involves hitting the particle with a high-frequency photon, so its position can be observed. The particle is so small that when the photon collides with it, the particle's momentum changes. To observe momentum, the observer must use low-frequency light to avoid changing the momentum, but then position is sacrificed.
Another reason for the uncertainty principle is the probabilistic nature of quantum mechanics. We can model this uncertainty with pairs of dice:
There are four pairs listed above. They are each indexed 11 or 22 to indicate the first pair and second pair, respectively. The idea is to roll the A11 pair and the A22 pair n number of times, take the difference, and divide by n. Doing this gives us the average difference between A11 and A22. We do the same with B11 and B22.
We then multiply the difference in A with the difference in B. That gives a number that is greater than or equal to the constant K (assuming K is really small). When n is small, the result is greater than K. When n is big, the result can be less than K--as low as zero.
The difference in A and difference in B shrink as n grows, and vice versa. It is safe to say that when we only roll one or two pairs of dice, we get a result larger than K, and we get big A and B differences; i.e., A and B are more uncertain and harder to pin down. If we roll trillions of dice (large n), then A and B are easy to pin down, and the result drops to zero. If this all seems too familiar, it should. We can replace A and B and substitute momentum (p) and position (x) to get Heisenberg's Uncertainty Principle:
Caveat: There is a difference between the dice and momentum-position: We can know either position or momentum at the quantum scale; whereas, A and B are both unpredictable when n is small.
You will find a simple way to derive Heisenberg's Uncertainty Principle here.
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