About Heisenberg's Uncertainty Principle

Most of us have done sums about falling particles and how long it would take them to fall and things like that. Take this problem for example: If a ball falls from the top of a building (on earth) of height 20 metres at 12 o'clock, where is it at 12:02? The noticeable thing about the question is this : we are saying that if you know the position of the ball at some time, and the environment it is in ( is there something pulling or pushing it? in this case it's earth, pulling it through gravity) , you can tell exactly where you'll find it later. There will be a certain position where it will invariably turn up. It won't , for example, say ' no thank you' and go back to the top of the building. Or that's what we thought.



It turns out that we just can't have enough information to tell you exactly where the ball can be. If you do an experiment with a hundred different balls, it turns out that we get different results, even if all the controllable conditions are kept just same. In one experiment you get 1 (say), in the other 2 or something. These differences are usually small, which is why for large objects like a ball we can neglect it. But the smaller our object gets, the more these differences matter.
But are these things totally random? Can we make any predictions at all? Yes, we can. Tell me where it was initially, and what the environment is, and I can tell you is the probability of finding it at different places. There'll even be a probability of it jumping up again ( though I admit none of its actually saying ' no thank you') . Not as good perhaps as telling you exact locations, but that's the best that can be done. And it's not just true of positions, but any physical quantity that's there : momentum, angular momentum etc. The theory which provides the scheme for working out these probabilities is Quantum Mechanics.

Now to uncertainty. Uncertainty is just the standard deviation (or sometimes half the s.d)calculated from these probabilities. Larger the uncertainty, larger is the range of values we are likely to get for whatever it is that we are measuring. Smaller the uncertainty, smaller is this range of likely results. That's why it's called uncertainty, duh!

So what's theHeisenberg's Uncertainty Relation? It is a result in Quantum Mechanics which says that the product of position uncertainty and momentum uncertainty of our object is always greater than a certain number. So if you find (through experiments) that the standard deviation in position values is small for our object, then it always turns out that the standard deviation in momentum values will be large. It is a statistical rule; narrower the probability distribution for position, wider the pd of m, and vice versa.

So this tells us that there must be something funny between position and momentum; by knowing the uncertainty in one, I can say something about the uncertainty in the other. There is something funny between the two, they are 'observables that do not commute with each other'.

How the uncertainty principle follows from this commutation relation, I'll write in another post.