A Universe of Uncertainties

Sorry it's been so long since I've put up a post! I've been busy with my extended project, which goes by the title  "How are quantum-mechanical effects and phenomena used in modern technologies?", and so haven't had much time to put together a post. 

So, picking up where I left off, at the end of the last post I wrote (click here if you want to refresh your memory), I introduced one part of the uncertainty principle: "Any determination of the alternative taken by a process capable of following more than one alternative destroys the interference between alternative", which was illustrated in the experiment explored in the last blog post. However, the uncertainty principle is usually known in a different form, presented to us by Heisenberg back in the 20's:

where "delta x" is the uncertainty in position, "delta p" is the uncertainty in momentum and "h bar" is Planck's constant (which we met back in my first post) divided by 2pi. What this means is that the more certain you are about one of the two variables, the less certain you are about the other: if you're fairly certain you know how fast a particle is moving, you can't be sure exactly where it is in space. Although on a macroscopic scale, this inequality doesn't really make much difference to us (as Planck's constant is in the order of 10^-34), it makes a huge different at a microscopic scale, and it leads to a number of weird phenomena. So, why is this the case? 

To understand why there has to be uncertainty, we have to consider what the dual nature of particles really means. The best place to start is with wavefunctions, which are the mathematical way of describing particles. A wavefunction has a value for every point in the universe, and a value squared gives you the probability of the particle being at a specific position at a specific time. The wavelength of this wavefunction (the distance from one peak to another) corresponds to the momentum of the particle. 

Imagine we have an electron somewhere in room with us. If we start by considering the electron as purely a wave, we get the following wavefunction and corresponding probability :

where the electron has an equal probability of being absolutely anywhere in the universe, and is moving with a constant, known momentum. Of course, this isn't right, as the electron we're imagining is in the room with us somewhere. So, what do we do? Well, the wavefunction only represents one "state" of the electron, moving at some constant momentum, and really, as we don't know how fast the electron is moving, so it could be in any one of an infinite "states". For example:

In order to truly represent the electron that's somewhere in our room, we need to add all these wavefunctions together, so that we're considering the infinite number of probabilities involved. To do this, you have to use calculus, What we get is a wavefunction, when squared, that looks like this:

What's happened is in the places where the waves are in phase, you've had them build up, and in the places where they were out of phase, they've cancelled eachother out. This just leaves you with a "wave packet" where the probability of the position of the electron is no longer evenly spread over the entirety of the universe, but a small amount of space (i.e. your room), and at some momentum. However, by working out where the electron is to a much larger degree of certainty, we've had to add the probabilities of the electron being at lots of different momentums: we can no longer be certain of its momentum. Our knowledge of one of our variables increased at the expense of our knowledge about the other!

And with that, we've arrived back at the original inequality, which is basically just setting a limit on how much we can know, by saying the product of the uncertainties has to be greater than or equal to some constant, meaning neither uncertainty can actually reach zero. However, this limit is a  fundamental feature of our universe of dual-natured particles: it's more than just saying that we can't measure both at the same time, it's saying that the variables don't exist in an absolute sense! 

What's more, momentum and position aren't the only two things that this rule applies too. There are a number of other variables, known as conjugate variables, and include Time and Space, and Angular Momentum and Angular position. 

As weird as this all seems, the uncertainty principle is now many decades old, and yet no one has managed to defeat the limitations in measurements which it implies. Furthermore, the rest of quantum mechanics relies on the workings of the principles, and the predictions of quantum mechanics have been confirmed over and over again to the incredible degrees of accuracy. 

The Uncertainty Principle leads on to a variety of phenomena, some that we've touched on in this post (superpositions, for example) and others, such as particles popping in and out of existence at random, and particles tunnelling through barriers they shouldn't be able to, and that, I shall leave to another post. 

I hope you enjoyed this post, I should have another one coming fairly soon! I'm pretty excited to talk about the uses of quantum tunnelling, which has all sorts of practical uses (uses that I'm hoping to write a large part of my extended project on), and I'm sure that'll be the subject of one the next posts. 

Thanks for reading, 
-GM