What happens to wave packets?

So I’ve told you what de Sitter space is, and that I’m interested in studying it, but what am I interested in studying? Well, to start things off, we want to know what happens to particles moving in de Sitter space. When we’re talking about particles, we’re talking about quantum field theory (see previous entry).

What happens to particles in a non-expanding universe? Well, because of the Heisenberg uncertainty principle, its position and its momentum cannot be perfectly determined, but certainly it can start roughly at some position with roughly some momentum (as well as some uncertainty in both).  That particle then starts moving in the direction of it’s momentum. Because both it’s position and its momentum are uncertain, the region over where this particle might be also starts expanding. However, there is something interesting when you look at the equations: a particle that starts off with a lot of position uncertainty eventually has less position uncertainty than a particle that starts off with very little position uncertainty.

Let me explain: the Heisenberg uncertainty principle is

\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}~~

that the uncertainty in the position multiplied by the uncertainty in the momentum is greater than Planck’s (reduced) constant over two. This means that a particle with a very uncertain momentum can have a very well defined position, and a particle with a very uncertain position can have a very well defined momentum.

A particle that starts out with very little uncertainty in its position must have a lot of uncertainty in its momentum, which actually makes it spread out faster than a particle that starts out with a very uncertain position but a very well defined momentum!

This might make sense intuitively, and can be pictured as follows:wave packets

The particle that starts out with more momentum uncertainty spreads out quicker than the one that starts out with a very little.

However, this fact is backed up by a mathematical calculation which I am not showing you here.

Why do we care about this?  Well think about this: a particle with one exact momentum value must have infinite position uncertainty. That is: if you know for 100% certainty what the momentum is, you have absolutely no idea where it is. However, it is very important, quantum mechanically speaking, that states with different momenta do not overlap. This is the same thing as saying that a state with a definite momentum will never be measured to have a different momentum. But how can this be if definite momentum states overlap with the entire universe?

Well, what if instead of a definite momentum we mearly had an almost definite momentum with almost no uncertainty? Well, then two states with different momenta would separate out and eventually have no overlap, so they are indeed different states.

This may seem like word juggling the way I am explaining it but I can assure you that it’s not, and this property of momentum states is very important. We can define perfect momentum states by thinking about merely very good momentum states and then pretend that they are perfect.

But what happens in an expanding universe? This took longer to explain than I thought so I will tell you next post…

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