# Uncertainty Principle

1. Jun 11, 2012

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

We are often told one reason why an electron does not fall to the centre of atoms is because it would then have a well-defined position. Any two particle which comes into contact define each others positions and so their momentum becomes very large.

How then is it, according to the Exclusion principle two bosons can actually occupy the same space? Surely that would violate the UP?

2. Jun 11, 2012

### HallsofIvy

Staff Emeritus
I've never heard that. Could you give a reference?

Two Fermions in an atom cannot be in the same state while two bosons can. That has nothing to do with occupying the "same space".

3. Jun 11, 2012

I'll try and find one... I think I read in it Hawkings brief history of time. Surely when a photon hits off of an electron, they define each others positions and so they fly off into different directions very fast as a result?

Here is a reference... well more of a discussion on the electron and how the uncertainty principle forbids it to have a defined location in the center of atoms

''The electron cannot be confined to the nucleus...''

Again, I read this in Hawkings book, a brief history of time.

As for the Bosons, I am speaking about the Bose-Einstein statistics, which uses the literature that bosons fall into the same energy levels... or as you will find printed all over the net, the idea that they occupy the same ''space''.

So again, the question arises how two bosons can occupy the same space... or is this just a bad terminology?

4. Jun 11, 2012

### vela

Staff Emeritus
"Same space" doesn't mean the same thing as "small space." The reason you don't see electrons confined to the nucleus is because, to satisfy the uncertainty principle, its momentum would have to be so large it would no longer be bound to the atom.

Now consider the two electrons in a helium atom. As long as their spin state is antisymmetric, they can have the same spatial wave function.

5. Jun 11, 2012

Right... this is what I was getting at with the electron example... it can't be confined to the nuclei of atoms because it would violate the UP...

howsoever, I have not heard of ''small space'' before. Can you explain how we differentiate between something said to ''share the same space'' to whatever a ''small space''? Remember, this is about bosons, not so much electrons. I keep reading the bosons can share the same space, whatever that means... but if the terminology is as it sounds, then surely this would violate the UP, at least positional-wise.

6. Jun 11, 2012

### vela

Staff Emeritus
The uncertainty principle suggests the scale for the the size of the atom. The atom has to be a certain size or larger otherwise the electron has too much energy to stay bound to the nucleus.

The question about the bosons have absolutely nothing to do with the uncertainty principle. It has to do with the exclusion principle. Fermions obey the exclusion principle; bosons don't.

You seem to be conflating the two principles, but I'm not sure how or why.

7. Jun 11, 2012

No I understand the uncertainty and exclusion are different principles.

I am stuck on why one would say that bosons can ''share the same space''.

8. Jun 11, 2012

We are not talking about point space then?

9. Jun 11, 2012

### vela

Staff Emeritus
Why can't they? Just like two electrons in the ground state of helium share the same space.

10. Jun 11, 2012

For instance two electrons cannot share the same point positions, that defines their positions really well and defies the uncertainty principle. So when bosons are said to share the same space, this isn't what is meant?

11. Jun 11, 2012

The same point, not smeared over a region? Which is what I am guessing is going on in a helium atom, except they have energy levels to define it.

The very same position?

12. Jun 11, 2012

They surely can't occupy the same position

$$\Delta X \Delta P = \hbar/2$$

Remember, particles act as observers as well. You try and shove two particles into the same (the exact same locations) they will define each others positions - resulting in an extremely large momentum.

13. Jun 11, 2012

### vela

Staff Emeritus
"Same space" doesn't mean the particles are confined to one point in space.