# How can bosons made of fermions occupy the same quantum state?

1. Apr 2, 2013

### andrewkirk

I've been reading about Bose-Einstein condensates, in which multiple bosons can occupy the same quantum state. I thought I understood how that could work until I learned that some atoms, such as Helium-4, are bosons.

It seemed to me that if two He-4 atoms H1 and H2 occupy the same quantum state then the four electrons must occupy the available electron quantum states, of which there are only two, so at least one of the electron quantum states must be occupied by more than one electron, which would violate the Pauli Exclusion Principle (as electrons are fermions).

Since this doesn't happen, there must be something wrong with my attempt at reasoning.

I would be grateful if someone can help me understand where I've gone wrong.

2. Apr 2, 2013

### crissyb1988

I would say its to do with the He-4 atoms being in a bound state. The spin of this bound state is a whole integer value (boson) and it doesn't matter about the constituent parts (fermions).

If you look at low temp superconductors their electrons pair up (cooper pairs) and this pairing acts as a boson. Due to this pairing superconductors act like Bose-Einstein condensates

3. Apr 2, 2013

### Charles Wilson

Edit: I'm pulling some of what I posted. Wrote something when I was very tired and made an error. Will repost tomorrow!

Find SciAm, June 1990, "The Helium3 Superfluids".

[[ZZzzz.....]]

The 2 Helium3 atoms begin orbiting each other and the behavior becomes as a Boson. Helium3 in this state behaves as a Boson and can now drop into a lowest State. It is a Superfluid as is Helium 4.

The SciAm article is absolutely first rate and goes a long way to explaining a very rich segment of low temperature physics, from Dirac Seas to Cooper Pairs.

CW

Last edited: Apr 2, 2013
4. Apr 2, 2013

### Bill_K

When you say the atoms are in the "same state", you're talking about the coordinates that describe the atom, namely the center-of-mass X, P, and total spin. The state of one of the electrons, on the other hand, also involves internal coordinates, e.g. its position x relative to its nucleus. Furthermore, the relative x in one atom is a different coordinate from the relative x in the other. So even if the atoms have the same external coordinates, and even if you assume the electrons of both are in the respective ground states, the electrons still have four states available.

5. Apr 2, 2013

6. Apr 2, 2013

### Staff: Mentor

My oath. When I saw it I thought - gee why didn't that occur to me. Brian Cox said in one of his TV shows since fermions can't occupy the same state, so if you move fermions about since it changes its state you are, in principle, affecting the state of fermions everywhere. When I heard Brian say that my hand went over my face and I thought - Brian - you know that applies only to interacting Fermions - Fermions on the other side of the universe are not affected - but he may counter - as he did - the universe can be considered to be in one big box so you have to consider the system of all Fermions - I still think is basically populist BS but its a bit harder to counter. Most certainly this question and its answer is something that really makes it hard for Brian to counter.

A really really nice question

Thanks
Bill

7. Apr 3, 2013

8. Apr 3, 2013

### Charles Wilson

I guess you get one edit per post... Let me start over sorta.

What a beautiful question!

Let me quote from the SciAm article:

"The particular ordering that takes place in liquid helium is a consequence of a fundamental division that exists in quantum mechanics between fermions...and bosons. Bosons comprise such force-carrying particles as photons and pions. Their spin is an integer multiple of the fundamental quantum of angular momentum, h-bar, Planck's constant divided by 2 pi.
"A He4 atom consists of two electrons, two protons and two neutrons, each with half integer spins. As a result, the atom is a boson." [[At 2.17 k, He4 starts to condense into its lowest energy state.]]
"Superfluidity in He3 has a different character. It's atoms contain an odd number of neutrons and so an odd number of particles in sum. Thus they are fermions and are unable to condense into a common ground state. As a result, He3 cannot become superfluid as easily as its boson sibling can. Instead, at transition temperature roughly 1000 times lower than that of He4, a weak interaction between He3 atoms begins to make itself evident. Atoms with equal and opposite momenta tend to form pairs in which the two particles orbit each other at a distance. These pairs (called Cooper pairs...) are bosons - their half integer angular momenta add up to an integer value. Therefore, they can condense to a common ground state and form a superfluid."

Better?

CW