Superconductivity and Pauli exclusion principle

[haael]

I have the following problem understanding Pauli exclusion principle.

Two identical fermions can't share the same quantum state. Two bosons can.
Now Cooper pairs are bosons made up from fermions. Everything clear up to this point.
Now several Cooper pairs can share the same quantum state, since they are bosons.

And now: how do electrons inside particular Cooper pairs consider them different?

I mean: if every Cooper pair is identical to any other and all of them are in the same quantum state, then every such Cooper pair is the same quantum state.

How do electrons know they belong to different Cooper pairs when all of them are the same?

I bet the interactions inside a pair have something to do with it. But how it is mathematically described? How come electrons can tell apart different pairs from inside and still each of them looks the same from outside?

My questions also extend to superfluids and all bosons composed of fermions that share the same quantum state.

 

[Ben Niehoff]

Cooper pairs are very large, on the order of hundreds of atoms in diameter. This is what allows two Cooper pairs to effectively share states. In reality, two overlapping Cooper pairs cannot share exactly the same state, because the pairs are composed of fermions that obey Pauli exclusion. However, since the Cooper pairs are so large, very minute differences in quantum state can be effectively ignored, and so they behave more or less like bosons. Cooper pairs do not exactly follow a Bose-Einstein distribution, but they come quite close (and can be made arbitrarily close under appropriate conditions).

 

[haael]

Cooper pairs do not exactly follow a Bose-Einstein distribution, but they come quite close

Wait, does that mean that there are no true composite bosons and only elementary particles can have Bose-Einstein statistics?

 

[ZapperZ]

I have the following problem understanding Pauli exclusion principle.

Two identical fermions can't share the same quantum state. Two bosons can.
Now Cooper pairs are bosons made up from fermions. Everything clear up to this point.
Now several Cooper pairs can share the same quantum state, since they are bosons.

And now: how do electrons inside particular Cooper pairs consider them different?

I mean: if every Cooper pair is identical to any other and all of them are in the same quantum state, then every such Cooper pair is the same quantum state.

How do electrons know they belong to different Cooper pairs when all of them are the same?

I bet the interactions inside a pair have something to do with it. But how it is mathematically described? How come electrons can tell apart different pairs from inside and still each of them looks the same from outside?

My questions also extend to superfluids and all bosons composed of fermions that share the same quantum state.

The problem here is two-fold.

1. Remember that via the indistinguishably concept, you can't really track which electron does what.

2. There is a continuous scattering in and out of the [itex](k_1,-k_1), (k_2, -k_2), ...[itex] states that make up all of these Cooper Pairs. In other words, one can imagine that electrons keep on scattering in and out of the states. When one goes out, another comes into fill that place (since there's a lot of electrons). But since we can't distinguish one with the other, it is the same as having the same two fixed electrons sticking together to form such a pair.

So no, they don't really know, or even care, which composite boson it goes to. All an electron knows is that such a state is open and in it goes.

Zz.

 

[haael]

1. Remember that via the indistinguishably concept, you can't really track which electron does what.

2. There is a continuous scattering in and out of the [itex](k_1,-k_1), (k_2, -k_2), ...[itex] states that make up all of these Cooper Pairs. In other words, one can imagine that electrons keep on scattering in and out of the states. When one goes out, another comes into fill that place (since there's a lot of electrons). But since we can't distinguish one with the other, it is the same as having the same two fixed electrons sticking together to form such a pair.

So no, they don't really know, or even care, which composite boson it goes to. All an electron knows is that such a state is open and in it goes.

You are cheating me somewhere here :).

Suppose we have a superconducting loop with current flowing round. We can measure the count of Cooper pairs by checking the current or charge. All Cooper pairs sit on the same ground state (am I right?).
Now: what are the momenta of the electrons? When we think of it, they are halves of Cooper pair's momenta. And all pairs have the same momentum, so all electrons have the same momentum.
Something isn't right.

Putting superconductivity aside, what happens with protons of helium when it reaches superfluidity state?

 

[xepma]

The momentum is a vector, so it is not the same for all electrons. In 3D the electrons near or at the surface of the Fermi sphere form Cooper pairs which condense into the lowest-energy state. The momentum of the Cooper pairs is -- in some approximation -- zero, as it pairs electrons with opposite momentum. However, that does not imply the momenta of the electrons participating in the Cooper-pair condensate are all the same.

The electrons still obey the Pauli principle and do not occupy the same quantum state.

 

[haael]

The momentum of the Cooper pairs is -- in some approximation -- zero, as it pairs electrons with opposite momentum. However, that does not imply the momenta of the electrons participating in the Cooper-pair condensate are all the same.

Do I understand you right:
Electron pairs of different momenta can couple into Cooper pairs of the same quantum state?
In other words, Cooper pairs "forget" some information about their electrons and have the same state outside, despite electrons inside them have different states?

 

[ZapperZ]

You are cheating me somewhere here :).

Suppose we have a superconducting loop with current flowing round. We can measure the count of Cooper pairs by checking the current or charge. All Cooper pairs sit on the same ground state (am I right?).
Now: what are the momenta of the electrons? When we think of it, they are halves of Cooper pair's momenta. And all pairs have the same momentum, so all electrons have the same momentum.
Something isn't right.

Putting superconductivity aside, what happens with protons of helium when it reaches superfluidity state?

Er.. no. All the electrons cannot have the same momentum, etc.. The Fermi-Dirac distribution still applies to them, even when they form the Cooper Pairs.

Note that I labeled the momentum with different subscripts. If you construct the BCS ground state wave function, you'll notice that it is a series of plane-waves, each one over different k values. So for an isotropic s-wave pairing, you have $k_n, -k_n$ pairs, summed over all n's.

Zz.