# How do Cooper pairs carry a current with zero net-momentum?

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• alberliu
In summary, the BCS theory in superconductors introduces the concept of zero-momentum for the ground state of a quantum system, where electrons must pair up with opposite spins to form Cooper pairs. This leads to the question of how a finite supercurrent can exist in this state. However, the misconception lies in confusing spin with linear momentum, as the electrons will still have small linear momenta even in the paired state. In superconductors, it is not the individual electron speeds that matter but the overall movement of a mass of electrons that results in electricity.
alberliu
TL;DR Summary
How do Cooper pairs carry a current with zero net-momentum?
One of the first starting points of introducing BCS theory in a superconductor is applying a theorem stating that the ground-state of a quantum system has an expectation value for its momentum of zero. You then use this to say that an electron must pair with another electron of equal and opposite momentum to form a Cooper pair.

If this is the case, how does this zero-momentum state result in a finite supercurrent? Maybe this is a frame-of-reference issue, but it's not clear to me how you get net charge movement in one direction with a net momentum of zero.

That's simply not true. In a state with finite current, the Cooper pairs don't have finite momentum and the electrons in a pair don't have exactly opposite momentum. In the book by Schrieffer "Theory of superconductivity", this is explained in detail.

alberliu said:
Summary:: How do Cooper pairs carry a current with zero net-momentum?

One of the first starting points of introducing BCS theory in a superconductor is applying a theorem stating that the ground-state of a quantum system has an expectation value for its momentum of zero. You then use this to say that an electron must pair with another electron of equal and opposite momentum to form a Cooper pair.

If this is the case, how does this zero-momentum state result in a finite supercurrent? Maybe this is a frame-of-reference issue, but it's not clear to me how you get net charge movement in one direction with a net momentum of zero.
I think you've mistaken electron spin with linear momentum. The Cooper pairs are of electrons with opposite spins, not opposite momenta. Now, the electrons will be very close to absolute zero, so their linear momenta will be very close to zero as well, but it's not a requirement that they stay at zero momentum once they've already paired up.

Electricity is actually a very slow movement of electrons in non-superconductive situations, but in superconductive situations, it's likely just as slow if not slower, but it's not the individual electron speeds that matter but the overall movement of a mass of electrons that give you your power.

## 1. What are Cooper pairs?

Cooper pairs are pairs of electrons that are bound together in a superconductor at very low temperatures. They are formed due to the interaction between electrons and the crystal lattice of the material, resulting in a net attraction between the electrons.

## 2. How do Cooper pairs form?

Cooper pairs are formed through a process called electron-phonon coupling. This occurs when an electron interacts with the vibrations of the crystal lattice, resulting in a decrease in energy and the formation of a Cooper pair.

## 3. How do Cooper pairs carry a current?

Cooper pairs carry a current by moving through the superconductor as a single entity. This is possible because the electrons in a Cooper pair are bound together and move in a coordinated manner, allowing for the transfer of energy and momentum without any resistance.

## 4. Why do Cooper pairs have zero net-momentum?

Cooper pairs have zero net-momentum because the electrons within the pair have opposite momenta, canceling each other out. This results in a collective motion of the pair with no overall momentum.

## 5. How does the zero net-momentum of Cooper pairs contribute to superconductivity?

The zero net-momentum of Cooper pairs is essential for superconductivity because it allows for the flow of electrons without any resistance. This results in the phenomenon of zero electrical resistance and perfect conductivity, which are key characteristics of superconductors.

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