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We talked about the Bose-Einstein-Condensate (BEC) in our thermodynamics lecture and are discussing superconductivity in the condensed matter lecture where we also talked about a condensate wavefunction in order to describe charges in superconducters.

I also read in the same context about Cooper-Pairs condensating into the same state.

My only knowledge of condensation so far is that it is the transition from gas to liquid. As far as I understand, in the context of quantum mechanics, it has to do with multiple particles occupying the same state (BEC, Cooper-Pairs).

I never really thought about the meaning of "condensation", but am really curious to find out what it means.

Any explanation is appreciated.

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Condensation is the phenomena in which a macroscopic number of bosons occupy the same microscopic quantum state. Usually, the number of bosons occupying a state with energy $E$ in a system held at a fixed temperature $T$ is given by the Bose-Einstein function $$n(E) = \frac{1}{e^{E/T}-1}$$ However, another restriction is of course that all bosons in the system are at some state, therefore $\sum_E n(E) = N$ where $N$ is the total number of bosons. In the thermodynamic limit of large systems and continuous energy, the sum is replaced by an integral. This restriction introduces an energy shift to the system, if the number of bosons is conserved - the chemical potential $\mu(N,T)$, and the energies are calculated with respect to it.

As the temperature is lowered, more and more bosons shift to the low-energy states, but the general form of the Bose-Einstein function is respected. If the temperature, is lowered below a certain level, however, it might be that there are "too many" bosons to occupy the states, and so a large number of them condense into the state with the lowest energy.

In BCS superconductivity, the electrons (which are fermions) form Cooper pairs, and as each pair consist of two fermions, the pair itself is a boson. These pairs are then described by the Bose-Einstein statistic, and at low temperatures, depending on the geometry of the system, may condense such that almost all of them occupy the same quantum state.

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