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How does the macroscopic wavefunction (the order parameter) builds up from zero value to the a finite value when liquid He undergoes a transition from normal to the superfluid state? How does it physically happen that the whole system ends up in a single quantum state? Does it have anything to do with the overlapping of de broglie waves of different particles because as the temperature is reduced further and further the de broglie wavelength increases?

Does this wavefunction grow in magnitude further and further as we lower the temperature? If yes, can it be understood as more and more atoms are coming to the same quantum state?

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Because all or most of the He atoms occupy the lowest possible quantum state; do en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate and the links therein answer your question? –  Kvothe Mar 26 '14 at 14:32

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