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The short answer is, in the thermodynamic limit, k could always be defined as a good quantum number for bulk states despite of different boundary conditions. This can be shown by considering a finite potential of several equally distributed wells and infinite high walls as boundary. The solution of wave function is then planewave-like and has some phase ...

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The interaction is needed for the validity of the mean field approximation, which assumes that the gap remains intact. It is clear that without interaction the argument fails for reasons you have mentioned (no gap). It is unfortunately not possible to derive analytically for which interaction the gap will survive, so the argument only gives a possible way ...

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In transition-edge sensors (TES), the superconducting detector is held very close to its critical temperature, so that it's resistance is a very steep function of the temperature. The figure below, from a review paper on TES, demonstrates the resistance vs. temperature in the right panel. In operation, the sensor temperature will be kept somewhere on the ...

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See the following link for a beautiful explanation: https://stanford.edu/~jeffjar/statmech/lec4.html#solving

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I see this post is old but I hope I can help those who read it in the future: In the string model that (I assume) is being explained, one is only interested in the vertical component of the spring's force (it is easy to see that the horizontal one must be, at least on average, always zero), which is obtained by multiplying the net force times the sine of the ...

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