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You miss first of all that Cooper pairs do not exist as some physical quantities. If you prefer, they are not particles as electrons. They are just correlations. The current is a collective response to a gradient of phase. You can generate such a gradient by a magnetic field, a voltage, a break of the condensate (like in Josephson system), with different ...

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You probably (you should really give more details when posting questions, so that those without your book can still help you) came across this in the Einstein model of the harmonic oscillators for the heat capacity estimation, in which case the energy eigenvalues in 1D (say for x-component) are given by: $$E_{nx} =\hbar \omega (1/2+n_x)$$ With $n_x\ge 0$ ...

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The statement is not true. Counter example: quantum spin ice or U(1) spin liquid. In gapless spin liquid phase, the boson (spin excitations) are emergent U(1) photons in the deconfined phase, which are gapless but not condensed.

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Let us understand Brownian motion in liquids before we look at the motion in solids. If you observe a glass of water at rest on a table, it "appears" to be motionless. However, all we need is a magnifying glass to observe the random, incessant motion of water on the surface. This random motion is a manifestation of heat. The same thing happens in a solid. A ...

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Lets get something out of the way first: The threshold, or turn-on voltage, is not really an intrinsic device property per se. It originates more from a desire by circuit designers to have a rule of thumb about how much a diode has to be forward biased to get it into conduction mode. As such, one takes the inherently non-linear current vs voltage response ...

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Exact statements The Hohenberg-Kohn theorems, which are the theoretical foundation of DFT, essentially say that the ground state properties of a many-electron system are only a function of the electron density. Any quantity you want to calculate can be re-expressed in terms of the electron density $n(r)$, including the many-body ground state wavefunction, ...

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