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Planck didn't know Bose-Einstein statistics at the time around 1900. With the existence of minimal unit, or quantization $E=hf$, in mind, he derived the Planck's law which describe the black body radiation. Two decades late, after the establishment of the Bose-Einstein statistics, then it is known that Plank's law is a special case of Bose-Einstein ...

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The reason for using microstates is that it is the only way to come up with quantum statistics, but the grand-canonical potential is defined also for a classical system. And you are right, one should take into acount the state with zero particles, let me show why on a simple example. Let us take for instance a system in which $N$ indistiguishable ...

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In general, both IQHE and FQHE are rigid quantum states, whose rigidness is protected by the finite energy gap ($h\omega$ for IQHE) between the ground state(s) and the exited states. Finite temperature can support excitations to overcome the gap, which destroys the rigidness of the state. Under finite temperature, the quantization of the Hall conductivity is ...

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Plank's distribution (law) is a specific application of the Bose-Einstein distribution. For example, there is no chemical potential, $\mu$, for photons, so it is missing from Planck's law, although it's in the Bose-Einstein distribution. (The chemical potential only comes into play when you have a fixed number of particles; there is no such restriction for ...

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The uncertainty principle is a fundamental property of quantum systems, and is not a statement about observational success. No particle either free or in crystal can have zero momentum otherwise a nonsensical infinity is required for the standard deviation of position $\Delta x$, in the uncertainty principle $\Delta x \Delta p \geq \hbar / 2$. $0 \cdot ... 1 I'm assuming that this section of the book is talking about the ultraviolet catastrophe, where an ideal black body in thermal equilibrium will emit an infinite amount of power through radiative means. The source goes on to say: The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all ... 1 Check the derivation of the Boltzmann distribution from the microcanonical ensemble on the Wikipedia page "Maxwell-Boltzmann Statistics". We suppose that the "wealth classes" of individuals are discretised, so that, for example, we find the number of individuals with$m_1 = \$500$, the number with $m_2 = \$10000$and so forth: as an approximation we ... 1 As I had written in the comments, it is the second term that you have gotten incorrect. Focusing exclusively on this term (leaving aside the$1/8\$ factor), we have $$|\psi_2\rangle \langle \psi_2| = \frac{1}{3}(|00\rangle \langle 00|+ |10\rangle \langle 10|+ |11\rangle \langle 11| - |00\rangle \langle 10|-|10\rangle \langle 00| + \ ...),$$ where I have ...

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