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Suppose a gas is kept at a temperature of $7000 \text{ K}$ and has a particle density of $2.7 \times 10^{34} \text m^{-3}$. Do we need to treat it quantum mechanically or will classical treatment be sufficient for describing its dynamics?

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    $\begingroup$ Classical treatment should be sufficient since, at such high temperatures, the quantum effects would effectively be classic (Fermi-Dirac or Bose-Einstein would tend to Boltzmann statistics) $\endgroup$ – exp ikx Oct 2 '19 at 10:03
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    $\begingroup$ What are the units of particle density? $\endgroup$ – Amey Joshi Oct 2 '19 at 10:34
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    $\begingroup$ That density is roughly 6 orders of magnitude greater than typical solid densities near STP. $\endgroup$ – Jon Custer Oct 2 '19 at 13:06
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You haven't mentioned the unit of particle density. I assume that it is $m^{-3}$. In that case, the volume available to a single particle is $10^{-34}/2.7$. Equivalently, every particle is confined to a sphere of radius $2.1 \times 10^{-12}$ $m$, which is lesser than typical atomic size. Such a dense material needs a quantum mechanical treatment.

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  • $\begingroup$ Thanks for your answer It's really useful, and yeah unit of particle density is in m^3. Also Could you please elaborate on how quantum mechanical treatment depends on density. Even solids at high temperatures show classical behavior. $\endgroup$ – shashank mishra Oct 2 '19 at 17:08
  • $\begingroup$ Solids at room temperature do need a quantum mechanical treatment when one wants to study their electronic properties. Solid state physics is a quantum theory. If we restrict ours study only to macroscopic properties like elasticity and ignore the atomic structure, a classical treatment suffices. That's why they theory of elasticity doesn't demand a quantum treatment. $\endgroup$ – Amey Joshi Oct 2 '19 at 22:03
  • $\begingroup$ Consider a gas at room temperature and pressure. The distance between the individual gas molecules is very large compared to their de Broglie wavelengths. Their molecular wavefunctions are far apart from each other. We can safely apply classical statistical mechanics to such a gas. $\endgroup$ – Amey Joshi Oct 2 '19 at 22:14

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