# “Quantum version” of Maxwell-Boltzmann speed distribution

Is there a "quantum mechanical version" of the classical Maxwell-Boltzmann speed distribution for monatomic ideal gases? I'm quite new to thermodynamics, but I just wanted to know. If there is such a distribution, can it be used to calculate the average speeds (such as the most-probable speed, or rms speed classically) of particles, given a mass? Or is it more subtle, dealing with quantum states and such?

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## 1 Answer

Yes, these are called Bose-Einstein and Fermi-Dirac distributions of energy, they apply respectively to particles with integer spin (bosons) and half-integer spin (fermions). The difference is that multiple bosons can be in a single quantum state while fermions can not. In the limit of high temperature both distributions go to the Maxwell-Boltzmann distribution. The criterion for applicability of classical (non-quantum) statistics is that the phase volume occupied by a particle, $\delta x^3 \delta p^3 \sim V (mT)^{3/2}$, divided by the minimum quantum phase volume $\hbar^3$ has to be large compared to the number of particles N in the system. This can be written as $n \hbar^3 /(mT)^{3/2} \ll 1$. Here $V$ is the system volume, $n$ is the number density, $m$ is the particle mass, $T$ is the temperature. If numbers are substituted here then it turns out that the density has to be very high; therefore quantum statistics is not relevant to usual atomic or molecular gases. An exception is super-cooled systems, e.g., see http://arxiv.org/abs/cond-mat/0311617

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Ok. So can these be applied to atoms, or just fermions/bosons? And at high temperatures, the limit tends to the M-B distribution? So at high temperatures, we don't ned to really worry about quantum effects, because the limit of both the B-E and F-D distributions turn into the classical one? Just clarifying. –  user86111 Jul 23 '13 at 0:09
Composite particles can be either bosons or fermions, depending on what they are composed of. But for usual atomic or molecular gases there is no need to apply quantum statistics, they are usually well described by the classical (Boltzmann) statistics while quantum statistics is usually applied for gases of electrons, photons, neutrons. In the limit of high temperature the statistics become classical. But "quantum effects" may still be there, playing a role in something other than statistics - say, line radiation emitted by a hot classical gas is described by quantum physics, right? –  Maxim Umansky Jul 23 '13 at 17:08
@user86111 Atoms and molecules, too. Their statistics, i.e. whether they are a boson or a fermion, corresponds to whether the total spin of the particle is a whole or half integer. You can estimate the temperature at which quantum effects become important from the mean separation of the particles $d$ and their mass $m$: $T_{Q} \approx \frac{h^2}{2 k_{B} m d^2}$. Feel free to swap $k_B$ and $m$ for $R$ the gas constant and molar mass if you like. –  Chay Paterson Jul 23 '13 at 18:41
Yes, by "composite particles" here I meant atoms and molecules but actually this term is primarily used for subatomic particles, sorry for the lack of clarity with this. –  Maxim Umansky Jul 23 '13 at 21:21