# “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?

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
• @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