PX284 - H2 - single-particle partition function

the sum is now written as an integral:

Z_{1} = \int_{0}^{\infty} e^{- β E (k)} g (k) d k

where, $e^{- β E (k)}$ is the boltzmann factor, and $g (k)$ is DOS

for a particle with a wavevector, $k$ , and mass, $m$ , the momentum, $p = ℏ k$ , and the energy, $E (= \frac{p^{2}}{2 m}) = \frac{ℏ^{2} k^{2}}{2 m}$
so,

Z = \int_{0}^{\infty} \exp (- β \frac{ℏ^{2} k^{2}}{2 m}) \frac{V k^{2}}{2 π^{2}} d k

using the standard integral:

\int_{\infty}^{\infty} x^{2} e^{- α x^{2}} d x = \frac{1}{2} \sqrt{\frac{π}{α^{3}}}

\begin{aligned} Z & = V {(\frac{m k_{B} T}{2 π ℏ^{2}})}^{3 / 2} \\ = V n_{Q} \\ = \frac{V}{λ_{t h}^{3}} \end{aligned}

where,
$n_{Q} = {(\frac{m k_{B} T}{2 π ℏ^{2}})}^{3 / 2}$ is the quantum concentration,
$λ_{t h} = \frac{h}{\sqrt{2 π m k_{B} T}}$ is the thermal wavelength

$λ_{t h}$ is the de broglie wavelength of a particle at temperature, $T$ , ie. it is the characteristic size of the particle's wavefunction
$n_{Q} = \frac{1}{λ_{t h}^{3}}$ is the concentration at which the particles are separated by $λ_{t h}$
if the actual concentration, $n ≪ n_{Q}$ , the particles are dilute with no wavefunction overlap, and there are no quantum effects
this is known as a 'classical gas'
if $n ≫ n_{Q}$ , the particles are concentrated, there are wavefunction overlaps, giving rise to quantum effects
this is known as a 'quantum gas'
eg: H $_{2}$ molecule at RTP
- $n_{Q} ≃ 10^{30}$ m $^{- 3}$
- $n ≃ 10^{25}$ m $^{- 3}$
- no overlap, so classical
eg: electrons in a metal at $300$ K
- $n_{Q} ≃ 10^{25}$ m $^{- 3}$
- $n ≃ 10^{29}$ m $^{- 3}$
- significant overlap, so quantum