A Hint of Quantum Theory — Question 10

Question 10

Quantum overlap tests the validity of classical translational statistics

Use thermal wavelength $\lambda_T=h/\sqrt{2\pi m k_BT}$ and overlap parameter $\eta=\nu\lambda_T^3$, where $\nu$ is number density. Classical dilute-gas statistics require $\eta\ll1$. Take $m=6.64\times10^{-27}$ kg, $\nu=2.00\times10^{25}\,\mathrm{m^{-3}}$, $h=6.626\times10^{-34}$ J s and $k_B=1.381\times10^{-23}$ J/K.

1. Calculate $\lambda_T$ and $\eta$ at 300 K. At unchanged density, what would $\eta$ be at .0300 K?

2. Find the temperature at which $\eta=.100$ under this fixed-density model. Explain why this calculation diagnoses classical-statistics validity without proving that a real substance remains a dilute gas.

Question 10 - Solution

Quantum overlap tests the validity of classical translational statistics

(a) Method and calculation

$$\lambda_T(300)=\frac{6.626\times10^{-34}}{\sqrt{2\pi(6.64\times10^{-27})(1.381\times10^{-23})(300)}}\approx5.03988\times10^{-11}\,\mathrm m.$$$$\eta(300)\approx2.56029\times10^{-6},\quad\eta\propto T^{-3/2}.$$$$\eta(.0300)=\eta(300)(10^4)^{3/2}\approx2.56029.$$

(b) Interpretation and further calculation

$$.100=\eta(300)(300/T)^{3/2}\ \Rightarrow\ T=300\left(\frac{2.56029\times10^{-6}}{.100}\right)^{2/3}\approx.260604\,\mathrm K.$$$$\eta\sim1\ \Rightarrow\lambda_T\sim\nu^{-1/3}.$$Interactions and phase changes require separate analysis. The extrapolation tests a statistical assumption, not the full phase diagram.

Physical check

Internal-mode freeze-out and overlap of translational wave packets are distinct quantum effects.