A Hint of Quantum Theory — Question 8

Question 8

A low-temperature rotor approximation must be checked against omitted levels

A simple rigid rotor has $E_J/k_B=\Theta_rJ(J+1)$ and degeneracy $g_J=2J+1$, with $\Theta_r=50.0$ K. Ignore spin statistics. Approximate the level sum by $J=0,1,2$ with weights $w_J=g_Je^{-E_J/(k_BT)}$.

1. At 25.0 K, calculate $P_1$ and $P_2$ using the three-level normalization and assess dropping $J=2$.

2. Find the temperature where $w_1=w_0$. At that temperature evaluate $w_2/w_0$ and explain why a two-level approximation is less reliable.

Question 8 - Solution

A low-temperature rotor approximation must be checked against omitted levels

(a) Method and calculation

$$w_0=1,\quad w_1=3e^{-4}\approx.0549469,\quad w_2=5e^{-12}\approx3.07211\times10^{-5}.$$$$Z_{012}=1+w_1+w_2,\quad P_1\approx.0520835,\quad P_2\approx2.91201\times10^{-5}.$$The omitted $J=2$ population in a two-level treatment is very small at 25 K.

(b) Interpretation and further calculation

$$3e^{-100/T}=1\ \Rightarrow\ T=100/\ln3\approx91.0239\,\mathrm K.$$$$w_2/w_0=5e^{-300/T}=5e^{-3\ln3}=5/27\approx.185185.$$The third level is now a substantial term in the normalization; higher levels should also be checked before trusting a truncated model.

Physical check

Degeneracy and level spacing both enter the test of whether an omitted level matters.