Energies of a Trapped Electron — Question 9

Question 9

Thermal population ratios provide a temperature scale for confinement

For a single electron weakly equilibrated with a heat bath, assume spatial mode probabilities satisfy $P_n/P_1=\exp[-(E_n-E_1)/(k_BT)]$, with $E_n=n^2E_1$. A measured ratio is $P_2/P_1=.100$.

1. Infer $k_BT$ in units of $E_1$.

2. Predict $P_3/P_1$ and assess whether a two-level population model is a useful approximation.

Question 9 - Solution

Thermal population ratios provide a temperature scale for confinement

(a) Method and calculation

$$.100=e^{-3E_1/(k_BT)}\Rightarrow\boxed{k_BT=3E_1/\ln10=1.30288E_1}.$$$$\frac{E_3-E_1}{k_BT}=\frac{8\ln10}{3}.$$

(b) Interpretation and further calculation

$$\boxed{P_3/P_1=10^{-8/3}=.00215443}.$$$$\boxed{P_3/P_2=10^{-5/3}=.0215443}.$$

Physical check

The third mode is small but not exactly absent. Higher levels are even more suppressed; a two-level approximation is reasonable only at a tolerance consistent with this residual population.