Charge Is Quantized — Question 8

Question 8

Random single-electron arrival produces a noninteger mean but integer outcomes

In each of 100 independent trials, a collector gains one electron with probability $p=.300$ and none otherwise. Let $X_i=1$ for capture and 0 for no capture, so deposited charge is $Q=-e\sum_iX_i$. Use $\operatorname{Var}(X_i)=p(1-p)$ and add variances for independent trials.

1. Find the mean deposited charge, its variance and standard deviation in units of $e$.

2. Find the probability of no deposited charge. Explain how an average charge of $-.300e$ per trial is consistent with integer electron transfer.

Question 8 - Solution

Random single-electron arrival produces a noninteger mean but integer outcomes

(a) Method and calculation

$$\langle Q\rangle=-100pe=-30e,\quad\operatorname{Var}(Q)=100e^2p(1-p)=21e^2.$$$$\sigma_Q=\sqrt{21}e\approx4.58258e.$$

(b) Interpretation and further calculation

$$P(Q=0)=(1-p)^{100}=.7^{100}\approx3.23448\times10^{-16}.$$$$Q/e\in\{0,-1,\ldots,-100\},\quad\langle Q_i\rangle/e=-.300.$$An ensemble mean is a probability-weighted average, not the charge transferred in a particular trial.

Physical check

The square-root variance rule depends on independence; correlated captures would change the fluctuation size.