Wave Functions of a Trapped Electron — Question 10

Question 10

A narrow detector near a node collects probability cubically in its half-width

An electron is in mode 2. Compare narrow windows of half-width $\epsilon\ll L$ centered on the node $x=L/2$ and the antinode $x=L/4$. Both windows lie inside the well. The infinite well occupies $0<x<L$; wave functions vanish outside. Its normalized modes are $u_n(x)=\sqrt{2/L}\sin(n\pi x/L)$ and energies $E_n=n^2E_1$.

1. Find the leading probability in the node-centered window.

2. Find the leading antinode-window probability and their ratio.

Question 10 - Solution

A narrow detector near a node collects probability cubically in its half-width

(a) Method and calculation

$$x=L/2+\delta:\quad u_2\simeq-\sqrt{2/L}\,(2\pi\delta/L).$$$$\boxed{P_{\mathrm{node}}\simeq\frac{8\pi^2}{L^3}\int_{-\epsilon}^{\epsilon}\delta^2d\delta=\frac{16\pi^2\epsilon^3}{3L^3}}.$$

(b) Interpretation and further calculation

$$x\simeq L/4:\quad|u_2|^2\simeq2/L\Rightarrow\boxed{P_{\mathrm{anti}}\simeq4\epsilon/L}.$$$$\boxed{P_{\mathrm{node}}/P_{\mathrm{anti}}\simeq\frac{4\pi^2}{3}(\epsilon/L)^2}.$$

Physical check

A finite detector centered on a node still has nonzero probability. The suppression is stronger than the linear width scaling near an antinode.