Wave Functions of a Trapped Electron — Question 3

Question 3

A quarter-well probability can be equal even when the density is nonuniform

An electron occupies mode $n=2$. 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. Calculate the probability in $0<x<L/4$.

2. Given a detection in the left half, calculate the probability that it lies in the leftmost quarter.

Question 3 - Solution

A quarter-well probability can be equal even when the density is nonuniform

(a) Method and calculation

$$P_q=\frac2L\int_0^{L/4}\sin^2(2\pi x/L)dx=\left[\frac xL-\frac{\sin(4\pi x/L)}{4\pi}\right]_0^{L/4}.$$$$\boxed{P_q=1/4}.$$

(b) Interpretation and further calculation

$$P(0<x<L/2)=1/2.$$$$\boxed{P(x<L/4\mid x<L/2)=\frac{1/4}{1/2}=1/2}.$$

Physical check

The density is not uniform within either interval. The integrated equality follows from the symmetry of each density lobe, not from a constant density.