Wave Functions of a Trapped Electron — Question 2

Question 2

The ground state favors the central half more than a uniform distribution does

An electron occupies the ground mode. 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. Integrate the density over $L/4<x<3L/4$.

2. Find the probability in the two outer quarters and compare with a uniform spatial model.

Question 2 - Solution

The ground state favors the central half more than a uniform distribution does

(a) Method and calculation

$$P_c=\frac2L\int_{L/4}^{3L/4}\sin^2(\pi x/L)dx=\left[\frac xL-\frac{\sin(2\pi x/L)}{2\pi}\right]_{L/4}^{3L/4}.$$$$\boxed{P_c=\frac12+\frac1\pi=.818310}.$$

(b) Interpretation and further calculation

$$\boxed{P_{\mathrm{outer}}=1-P_c=\frac12-\frac1\pi=.181690}.$$$$P_{c,\mathrm{uniform}}=1/2\Rightarrow\boxed{P_c-P_{c,\mathrm{uniform}}=1/\pi}.$$

Physical check

The ground-state density vanishes at the walls and peaks at the center. Equal interval lengths need not have equal probability.