Wave Functions of a Trapped Electron — Question 7

Question 7

A coherent dark point disappears when the same populations are mixed incoherently

Compare the pure states $\psi_\pm=(u_1\pm u_3)/\sqrt2$ with an incoherent mixture containing equal populations of modes 1 and 3. 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 three probability densities at $x=L/2$.

2. Explain mathematically why equal mode populations do not specify the local density.

Question 7 - Solution

A coherent dark point disappears when the same populations are mixed incoherently

(a) Method and calculation

$$u_1(L/2)=\sqrt{2/L},\quad u_3(L/2)=-\sqrt{2/L}.$$$$\boxed{\rho_+(L/2)=0,\quad\rho_-(L/2)=4/L,\quad\rho_{\mathrm{mix}}(L/2)=2/L}.$$

(b) Interpretation and further calculation

$$\rho_\pm=\tfrac12(u_1^2+u_3^2)\pm u_1u_3,\qquad\rho_{\mathrm{mix}}=\tfrac12(u_1^2+u_3^2).$$$$\boxed{\rho_\pm-\rho_{\mathrm{mix}}=\pm u_1u_3}.$$

Physical check

Density at a point is not a point probability; a finite detector integrates over a finite interval. Coherence controls the cross term even when energy populations agree.