Two- and Three-Dimensional Electron Traps — Question 4

Question 4

Separable densities turn rectangular detection probabilities into products

A two-dimensional ground state is $\psi=A\sin(\pi x/L_x)\sin(\pi y/L_y)$ inside a rectangular infinite well. Use $\int_0^L\sin^2(\pi x/L)dx=L/2$.

1. Normalize the state with A positive.

2. Find the probability in the central rectangle $L_x/4<x<3L_x/4$, $L_y/4<y<3L_y/4$.

Question 4 - Solution

Separable densities turn rectangular detection probabilities into products

(a) Method and calculation

$$1=A^2(L_x/2)(L_y/2)\Rightarrow\boxed{A=2/\sqrt{L_xL_y}}.$$$$\rho(x,y)=\left[\frac2{L_x}\sin^2\frac{\pi x}{L_x}\right]\left[\frac2{L_y}\sin^2\frac{\pi y}{L_y}\right].$$

(b) Interpretation and further calculation

$$P_x=P_y=\frac12+\frac1\pi\quad\Rightarrow\quad P_{\mathrm{central}}=P_xP_y.$$$$\boxed{P_{\mathrm{central}}=\left(\frac12+\frac1\pi\right)^2=.669631}.$$

Physical check

The central rectangle occupies one quarter of the area, but contains more than half the probability. Product factorization relies on the separable state.