Wave Functions of a Trapped Electron — Question 4

Question 4

Box eigenstates have nonzero momentum spread despite zero mean momentum

For a normalized box mode, use $\langle x\rangle=L/2$, $\langle x^2\rangle=L^2[1/3-1/(2n^2\pi^2)]$, and $\langle p\rangle=0$. Evaluate the kinetic-energy second moment with $\langle p^2\rangle=\hbar^2\int_0^L|u_n\prime|^2dx$. 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 $\sigma_x$ and $\sigma_p$.

2. Find their product and evaluate it for the ground mode in units of $\hbar$.

Question 4 - Solution

Box eigenstates have nonzero momentum spread despite zero mean momentum

(a) Method and calculation

$$\boxed{\sigma_x^2=L^2\left(\frac1{12}-\frac1{2n^2\pi^2}\right)}.$$$$u_n\prime=\sqrt{2/L}\frac{n\pi}{L}\cos(n\pi x/L)\Rightarrow\boxed{\sigma_p=n\pi\hbar/L}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_x\sigma_p=\hbar\sqrt{n^2\pi^2/12-1/2}}.$$$$n=1:\quad\boxed{\sigma_x\sigma_p=.567862\hbar>\hbar/2}.$$

Physical check

A stationary standing wave has no mean flow, but a momentum measurement still has a spread. The hard-wall eigenstate is not a minimum-uncertainty Gaussian.