Wave Functions of a Trapped Electron — Question 6

Question 6

A coherent two-mode state has an oscillating mean position

An electron is in $\Psi=(u_1e^{-iE_1t/\hbar}+u_2e^{-iE_2t/\hbar})/\sqrt2$. Use $\langle1|x|1\rangle=\langle2|x|2\rangle=L/2$ and $x_{12}=-16L/(9\pi^2)$. 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. Derive the mean position as a function of time.

2. Find its oscillation amplitude, period and value after one quarter period.

Question 6 - Solution

A coherent two-mode state has an oscillating mean position

(a) Method and calculation

$$\langle x\rangle=\tfrac12(x_{11}+x_{22})+x_{12}\cos[(E_2-E_1)t/\hbar].$$$$\boxed{\langle x\rangle=\frac L2-\frac{16L}{9\pi^2}\cos(3E_1t/\hbar)}.$$

(b) Interpretation and further calculation

$$\boxed{A_x=16L/(9\pi^2)=.180127L},\quad\boxed{T=2\pi\hbar/(3E_1)}.$$$$t=T/4:\quad\boxed{\langle x\rangle=L/2}.$$

Physical check

The oscillation is an expectation value of a superposition, not a definite classical trajectory of an electron bouncing between the walls.