String Waves and Matter Waves — Question 9

Question 9

At half a revival a box wave packet becomes its mirror image

Use $u_n(x)=\sqrt{2/L}\sin(n\pi x/L)$, $E_n=n^2E_1$ and $T_{\mathrm{rev}}=2\pi\hbar/E_1$. A finite superposition has arbitrary coefficients $c_n$.

1. Find the phase of each mode at $T_{\mathrm{rev}}/2$.

2. Use reflection about the center of the box to express the resulting wave function and density.

Question 9 - Solution

At half a revival a box wave packet becomes its mirror image

(a) Method and calculation

$$e^{-iE_nT_{\mathrm{rev}}/(2\hbar)}=e^{-i\pi n^2}=(-1)^n.$$$$u_n(L-x)=(-1)^{n+1}u_n(x).$$

(b) Interpretation and further calculation

$$\Psi(x,T_{\mathrm{rev}}/2)=\sum_nc_n(-1)^nu_n(x)=\boxed{-\Psi(L-x,0)}.$$$$\boxed{|\Psi(x,T_{\mathrm{rev}}/2)|^2=|\Psi(L-x,0)|^2}.$$

Physical check

The overall minus sign is unobservable in the density. The reflected packet generally differs from its initial position distribution.