Heisenberg's Uncertainty Principle — Question 9

Question 9

Optimizing the initial width minimizes free-packet spreading at a chosen time

An unchirped minimum-uncertainty Gaussian of initial rms width a evolves freely for a fixed time $T>0$. Its final position variance is $f(a)=a^2+\hbar^2T^2/(4m^2a^2)$. Only a may be varied; the initial covariance is fixed at zero.

1. Find the initial width that minimizes f.

2. Find the smallest final width and compare the two terms in f at the optimum.

Question 9 - Solution

Optimizing the initial width minimizes free-packet spreading at a chosen time

(a) Method and calculation

$$\frac{df}{da}=2a-\frac{\hbar^2T^2}{2m^2a^3}=0\quad\Rightarrow\quad a^4=\frac{\hbar^2T^2}{4m^2}.$$$$\boxed{a_{\mathrm{opt}}^2=\hbar T/(2m)}.$$

(b) Interpretation and further calculation

$$f_{\min}=\frac{\hbar T}{2m}+\frac{\hbar T}{2m}=\frac{\hbar T}{m}.$$$$\boxed{\sigma_{x,\min}(T)=\sqrt{\hbar T/m}=\sqrt2\,a_{\mathrm{opt}}}.$$

Physical check

Tighter initial localization increases momentum spread. This optimization applies to the stated unchirped family; allowing initial correlations changes the focusing problem.