A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured.  (b) Find the probability that the measured value is between $x=0$ and $x=2 \mathrm{~nm}$.
(b) Use of Eq. (1.23) and $|x|=x$ for $x \geq 0$ gives
$$
\begin{aligned}
\operatorname{Pr}(0 \leq x \leq 2 \mathrm{~nm}) & =\int_0^{2 \mathrm{~nm}}|\Psi|^2 d x=a^{-1} \int_0^{2 \mathrm{~nm}} e^{-2 x / a} d x \\
& =-\left.\frac{1}{2} e^{-2 x / a}\right|_0 ^{2 \mathrm{~nm}}=-\frac{1}{2}\left(e^{-4}-1\right)=0.4908
\end{aligned}
$$