Modulus Square of the Gaussian Wave Packet for uncertainty in $p$ Upon evaluating the integral (2.67) and obtaining the complex valued equation given in box 2.4, the author performs the modulus square to obtain the Gaussian distribution (2.68). How does one go about taking the modulus square of such a function and how are we able to ascertain the Gaussian dispersion inferred therein?


 A: If we ignore the overall normalization constant, then the square-modulus of the wave function you presented can be expressed as:
$$\left|\Psi(x,t)\right|^2 = \bigg|\exp\bigg\{\frac{ip_0^2t}{2m\hbar}\bigg\}\bigg|^2 \; \bigg|\exp\bigg\{\frac{ip_0}{\hbar}\bigg(x-\frac{p_0t}{m}\bigg)\bigg\}\bigg|^2 \; \bigg|\exp\bigg\{-\frac{(x-p_0t/m)^2}{4\hbar^2b^2}\bigg\}\bigg|^2$$
Remember that $\forall\alpha\in\mathbb{R}: |\exp(i\alpha)| = 1$. Since the first two exponentials in the expression above have imaginary arguments, the expression simplifies to:
$$\left|\Psi(x,t)\right|^2 = \bigg|\exp\bigg\{-\frac{(x-p_0t/m)^2}{4\hbar^2b^2}\bigg\}\bigg|^2$$
Now, the tricky part is that $b^2$ is a complex number. Let's explicitly write out what $1/b^{2}$ is:
$$b^2 = \frac{\sigma^2}{\hbar^2} + \frac{it}{2m\hbar} = \frac{\sigma^2 + it\hbar/2m}{\hbar^2}$$
$$\frac{1}{b^2} = \frac{\hbar^2}{\sigma^2 + it\hbar/2m} = \frac{\hbar^2\sigma^2}{\sigma^4-\hbar^2t^2/4m^2} - \frac{it\hbar^3/2m}{\sigma^4-\hbar^2t^2/4m^2}$$
But since $|\exp(\alpha+i\beta)| = |\exp(\alpha)||\exp(i\beta)| = |\exp(\alpha)|$, we can discard the imaginary term above, and only insert the real part into our expression for $\left|\Psi(x,t)\right|^2$:
$$\left|\Psi(x,t)\right|^2 = \bigg|\exp\bigg\{-\frac{(x-p_0t/m)^2}{4\sigma^2-\hbar^2t^2/\sigma^2m^2}\bigg\}\bigg|^2$$
But $\forall \alpha\in\mathbb{R}: |\exp(\alpha)|^2 = \exp(\alpha)^2 = \exp(2\alpha)$, so we get:
$$\left|\Psi(x,t)\right|^2 = \exp\bigg\{-\frac{(x-p_0t/m)^2}{2\sigma^2-\hbar^2t^2/2\sigma^2m^2}\bigg\}$$
Now remember that a Gaussian distribution with mean $\mu$ and variance $\sigma^2$ is defined as the following (up to a  normalization constant):
$$\mathcal{N}(x; \mu, \sigma) = \exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}$$
Just compare the two expressions above, and you get the parameters that you want.
