Simple examples of position-momentum correlated Gaussian free-particle wavepackets in one-dimension with the general form of the time-dependent spread in position

We provide simple examples of closed-form Gaussian wavepacket solutions of the free-particle Schrodinger equation in one dimension which exhibit the most general form of the time-dependent spread in position, namely (Delta x_t)^2 = (Delta x_0)^2 + At + (Delta p_0)^2t^2/m^2, where A = <(xhat - <xhat>_0)(phat - <phat>_0) + (phat - <phat>_0)(xhat - <xhat>_0)>_0 contains information on the position-momentum correlation structure of the initial wave packet. We exhibit straightforward examples corresponding to squeezed states, as well as quasi-classical cases, for which A < 0 so that the position spread can (at least initially) decrease in time because of such correlations. We discuss how the initial correlations in these examples can be dynamically generated (at least conceptually) in various bound state systems. Finally, we focus on providing different ways of visualizing the x-p correlations present in these cases, including the time-dependent distribution of kinetic energy and the use of the Wigner quasi-probability distribution. We discuss similar results, both for the time-dependent Delta x_t and special correlated solutions, for the case of a particle subject to a uniform force.