Rotating quantum Gaussian packets

We study two-dimensional quantum Gaussian packets with a fixed value of mean angular momentum. This value is the sum of two independent parts: the 'external' momentum related to the motion of the packet center and the 'internal' momentum due to quantum fluctuations. The packets minimizing the mean energy of an isotropic oscillator with the fixed mean angular momentum are found. They exist for 'co-rotating' external and internal motions, and they have nonzero correlation coefficients between coordinates and momenta, together with some (moderate) amount of quadrature squeezing. Variances of angular momentum and energy are calculated, too. Differences in the behavior of 'co-rotating' and 'anti-rotating' packets are shown. The time evolution of rotating Gaussian packets is analyzed, including the cases of a charge in a homogeneous magnetic field and a free particle. In the latter case, the effect of initial shrinking of packets with big enough coordinate-momentum correlation coefficients (followed by the well known expansion) is discovered. This happens due to a competition of 'focusing' and 'de-focusing' in the orthogonal directions.