  The generation of angular momentum in fissioning nuclei is not well
understood. The predictions of different models disagree, particularly
concerning the correlation between the fragments' angular momenta. In this
article, a time-dependent collective Hamiltonian model is proposed to treat the
generation of the angular momentum in the fission fragments due to the quantum
uncertainty principle as well as the dynamics of the collective wave function
during and after scission. The model is constructed in the framework of the
frozen Hartree-Fock approximation using a Skyrme energy functional to extract
deformations of the fission fragments as well as the interactions in a derived
collective Hamiltonian. The fission reactions studied are $^{240}$Pu
$\rightarrow$ $^{132}$Sn+$^{108}$Ru and $^{240}$Pu $\rightarrow$
$^{144}$Ba+$^{96}$Sr. The model can account for a large part of the angular
momentum found in experimental data. The orientation of the angular momentum of
each fragment is found to be mainly in the plane perpendicular to the fission
axis, in agreement with the experiment. The magnitudes of the angular momenta
in the two fragments are nearly uncorrelated, in agreement with the recent
experimental data of Wilson et al., Nature (London) 590, 566 (2021). Some of
the conclusions of the traditional collective vibration model are supported by
the present model but some are not. Surprisingly, it is found that the angular
momenta of the fragments are slightly correlated positively as in a wriggling
mode. It is also found that the presence of an octupole deformation in a
fragment can significantly increase the generated angular momentum.
