  In this paper, we revisit the Cauchy problem for the three dimensional
nonlinear Schr\"odinger equation with a constant magnetic field. We first
establish sufficient conditions that ensure the existence of global in time and
finite time blow-up solutions. In particular, we derive sharp thresholds for
global existence versus blow-up for the equation with mass-critical and
mass-supercritical nonlinearities. We next prove the existence and orbital
stability of normalized standing waves which extend the previous known results
to the mass-critical and mass-supercritical cases. To show the existence of
normalized solitary waves, we present a new approach that avoids the celebrated
concentration-compactness principle. Finally, we study the existence and strong
instability of ground state standing waves which greatly improve the previous
literature.
