  The kinematical and dynamical symmetries of equations describing the time
evolution of quantum systems like the supersymmetric harmonic oscillator in one
space dimension and the interaction of a non-relativistic spin one-half
particle in a constant magnetic field are reviewed from the point of view of
the vector field prolongation method. Generators of supersymmetries are then
introduced so that we get Lie superalgebras of symmetries and supersymmetries.
This approach does not require the introduction of Grassmann valued
differential equations but a specific matrix realization and the concept of
dynamical symmetry. The Jaynes-Cummings model and supersymmetric
generalizations are then studied. We show how it is closely related to the
preceding models. Lie algebras of symmetries and supersymmetries are also
obtained.
