The SU(1,1) generators are considered in order to investigate the uncertainty relation in the Barut–Girardello coherent state for the time-dependent singular oscillator. The corresponding uncertainty product is obtained by evaluating the expectation values of the canonical variables and their squares.

The problem of a charged particle in the presence of a variable magnetic field is considered. Using the linear and the quadratic invariants as a tool, the wave functions in Fock state as well as in coherent state are obtained. The corresponding propagators which propagate the wave functions in the space–time are derived. Using numerical computations we have managed to draw some plots for the real, imaginary, and absolute values of the propagators. This has been used to analyze the properties of the propagators associated with both of the linear and the quadratic invariants. It has been shown that there is no essential difference between the behavior of the absolute value of the propagators in both of the linear and the quadratic invariants.