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Making use of the expression for the propagator in terms of path integrals, we study the effect of certain variational symmetries of a Lagrangian on the corresponding propagator. We also show that by considering a point transformation that relates two different Lagrangians one can obtain a relation between the corresponding propagators.

A simple procedure to derive the transformation of a wavefunction under a change of reference frame is applied to some examples and its relation with the transformation of the Hamilton principal function is studied.

We find the form of the potential depending on the coordinates and the time such that a solution, $S$, of the Hamilton--Jacobi equation yields an exact solution, $\exp ({\rm i} S/\hbar)$, of the corresponding Schr\"odinger equation.

It is shown that, when there is a magnetic field present, in the framework of classical or quantum mechanics, the active translations differ from the passive ones and that the canonical momentum is not the generator of them. It is also shown that an infinitesimal generator of passive translations or rotations exists only if the magnetic field is invariant under these transformations.

It is shown that, in the framework of non-relativistic quantum mechanics, any conserved Hermitian operator (which may depend explicitly on the time) is the generator of a one-parameter group of unitary symmetries of the Hamiltonian and that, conversely, any one-parameter family of unitary symmetries of the Hamiltonian is generated by a conserved Hermitian operator.

It is shown that if a complete set of mutually commuting operators is formed by constants of motion, then, up to a factor that only depends on the time, each common eigenfunction of such operators is a solution of the Schr\"odinger equation. In particular, the operators representing the initial values of the Cartesian coordinates of a particle are constants of motion that commute with each other and from their common eigenfunction one readily obtains the Green function.

We analyze the quantum dynamics of the non-relativistic two-dimensional isotropic harmonic oscillator in Heisenberg's picture. Such a system is taken as toy model to analyze some of the various quantum theories that can be built from the application of Dirac's quantization rule to the various symplectic structures recently reported for this classical system. It is pointed out that that these quantum theories are inequivalent in the sense that the mean values for the operators (observables) associated with the same physical classical observable do not agree with each other. The inequivalence does not arise from ambiguities in the ordering of operators but from the fact of having several symplectic structures defined with respect to the same set of coordinates. It is also shown that the uncertainty relations between the fundamental observables depend on the particular quantum theory chosen. It is important to emphasize that these (somehow paradoxical) results emerge from the combination of two paradigms: Dirac's quantization rule and the usual Copenhagen interpretation of quantum mechanics., Comment: 8 pages, LaTex file, no figures. Accepted for publication in Phys. Rev. A