  A certain notion of canonical equivalence in quantum mechanics is proposed.
It is used to relate quantal systems with discrete ones. Discrete systems
canonically equivalent to the celebrated harmonic oscillator as well as the
quartic and the quasi-exactly-solvable anharmonic oscillators are found. They
can be viewed as a translation-covariant discretization of the (an)harmonic
oscillator preserving isospectrality. The notion of the $q-$deformation of the
canonical equivalence leading to a dilatation-covariant discretization
preserving polynomiality of eigenfunctions is also presented.
