  We revise the construction of creation/annihilation operators in quantum
mechanics based on the representation theory of the Heisenberg and symplectic
groups. Besides the standard harmonic oscillator (the elliptic case) we
similarly treat the repulsive oscillator (hyperbolic case) and the free
particle (the parabolic case). The respective hypercomplex numbers turn to be
handy on this occasion, this provides a further illustration to Similarity and
Correspondence Principle.
  Keywords: Heisenberg group, Kirillov's method of orbits, geometric
quantisation, quantum mechanics, classical mechanics, Planck constant, dual
numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics,
interference, Fock--Segal--Bargmann representation, Schr\"odinger
representation, dynamics equation, harmonic and unharmonic oscillator,
contextual probability, symplectic group, metaplectic representation,
Shale--Weil representation
