The generalized harmonic oscillator and the infinite square well with a moving boundary

For the one-dimensional generalized harmonic oscillator we obtain in this paper its wavefunctions in closed form by means of two independent methods which are based respectively in (i) the algebraic properties of the dynamical symmetry of the system and (ii) the construction of an invariant operator. The total equivalence of these two formulations is shown and quantal properties are described in terms of a classical solution of the equations of motion. Two possible reductions for the system exist: the static harmonic oscillator and the free particle. In the latter case the quantum system becomes a Fermi oscillator or equivalently it can describe a free particle in a well with one moving boundary which in turn follows certain classical rules. The time-dependent boundary conditions in the well play the role of an effective interaction acting on the particle. The formalism is shown to be compatible with the gauge principle of minimal coupling and several different gauges are constructed and analysed.

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