Quantum-mechanical oscillator in a uniform force field

The symmetry group of an isotropic oscillator explaining additional degeneracy of its energy levels was for the first time considered by Demkov [1–3]. He directed attention to the fact that harmonic oscillator Hamiltonian symmetry was beyond the scope of usual geometric concepts and could be expressed using canonical transformations relating coordinates and momenta. We show in this paper how the symmetry properties of the harmonic oscillator Hamiltonian can be used to solve the dynamic problem of an oscillator in a uniform force field.