Time-Dependent 4<i>D</i> Quantum Harmonic Oscillator and Reacting Hydrogen Atom

Abstract

With the help of low-dimensional reference equations (ordinary differential equations) and the corresponding coordinate transformations, the non-stationary 4D quantum oscillator in an external field is reduced to an autonomous form. The latter, in particular, reflects the existence of a new type of dynamical symmetry that reduces the equation of motion of a non-stationary oscillator to an autonomous form that does not change with time. By imposing an additional constraint on the wave function of the isotropic oscillator, we have obtained the total wave functions of the reacting hydrogen atom in two different cases: (a) when the non-stationary frequency has two asymptotic values and there is no external field; and (b) when, in addition to the non-stationary frequency, an external force acts on the hydrogen atom. The transition S-matrix elements of various elementary atomic–molecular processes are constructed. The probabilities of quantum transitions of the hydrogen atom to others, including new bound states, are studied in detail, taking into account the influence of external forces.

Keywords

• time-dependent 4<i>D</i> quantum oscillator
• reference equation method
• dynamical symmetry
• hydrogen atom in external field
• transition <i>S</i>-matrix element