Feynman disentangling of noncommuting operators in quantum mechanics

Feynman’s disentangling theorem is applied to noncommuting operators in the problem of quantum parametric oscillator, which is mathematically equivalent to the problem of SU(1, 1) pseudospin rotation. The number states of the oscillator correspond to unitary irreducible representations of the SU(1, 1) group. Feynman disentangling is combined with group-theoretic arguments to obtain simple analytical formulas for the matrix elements and transition probabilities between the initial and final states of the oscillator. Feynman disentangling of time evolution operators is also discussed for an atom or ion interacting with a laser field and for a model Hamiltonian possessing the “ hidden” symmetry of the hydrogen atom.