Quantum States of the Kapitza Pendulum

The quantum states of the Kapitza pendulum are investigated within the framework of the effective potential obtained by the method of averaging over fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using a model potential. For the lowest states of an inverted pendulum, an expression is obtained for the spectrum in terms of energies of a harmonic oscillator corrected using perturbation theory. Tunneling effect corrections to the energies of resonance states in two wells of the effective potential are found. The results of calculations of the structure of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by the Numerov numerical algorithm are compared.