Oscillatory states of quantum Kapitza pendulum

TL;DR

This paper investigates the quantum behavior of the Kapitza pendulum, deriving eigenvalues, wavefunctions, and tunneling effects using complex analysis techniques to understand oscillatory states in an asymmetric double-well potential.

Contribution

It introduces a novel method of extending the angle coordinate into the complex plane to compute the spectrum and tunneling in the quantum Kapitza pendulum.

Findings

Eigenvalues and wavefunctions for localized oscillatory states are obtained.

Quantum tunneling between potential wells is quantitatively analyzed.

A complex contour integral approach is developed for spectrum calculation.

Abstract

We study quantum mechanics problem described by the Schr\"{o}dinger equation with Kapitza pendulum potential, that is the asymmetric double-well potential on the circle. For the oscillatory states spatially localize around the two stable saddle positions of the potential, we obtain the perturbative eigenvalues and corresponding piecewise wavefunctions. The spectrum is computed by extending the angle coordinate to the complex plane so that the quantization condition is formulated as contour integral along a path extending in the imaginary direction. Quantum tunneling between the wells is computed.