Exact solution of rigid planar rotor in external electric field

Abstract

In this paper we propose a new method for accurately solving the energy levels and analytical wave functions of a rigid planar rotor in an electric field. First, we use different forms of variable substitution to transform the corresponding stationary Schrödinger equations into a confluent Heun differential equation, and then according to the characteristics of confluent Heun differential equations, confluent Heun functions as well as the studied quantum system, we obtain the analytical solutions of two even-parity functions and two odd-parity functions simultaneously at φ = 0 (2π) and π. Due to symmetry, two analytical solutions corresponding to the same parity should be linearly dependent, so that the Wronskian determinant can be constructed separately and the energy spectrum equations for calculating odd and even parity states can be found. Using the plotting method determines each respective energy and the corresponding quantum number. Then relying on the Maple software to calculate the energy spectra of different eigenstates, we finally obtain the analytical wave function of the bound states expressed by the confluent Heun function and also discuss their linear dependencies.