Exact solution of Schr\"odinger equation with symmetric double-well potential versus WKB: accuracy for ground state splitting

TL;DR

This paper presents an exact analytical solution for the ground state splitting in a symmetric double-well potential, compares it with various WKB approximations, and enhances the theoretical tools for interpreting IR spectroscopy data of malonaldehyde.

Contribution

It introduces an exact solution using spheroidal functions and evaluates the accuracy of different WKB methods against this benchmark.

Findings

Exact solution provides high-precision ground state splitting values.

WKB approximations vary in accuracy, with some closely matching the exact results.

Results improve the theoretical understanding of IR spectra in malonaldehyde.

Abstract

The one-dimensional Schr\"odinger equation with symmetric trigonometric double-well potential (DWP) is exactly solved via angular oblate spheroidal function. The results of stringent analytic calculation for the ground state splitting of hydrogen bond in malonaldehyde are compared with several variants of approximate semiclassical (WKB) ones. This enables us to compare the accuracy of various WKB formulas suggested in the literature: 1. ordinary WKB, i.e., the formula from the Landau and Lifshitz textbook; 2. Garg's formula; 3. instanton approach. The results obtained provide a new theoretical tool for the precise quantitative description of experimental data on IR spectroscopy of malonaldehyde.