Analytic calculation of ground state splitting in symmetric double well potential

Abstract

The exact solution of the one-dimensional Schrödinger equation with symmetric trigonometric double-well potential (DWP) is obtained via angular oblate spheroidal function. The results of stringent analytic calculation for the ground state splitting of ring-puckering vibration in the 1,3-dioxole (as an example of the case when the ground state tunneling doublet is well below the potential barrier top) and 2,3-dihydrofuran (as an example of the case when the ground state tunneling doublet is close to the potential barrier top) are compared with several variants of approximate semiclassical (WKB) ones. This enables us to verify 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. We show that for the former case all three variants of WKB provide good accuracy while for the latter one they are very inaccurate. The results obtained provide a new theoretical tool for describing relevant experimental data on IR spectroscopy of ring-puckering vibrations.

21 p., 4 fig. arXiv admin note: text overlap with arXiv:1801.03733

References in corpus (10)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• The double well potential in quantum mechanics: a simple, numerically exact formulation
• Magnetic quantum dots and rings in two dimensions
• Semi-classical formula for quantum tunneling in asymmetric double-well potentials
• Exactly solvable Schrödinger equation with double-well potential for hydrogen bond
• Analytic description of inversion vibrational mode for ammonia molecule
• Two-electron atom with a screened interaction
• Localization or tunneling in asymmetric double-well potentials
• Tuning terahertz transitions in a double-gated quantum ring
• Exact solution of Smoluchowski's equation for reorientational motion in Maier-Saupe potential

Cited by in corpus (7)

• Calculation of IR absorption intensities for hydrogen bond from exactly solvable Schrödinger equation
• Analytic treatment of IR-spectroscopy data for double well potential
• Model for vibrationally enhanced tunneling of proton transfer in hydrogen bond
• Exactly solvable double-well potential in Schrödinger equation for inversion mode of phosphine molecule
• Robust entangling gate for capacitively coupled few-electron singlet-triplet qubits
• Schrödinger equation with Pauli-Fierz Hamiltonian and double well potential as model of vibrationally enhanced tunneling for proton transfer in hydrogen bond
• Non-standard quantum algebras and infinite-dimensional PT-symmetric systems