Energy splitting in symmetric double-well potentials

The quantum mechanical tunnelling in a smooth symmetric double-well potential is a long-standing and well-known problem. Three methods have been proposed to calculate the energy splitting: the instanton method @1,2#, the WKB approximation @3,4# and numerical calculation @4–6#. The instanton method is helpful to understand the physical insight of quantum tunnelling, but the validity is restricted to the case of large separation between the two potential minima. The WKB approximation is widely used for its simple mathematical form, but the result is known to be inaccurate due to its inherent defect in connection formula. One had taken the quadratic connection formula instead of the Airy function to modify the WKB result at ground state @1,7#. Recently, some refinements were developed to improve the accuracy of WKB by changing the phase loss at the classical turning points @8,9#. The anharmonicity is also taken into consideration in the case of small separation distance @9#. To the best of knowledge of the authors, no above approximations have provided the perfect results. Without doubt using numerical methods, one can get the solution up to the desired accuracy, but a considerable deal of physical insight is lost in these processes. In this paper, the quantization conditions to onedimensional Schrödinger equation in symmetric double-well potential are presented by using analytical transfer matrix method ~ATMM!, which has been applied to arbitrary potential wells successfully @10#. Taking into account the correct phase losses at the turning points and the phase contribution of the scattered subwaves, this analysis gives the explicit quantization conditions of the split energy levels. We also calculate the energy eigenvalues of the split energy levels and compare them with that of the exact numerical method @4#.