Tunneling and energy splitting in an asymmetric double-well potential

An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ωω and the difference of the minima is close to a multiple of ℏωℏω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential.