THE WENTZEL-KRAMERS-BRILLOUIN METHOD IN MULTIDIMENSIONAL TUNNELING - APPLICATION TO SCANNING TUNNELING MICROSCOPY

The Wentzel-Kramers-Brillouin method is commonly used for semiclassical approximations to the wave function in both the classically allowed and the forbidden regions of one-dimensional potentials. In a multidimensional space, the method can be adapted to construct wave functions in an “allowed” region from classical trajectories or wave normals. However, in a “forbidden” region, this construction is not very well understood. It has been argued that the approximation can be implemented by constructing a single set of semiclassical paths that satisfy in the forbidden region the same path equations as in the allowed regions. We show that such an approach is generally inapplicable. However, it is correct if the incident wave is normal to the turning surface. In general, the wave function in the forbidden region is specified by two sets of wave fronts, the equiphase and the equiamplitude surfaces. We present a formal scheme for constructing these wave fronts. This reveals that for non-normal incidence the paths are not the solutions of a single set of ordinary differential equations, such as Newton’s equations of motion for the trajectories in the allowed region. The analysis enables us to answer some of the basic questions concerning tunneling in multidimensional nonseparable potentials. © 1990, American Vacuum Society. All rights reserved.